Electromagnetic Relations between Materials and Fields for Microwave Chemistry

*Jun-ichi Sugiyama, Hayato Sugiyama, Chika Sato and Maki Morizumi*

## **Abstract**

We consider the application of microwave energy to a material. The effects of the electromagnetic field on the material and of the material on the electromagnetic field will be described, focusing on the dielectric relaxation phenomenon of the liquid. The dielectric permittivity of mixtures is discussed by extending Debye relaxation to explain how the material behaves with respect to an electric field. We will also consider the energy that the electric field imparts to the material, both thermally and nonthermally. We will develop this relation and describe what form it should take if there is a nonthermal effect in the chemical reaction field under microwave irradiation.

**Keywords:** microwave chemistry, complex permittivity, Debye relaxation, nonthermal effect, Arrhenius equation

## **1. Introduction**

There has been much debate about whether microwave irradiation acts as heat in chemical synthesis or whether it has a nonthermal effect [1–6]. This problem has been discussed in many cases based on changes in the reaction rate and in the selectivity of the difference between the results with and without the application of microwaves. What is particularly important here is whether the temperatures of the two conditions to be compared are exactly the same. In microwave irradiation, it is difficult to use a general thermometer such as a metal thermocouple or an alcohol thermometer. This is because the distribution of the electromagnetic field changes significantly due to the insertion of a metal material (thermocouple or mercury thermometer), or because the indicator material (alcohol) itself is heated. As an alternative, an indirect method such as measuring the temperature from radiation on the surface of the vessel is used. In the comparison between microwave irradiation and non-irradiation, if the measurement does not correctly indicate the internal temperature, the difference may be due to the microwave irradiation condition being higher than the non-irradiation condition.

The heat source in microwave irradiation is the loss of electromagnetic wave energy, i.e., loss of the electric or the magnetic fields due to undulation of the molecule itself. Therefore, the movement behavior varies depending on the molecular species of the irradiated material. A different momentum obtained for each molecule means that it is not in a thermal equilibrium state, meaning that it does not match the definition of temperature, which requires an isotropic equilibrium motion.

The aforementioned is an inductive argument that discusses differences due to microwave irradiation, such as changes in reaction rates and in selectivity. A lot of data are reported every year, but the interpretations are diverse, and there are cases where it is "both hard to explain and hard to ignore" [1].

On the other hand, in this review, the original physical meaning is examined based on the dielectric relaxation phenomenon and how the material behaves under microwave irradiation. Based on this, we will discuss deductions about what action should be generated if there is an effect other than heat based on principles, rather than data.

## **2. Material properties (relaxation properties)**

### **2.1 Permittivity and refractive index**

When an object is heated by irradiation with microwaves, the microwave energy is attenuated inside the object [7]. The Beer–Lambert law in optics can also be applied in the microwave region. The refractive index, n, and the attenuation factor, k, can be combined as a complex refractive index *n*\*, as shown by Eq. (1):

$$m^\* = n - jk,\tag{1}$$

where *j* is the square root of �1. When the wave has a cosine signal *s*(*t*, *x*) as a function of time *t* and position *x*, the *n* and *k* correspond to the propagation and attenuation velocities in the phasor formula (Eq. (2)) as shown in **Figure 1**:

**Figure 1.** *Undulation and complex refractive index at* t *= 0.*

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

$$s(\mathbf{x}, t) = A \cos \left( n^\* \mathbf{x} + \alpha t \right) = \operatorname{Re} \left( A e^{-j(n - jk) \mathbf{x}} e^{-j \alpha t} \right). \tag{2}$$

Physical properties in the microwave region are indicated by the complex permittivity *ε*\* [F/m] and complex permeability *μ*\* [H/m]. The meanings of these terms can be explained by definition of the values of basic physical constants. The speed of light propagation through the vacuum (*<sup>c</sup>* = 2.99792458 � <sup>10</sup><sup>8</sup> m/s) is defined value. After May 20, 2019, the magnetic constant *<sup>μ</sup>*<sup>0</sup> changed from the defined value (4<sup>π</sup> � <sup>10</sup>�<sup>7</sup> H/m) to the experimentally determined value (1.25663706212 (19) � <sup>10</sup>�<sup>6</sup> H/m) [8]. The electric constant *ε*<sup>0</sup> is derived from these constants (Eq. (3)):

$$
\varepsilon\_0 = \frac{1}{\mu\_0 c^2}.\tag{3}
$$

When Eq. (3) is transformed to Eq. (4), c represents the reciprocal of the square root of *ε*<sup>0</sup> and *μ*0:

$$
\varepsilon = \frac{1}{\sqrt{\varepsilon\_0 \mu\_0}}.\tag{4}
$$

To be precise, vacuum is not a material, but electromagnetic waves propagate through it. Since the same relationship applies to materials, they can be treated equally well in terms of mathematical expressions. The velocity *v* [m/s] of the electromagnetic wave propagating through a material is the reciprocal of square root of the product of the material's permittivity *ε* [F/m] and permeability *μ* [H/m] (Eq. (5)):

$$v = \frac{1}{\sqrt{\varepsilon\mu}}.\tag{5}$$

Therefore, the refractive index *n* is obtained by Eq. (6):

$$n = \frac{\varepsilon}{v} = \frac{\sqrt{\varepsilon\mu}}{\sqrt{\varepsilon\_0\mu\_0}} = \sqrt{\varepsilon\_r\mu\_r} \quad \text{where } \varepsilon = \varepsilon\_0\varepsilon\_r, \ \mu = \mu\_0\mu\_r. \tag{6}$$

Here, the relative permittivity *ε*<sup>r</sup> [nd] and the relative permeability *μ*<sup>r</sup> [nd] are coefficients based on *ε*<sup>0</sup> and *μ*0, and *n* is dimensionless. In the following, the dimensionless value is expressed as [nd].

When attenuation of electromagnetic waves occurs in a material, the complex relative permittivity *ε*r\* [nd] and the complex relative permeability *μ*r\* [nd] are used. Therefore, the relationship with *n*\* is as follows (Eq. (7)):

$$
\mu^\* = \sqrt{\varepsilon\_r^\* \mu\_r^\*} \quad \text{where } \varepsilon\_r^\* = \varepsilon\_r' - j\varepsilon\_r'' \quad , \ \mu\_r^\* = \mu\_r' - j\mu\_r''. \tag{7}
$$

The superscripts 'and "indicate a real part and an imaginary part, respectively.

Discussions dealing only with dielectrics generally introduce important assumptions here. Since dielectrics often do not exhibit magnetism, the permeability is considered to be the same as that of vacuum, and *μ*<sup>r</sup> \* is set to 1�*j*0. Devices that measure permittivity (actually complex relative permittivity) often base their calculations on this assumption, so one should be careful when measuring materials with magnetism or high conductivity. Under this assumption, Eq. (7) is approximated by Eqs. (8) and (9):

$$n^\* = \sqrt{\varepsilon\_r^\*},\tag{8}$$

$$(n - jk)^2 = \epsilon\_r' - j\epsilon\_r''.\tag{9}$$

Here, Eq. (1) is reviewed again. Since *n* is the ratio of the propagation velocity in the material to that in the vacuum, it can be regarded as the ratio of the wavelength *λ*<sup>0</sup> [m] in the vacuum to the wavelength *λ* [m] in the material (Eq. (10)):

$$m = \frac{c}{v} = \frac{\lambda\_0}{\lambda}.\tag{10}$$

On the other hand, since attenuation does not occur in a vacuum, it is difficult to understand *k* as a ratio. Therefore, a distance *δ* [m] at which an electric field intensity *E* [V/m] becomes 1/*e* = 36.8% is used. Here, *e* is the Napier number and *ω* [rad / *s*] is the angular frequency. *δ* is called the skin depth and has dimensions of length. As *δ* decreases, the amount of attenuation increases, indicating a large *k* (Eq. (11)):

$$k = \frac{c}{o\delta} = \frac{\lambda\_0}{2\pi\delta}.\tag{11}$$

Furthermore, when Eq. (9) is transformed, Eqs. (12) and (13) are obtained:

$$m = \left[\frac{\mathbf{1}}{2} \epsilon\_r' \left(\sqrt{\mathbf{1} + \left(\varepsilon\_r''/\varepsilon\_r'\right)^2} + \mathbf{1}\right)\right]^{1/2},\tag{12}$$

$$\boldsymbol{k} = \left[\frac{\mathbf{1}}{2} \boldsymbol{\varepsilon}\_r^{\prime} \left(\sqrt{\mathbf{1} + \left(\boldsymbol{\varepsilon}\_r^{\prime}/\boldsymbol{\varepsilon}\_r^{\prime}\right)^2} - \mathbf{1}\right)\right]^{1/2}.\tag{13}$$

From these equations, *n* and *k* are obtained from the *ε*r\* of the material. When *n* is large, the microwave that has progressed is greatly refracted. In particular, a cylindrical vessel collects power at the center as in the case of a lens, and heating may proceed locally. **Figure 2** shows an example simulating the electromagnetic field distribution

#### **Figure 2.**

*Simulation of the electromagnetic field of water in a cylindrical vessel in an oven-type furnace. A: Oven shape. B: PLD of 25°C water. C: PLD of 100°C water.*

of water in a cylindrical container. When the water temperature is uniform, the power loss density (PLD) is concentrated in the center [9].

#### **2.2 Penetration depth and skin depth**

When the same material is irradiated with electromagnetic waves having the same frequency, the applied power intensity *P* [W/m<sup>3</sup> ] is proportional to the square of the electric field intensity *E* [V/m]. When the attenuation is large, the microwave may not reach the deep part of the vessel. Although the skin depth, *δ*, has been described earlier, there is a penetration depth *L* [m] as a similar index [7]. The distance at which the power intensity *P* becomes 1/e due to the loss during propagation is expressed as *L*1/*<sup>e</sup>* [m]. In this case, penetration depth *L*1/*<sup>e</sup>* is half of the skin depth, *δ*. Microwaves are not absent beyond this depth.

There are two methods for describing *L*1/*e*, as shown in Eqs. (14) and (15):

$$L\_{\mathbb{A}\_{\ell}} = \frac{\lambda\_0}{4\pi} \left[ \frac{2}{\varepsilon\_r' \left( \sqrt{1 + \left( \varepsilon\_r''/\varepsilon\_r' \right)^2} - 1 \right)} \right]^{1/2},\tag{14}$$
 
$$\lambda\_0$$

$$L\_{\mathbb{Q}\_\ell} = \frac{\lambda\_0}{2\pi\sqrt{\varepsilon\_r'}\tan\delta}.\tag{15}$$

The *δ* in Eq. (15) is not a skin depth, but a value indicating dielectric loss in a narrow definition as tan *δ* (Eq. (16)):

$$\tan \delta = \frac{\varepsilon\_r''}{\varepsilon\_r'}.\tag{16}$$

Eq. (15) can be obtained from a modification in which the second and subsequent terms are ignored in the Maclaurin expansion when tan<sup>2</sup> *<sup>δ</sup>* ! 0 in Eq. (14). Therefore, when using Eq. (15), it is assumed that tan *δ* ! 0. On the other hand, during microwave heating, the material of tan *δ* ! 0 does not heat, as will be described later. Therefore, if the target is not tan *δ* ! 0, it is necessary to pay attention to whether the value based on the latter formula deviates from the premise.

#### **2.3 Plotting on bode and Nyquist diagrams**

The correlation with the horizontal axis representing frequency and the vertical axis representing complex permittivity is called a Bode diagram. The Bode diagram of the water is shown in **Figure 3**, indicating that the dielectric constants *ε*<sup>r</sup> <sup>0</sup> and *<sup>ε</sup>*r″ are functions of the (angular) frequency [10]. This is represented by the Eq. (17):

$$
\varepsilon = \varepsilon\_0 \varepsilon\_r^\* \left( \boldsymbol{\alpha} \right) = \varepsilon\_0 \left\{ \boldsymbol{\varepsilon}\_r'(\boldsymbol{\alpha}) - j \boldsymbol{\varepsilon}\_r''(\boldsymbol{\alpha}) \right\}. \tag{17}
$$

When the complex permittivity at each frequency is plotted on a Nyquist diagram with a real part on the horizontal axis and an imaginary part on the vertical axis, they draw a semicircular locus as shown in **Figure 3**. Such behavior is called Debye relaxation. Debye relaxation is a behavior commonly found in nonionic liquid materials.

**Figure 3.** *Nyquist diagram and bode diagram of water (200 MHz–14 GHz).*

Examples that cannot be applied include cases where the relaxation frequency is not single, and those where a conductive material is included (described at 2.10 and 2.11).

## **2.4 Relationship between the Debye relaxation formula and the bode/Nyquist diagram**

In the previous section, we described how many liquids show a characteristically semicircular geometric locus due to Debye relaxation. We will return to the basics to explain why and what information can be gleaned below. The characteristic behavior in the microwave band is called dielectric relaxation. The deformation of electron clouds and molecular structures is a response in the UV and IR bands and is faster than in the microwave band. These contributions are prompt responses to undulated fields.

On the other hand, molecular orientation is a phenomenon based on rotation of an electric dipole. A large moment like a molecule causes a time delay in orientation with respect to field changes. The time delay referred to here is a phase delay and does not vibrate at a different period from that of the applied external field. Since the molecule cannot rotate if the external field vibration is too fast, but it can follow a too-slow external field vibration without time delay, the behavior is distributed around a specific vibration frequency [11–14]. The prompt response is only a propagation delay and does not contribute to the loss. This is expressed as *ε*<sup>r</sup> (∞) only in the real part. At the current time t, the application of an external electric field *E*(*t*) generates an electric flux density *D*(t) in the material.

The part of the electric flux density in the material *D*p(*t*) (p: prompt) pertaining to prompt response is expressed by Eq. (18):

$$D\_p(t) = \varepsilon\_0 \varepsilon\_r(\infty) E(t). \tag{18}$$

On the other hand, the contribution of the delayed response includes not only the electric field *E* (*t*) at the current time t but also the influence of the electric field *E* (*u*) at the previous time *u*. Therefore, the electric flux density *D*<sup>d</sup> (*t*) (d: delayed) of the dielectric following the delay is expressed by Eq. (19), integrated from start time 0 to the current time *t*:

$$D\_d(t) = \int\_0^t E(u)f(t-u)du. \tag{19}$$

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

In Eq. (19), *f* (*t*-*u*) represents a response function in terms of the previous time u and the current time *t*. From Eqs. (18) and (19), the electric flux density *D* (*t*) of the dielectric at the current time *t* is expressed by Eq. (20):

$$D(t) = \epsilon\_0 \varepsilon\_r(\infty) E(t) + \int\_0^t E(u)f(t-u) du. \tag{20}$$

*D*(*t*) in Eq. (20) can also be expressed as the product of the applied electric field *E*(*t*) and the dielectric constant *ε*\*(*ω*) of the material. Therefore, Eq. (21) can be obtained:

$$D(t) = \varepsilon\_0 \varepsilon\_r^\* \left( a \right) E(t). \tag{21}$$

Substituting Eq. (21) into Eq. (20) and rewriting the response function to *f* (*x*) yields Eq. (22):

$$
\varepsilon\_0 \left\{ \varepsilon\_r^\* \left( a \right) - \varepsilon\_r (\infty) \right\} = \int\_0^\infty e^{-j\alpha x} f(\varkappa) d\varkappa. \tag{22}
$$

Next, an appropriate expression is set for a response function. In Debye-type relaxation, Eq. (23) is used as a response function to Eq. (7):

$$f(\mathbf{x}) = \varepsilon\_0 \{ \varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\infty) \} \frac{e^{-\mathbf{x}/\tau}}{\tau}. \tag{23}$$

Here, *τ* [s/rad] is the relaxation time, which is the reciprocal of the angular frequency *ω*<sup>0</sup> [rad/s] at which the vibration phase is delayed by π/2 (90 degrees). A very slowly undulating field is effectively the same as a static field. Therefore, the complex relative dielectric constant is also only the real part, and this is represented by *ε*r(0). As shown in the Nyquist diagram of **Figure 3**, *ε*<sup>r</sup> 0 (*ω*), which is the real part of *ε*r\*(*ω*), is between *ε*r(0) and *ε*r(∞). Therefore, *f* (*x*) can be interpreted as having a proportional coefficient of *ε*r(0)-*ε*r(∞), which is the difference between the real parts.

When Eq. (23) is substituted into Eq. (22) and transformed, *ε*r\*(*ω*) is expressed by Eqs. (24)–(26). These are Debye's dispersion equations:

$$
\varepsilon\_r^\*\left(\boldsymbol{\omega}\right) = \varepsilon\_r(\infty) + \frac{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\infty)}{1 + j\alpha\tau};\tag{24}
$$

$$\varepsilon\_r'(\boldsymbol{\omega}) = \operatorname{Re}\varepsilon\_r^\*\left(\boldsymbol{\omega}\right) = \varepsilon\_r(\boldsymbol{\infty}) + \frac{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\boldsymbol{\infty})}{\mathbf{1} + \boldsymbol{\alpha}^2 \boldsymbol{\pi}^2};\tag{25}$$

$$\varepsilon\_r''(\boldsymbol{\omega}) = \text{Im}\varepsilon\_r^\*\left(\boldsymbol{\omega}\right) = \frac{\{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\boldsymbol{\infty})\}\boldsymbol{\alpha}\boldsymbol{\tau}}{\mathbf{1} + \boldsymbol{\alpha}^2 \boldsymbol{\tau}^2}. \tag{26}$$

Eliminating *ωτ* from Eqs. (25) and (26) leads to the relationship of Eq. (27). Therefore, when *ε*<sup>r</sup> 0 (*ω*) is on the horizontal axis and *<sup>ε</sup>*r″(*ω*) is on the vertical axis, a semicircle is drawn as shown in **Figure 3**:

$$\left\{\varepsilon\_r'(o) - \frac{\varepsilon\_r(\mathbf{0}) + \varepsilon\_r(\mathbf{\infty})}{2}\right\}^2 + \varepsilon\_r''(o)^2 = \left\{\frac{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\mathbf{\infty})}{2}\right\}^2. \tag{27}$$

From the aforementioned, if the measured values are plotted on a semicircle, it indicates that the material has a response that can be explained by Debye relaxation theory and is not in a special state.

## **2.5 Three angles on the semicircle:** *φ***,** *θ***, and** *δ*

The dielectric loss is also written as tan *δ*. This is caused by the phase delay angle *δ* of the voltage and current when an alternating electric field is applied to the dielectric; it is defined by Eq. (16). The complex dielectric constant is calculated by measuring the loss and the propagation delay. The physical meaning of this value is a tangent, where *δ* is the angle between the horizontal axis and the line segment from the origin to the circumference (**Figure 4**).

Substituting Eqs. (25) and (26) into Eq. (16) shows that tan *δ* is a function of *ω*:

$$\tan \delta = \frac{\{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\infty)\} a \tau}{\varepsilon\_r(\mathbf{0}) + \varepsilon\_r(\infty) a^2 \tau^2}. \tag{28}$$

If the central angle *φ* is determined at an arbitrary point on the semicircle when *ε*r(0) is *φ* = 0°, *ε*r(∞) shows *φ* = 180° [15]. Therefore, *φ* indicates a phase delay of molecular motion in the delayed response. The tan *φ* at an arbitrary frequency is obtained geometrically by the following equation:

$$\tan \varphi = \frac{\varepsilon\_r''(o)}{\varepsilon\_r'(o) - \frac{\varepsilon\_r(0) + \varepsilon\_r(\infty)}{2}}. \tag{29}$$

Substituting Eqs. (25) and (26) into Eq. (29) eliminates *ε*r(0) and *ε*r(∞), as shown in Eq. (30). Therefore, the influence of *φ* upon temperature change means that it is

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

based only on the change of the dielectric relaxation time *τ* when the frequency is constant:

$$\tan \varphi = \frac{2\alpha \tau}{1 - \alpha^2 \tau^2} = -\frac{1}{\sinh \left( \ln \left( \alpha \tau \right) \right)} = -\text{csch}(\ln \left( \alpha \tau \right)).\tag{30}$$

As shown in **Figure 4**, if *θ* is defined as an angle formed by a straight line extending from the left intersection of the semicircle and the horizontal axis toward the measurement point and the horizontal axis, then *θ* is a circumferential angle of *φ*. Here, tan *θ* is expressed by Eq. (31):

$$\tan \theta = \frac{\varepsilon\_r''(o)}{\varepsilon\_r'(o) - \varepsilon\_r'(\infty)}.\tag{31}$$

Substituting Eqs. (25) and (26) into Eq. (31) yields Eq. (32):

$$
\tan \theta = a\pi;
\tag{32}
$$

transforming it yields Eq. (33) for relaxation time *τ* [s/rad]:

$$\pi = \frac{1}{\alpha} \tan \theta = \frac{1}{\alpha} \frac{\varepsilon\_r''(\alpha)}{\varepsilon\_r'(\alpha) - \varepsilon\_r(\infty)}. \tag{33}$$

When calculating *τ*, in a region far from *θ* = 45°(*φ* = 90°), even if *ω* changes logarithmically, *φ* does not change significantly, and the calculation of *τ* has a large error. On the other hand, in the vicinity of *φ* = 90°, *ε*r\*(*ω*) changes sharply with respect to *ω*. Therefore, when calculating *τ* using Eq. (31), it is preferable to perform evaluation with a measurement point in the vicinity of 2*θ* = *φ* = 90°.

From Eq. (32), *<sup>ε</sup>*r″ (*ω*) is maximum at *<sup>ω</sup>* = 1/*τ*. In actual measurements, the function to be swept is often denoted by *f*[Hz] instead of *ω*[rad/s]. It should be noted that in many references, *τ* is sometimes written in terms of the reciprocal of the relaxation frequency *f*<sup>c</sup> which is defined in Eq. (58). In this case, the unit of the derived *τ* is [s], which is 2π times *τ*[s/rad]. In **Table 1**, the unit of τ is written in [s] instead of [s/rad].



*Entry 1: distilled water; 2: propylene carbonate; 3: dimethyl sulfoxide; 4: N,N-dimethylacetamide; 5: nitromethane; 6: N,N-dimethylformamide; 7: acetonitrile; 8: N-methylimidazole; 9: N- methyl-2-pyrrolidone; 10: nitrobenzene; 11: methanol; 12: benzonitrile; 13: acetone; 14: methyl ethyl ketone; 15: benzaldehyde; 16: 1,2-dichloroethane; 17: dichloromethane; 18: ethanol; 19: methyl iodide; 20: ethyl acetate; 21: chlorobenzene; 22: fluorobenzene; 23: bromobenzene; 24: chloroform; 25: iodobenzene; 26: diethyl ether; 27: 2-propanol; 28: cyclopentyl methyl ether; 29: 1 butanol; 30: 2-butanol; 31: isobutyl alcohol; 32:* tert*-butyl alcohol; 33: toluene; 34: benzene; 35: cyclohexane; 36:* n*hexane.* f*c: relaxation frequency obtained by fitting,* τ*: relaxation time, calcd.* ε*<sup>r</sup> (*f*): calculated* ε*<sup>r</sup> \*, found: measured* ε*<sup>r</sup> \*.*

#### **Table 1.**

*Physical property values of several liquids at room temperature [16].*

## **2.6 Loss calculation formula**

The propagation of the electromagnetic wave energy of the microwave is propagation of electromagnetic field vibration. Specifically, this vibration is caused by *Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

continuously and repeatedly converting the electric field energy into magnetic field energy and vice versa. The propagation equations are shown as follows (Eqs. (34)–(38)):

$$\mathbf{rotH} = \mathbf{i} + \frac{\partial \mathbf{D}}{\partial t};\tag{34}$$

$$\mathbf{B} = \mu \mathbf{H};\tag{35}$$

$$\text{rot}\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t};\tag{36}$$

$$\mathbf{D} = \epsilon \mathbf{E};\tag{37}$$

$$
\mathbf{i} = \sigma \mathbf{E}.\tag{38}
$$

Here, **E**, **H**, **D**, **B**, and **i** are the electric field intensity [V/m], magnetic field intensity [A/m], electric flux density [C/m2 ], magnetic flux density [Wb/m<sup>2</sup> ], and current density [A/m<sup>2</sup> ], all of which are vector quantities. In addition, the physical property coefficient of the material is transferred as a permittivity *ε* [F/m], a conductivity *σ* [S/m], and a permeability *μ* [H/m]. The loss of propagation energy means that the vector quantity has been lost when converted to the other field. This loss is mainly regarded as a conversion to heat. The loss equation calculated based on the physical property values is derived below [11, 12, 17].

The applied *E* is expressed by using a phasor as follows:

$$E = E\_0 e^{j\alpha t}.\tag{39}$$

D in the dielectric produced by aligning the directions of the dielectric molecules undulates with a phase shift (delay) of *δ*. *δ* is a value expressed by Eq. (16) and is not *φ*, which is an undulation phase delay of the molecule:

$$D = D\_0 \mathfrak{e}^{j(\alpha t - \delta)}.\tag{40}$$

From Eq. (37), the product of *E*, *ε*0, and *ε*r\* gives *D*. Therefore, the phase delay of *D* can also be expressed by the complex permittivity (Eq. (41)):

$$D = \varepsilon\_0 \varepsilon\_r^\* E. \tag{41}$$

Substituting Eqs. (39) and (40) into Eq. (41) and converting the exponential function to a trigonometric function using Euler's formula yields the Eqs. (42) and (43). Eq. (42) divided by Eq. (43) matches Eq. (16):

$$D\_0 \cos \delta = \varepsilon\_0 \varepsilon\_r' E\_0;\tag{42}$$

$$D\_0 \sin \delta = \varepsilon\_0 \varepsilon\_r'' E\_0. \tag{43}$$

Next, the energy consumption per unit volume when the electric flux density of the dielectric changes by *dD* is determined as *dU*. This *dU* is obtained by the product of *E* and *dD*. When *dD* is integrated over one period of vibration (1 / *f* = 2π / *ω*), the energy consumed by the dielectric during one period is obtained as *w*,

$$w = \int\_0^{2\pi/w} dU = \int\_0^{2\pi/w} EdD = \int\_0^{2\pi/w} E\frac{dD}{dt} dt.\tag{44}$$

In Eq. (44), *E* and *D* are the real parts:

$$E = E\_0 \cos at;\tag{45}$$

$$D = D\_0 \cos\left(at - \delta\right). \tag{46}$$

After transforming Eq. (46) with the cosine difference formula, differentiating with *t* gives Eq. (47):

$$\frac{d\mathbf{D}}{dt} = -a\boldsymbol{\alpha}\_0 \boldsymbol{\varepsilon}\_r^\prime \mathbf{E}\_0 \sin\alpha t + a\boldsymbol{\alpha}\_0 \boldsymbol{\varepsilon}\_r^\prime \mathbf{E}\_0 \cos\alpha t. \tag{47}$$

By substituting Eq. (47) into Eq. (44), we obtain Eq. (48):

$$
\omega = \pi \varepsilon\_0 \varepsilon\_r'' E\_0^2. \tag{48}
$$

Since w is repeated *f* times per second (= *ω*/2π times), the energy *W* received by the dielectric from the electric field per unit volume / unit time is expressed by Eq. (49):

$$\mathcal{W} = \frac{\alpha}{2\pi} \boldsymbol{w} = \frac{1}{2} \alpha \varepsilon\_0 \varepsilon\_r'' \boldsymbol{E}\_0^2 = \pi \oint \varepsilon\_0 \varepsilon\_r'' \boldsymbol{E}\_0^2. \tag{49}$$

Rewriting Eq. (49) yields Eq. (50). Here, when *E*<sup>0</sup> is rewritten to |**E**|, the electric field loss equation is obtained. In this paper, the dielectric loss based on *ε* obtained by Eq. (50) is defined as *P*ε\_loss [W/m<sup>3</sup> ]:

$$P\_{\varepsilon\_{\text{-}loss}} = \frac{1}{2} \alpha \varepsilon\_0 \varepsilon\_r'' \left| E \right|^2. \tag{50}$$

When the frequency is very low and the change in the electric field is very slow, the molecules align their dipole moments in a direction that cancels the electric field without delay in proportion to the electric field strength, and this flux density can follow without delay. Since *<sup>φ</sup>* ! 0 and *<sup>δ</sup>* ! 0, *<sup>ε</sup>*r″ ! 0 from Eq. (43), *<sup>P</sup>*ε\_loss ! 0 from Eq. (50), and the electromagnetic wave energy loss is small. On the other hand, if the frequency is very high and the change in the electric field is very fast, the movement of the molecules cannot respond to the alternating electric field, and the application direction reverses before aligning the dipole moments. As a result, since *<sup>φ</sup>* ! <sup>π</sup> and *<sup>δ</sup>* ! 0, *<sup>ε</sup>*r″ ! 0, so *<sup>P</sup>*ε\_loss! 0 and the electromagnetic wave energy loss is small.

## **2.7 Difference between tan** *<sup>δ</sup>* **and** *<sup>ε</sup>***r″ in loss**

If the target frequencies are the same, *ω* may be regarded as a constant. According to the aforementioned equation, the loss appears to be proportional to *<sup>ε</sup>*r″, but the actual loss is not determined solely by the difference in *<sup>ε</sup>*r″. The meaning of Eq. 50 indicates that if *E* is constant, the amount of power loss per unit volume is proportional to *<sup>ε</sup>*r″. However, the propagation speed of electromagnetic waves decreases with the refractive index *n*, which varies with *ε* (and *μ*).

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

**Figure 5.** *Model structure of heating considerations. Area 1:Air, area 2:Water, area 3:Air.*

**Figure 5** shows a model in which microwaves pass through media in the order of air, water, and air. Here, for the sake of simplicity, the electric flux is indicated by a straight arrow, and the reflected wave at the boundary is not considered. Wavelength reduction is considered first [18]. The electric flux density in air is *D*<sup>1</sup> [C/m2 ], and that in water is *D*<sup>2</sup> [C/m2 ]. Since wavelength shortening occurs in water with a large *ε*<sup>r</sup> 0 , the interval (density) of arrows in *D*<sup>2</sup> increases to (*ε*r) 0.5 times in *D*<sup>1</sup> according to Eq. (8). Next, the electric field strength is considered. The electric field strength in air is *E*<sup>1</sup> [V/m], and that in water is *E*<sup>2</sup> [V/m]. In water where *ε*<sup>r</sup> <sup>0</sup> is large, the dipole of water cancels the applied electric field so that the intensity decreases to 1/*ε*<sup>r</sup> according to the constitutive Eq. (37). Combining these two effects, the electric field strength *E*<sup>2</sup> in water attenuates to 1 / (*ε*r) 0.5 times the electric field strength *E*<sup>1</sup> in air as shown in Eq. (51):

$$\begin{cases} D\_2 = \sqrt{\varepsilon\_r'} D\_1\\ D\_1 = \varepsilon\_0 E\_1\\ D\_2 = \varepsilon\_0 \varepsilon\_r' E\_2 \end{cases} \quad \text{therefore } E\_2 = \frac{1}{\sqrt{\varepsilon\_r'}} E\_1. \tag{51}$$

In actual examinations, it is difficult to measure the electric field strength inside the irradiation target. Therefore, irradiation with a predetermined irradiation power is performed. Thus, the following interpretation is derived:

a. Eq. (52) indicates that the loss is proportional to *<sup>ε</sup>*r″ when the applied electric field is constant:

$$P\_{e\_\text{-}loss} = \frac{1}{2} a \rho \varepsilon\_0 \varepsilon\_r'' \left| E\_2 \right|^2. \tag{52}$$

b. Eq. (53) indicates that the loss is proportional to tan *δ* when the applied power is constant:

$$P\_{e\text{-}loss} = \frac{1}{2} \rho \epsilon \epsilon\_0 \epsilon\_r^{\prime\prime} \left| \frac{E\_1}{\sqrt{\epsilon\_r^{\prime}}} \right|^2 = \frac{1}{2} \rho \epsilon \epsilon\_0 \tan \delta |E\_1|^2. \tag{53}$$

The applied electric field and applied power referred to here are the net electric field and power applied to the materials. As will be described later, not all irradiation power is always applied. The irradiated and reflected powers can be measured in area 1. The passing through power can be measured in area 3.

## **2.8 Maximum of tan***δ*

From the Nyquist diagram, the maximum value of tan *δ* is the tangent point through the origin. Therefore, it is obtained from Eq. (54):

$$\tan \delta\_{\text{max}} = \frac{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\infty)}{2\sqrt{\varepsilon\_r(\mathbf{0})\varepsilon\_r(\infty)}}.\tag{54}$$

Furthermore, *ε*<sup>r</sup> 0 (*ω*) and *<sup>ε</sup>*r″(*ω*) when tan *<sup>δ</sup>* is maximal are represented by Eqs. (55) and (56):

$$
\varepsilon\_{r\tan\delta\max}^{\prime} = \frac{2\varepsilon\_r(\mathbf{0})\varepsilon\_r(\infty)}{\varepsilon\_r(\mathbf{0}) + \varepsilon\_r(\infty)},\tag{55}
$$

$$
\varepsilon\_{r\tan\delta\max}^{\prime\prime} = \frac{\varepsilon\_r(\mathbf{0}) - \varepsilon\_r(\infty)}{\varepsilon\_r(\mathbf{0}) + \varepsilon\_r(\infty)} \sqrt{\varepsilon\_r(\mathbf{0})\varepsilon\_r(\infty)}.\tag{56}
$$

The tan *θ* when tan *δ* becomes maximum is defined as tan *θ*tan*<sup>δ</sup>*max. This is geometrically determined from **Figure 4**. The angular frequency *ω*tan*δ*max at this time is given by Eq. (32). When both are combined, Eq. (57) is derived:

$$
\tan \theta\_{\tan \delta \max} = \omega\_{\tan \delta \max} \pi = \sqrt{\frac{\varepsilon\_r(\mathbf{0})}{\varepsilon\_r(\infty)}}.\tag{57}
$$

The frequency at which *<sup>ε</sup>*r″ is maximized is defined as *<sup>f</sup>*c. Since tan*<sup>θ</sup>* is 1 at *<sup>f</sup>*c, 2π*f*<sup>c</sup> *<sup>τ</sup>* is equal to 1 from Eq. (32) and **Figure 4**. Therefore, Eq. (57) becomes Eq. (58) by *f*tan*δ*max and *f*c:

$$f\_{\tan \delta \max} = \sqrt{\frac{\varepsilon\_r(\mathbf{0})}{\varepsilon\_r(\mathbf{os})}} \mathcal{f}\_{\varepsilon}. \tag{58}$$

The semicircle in the Nyquist diagram shows that the maximum values of *<sup>ε</sup>*r″ and tan *<sup>δ</sup>* do not match. For example, water has a maximum value of *<sup>ε</sup>*r″ at 18 to 22 GHz, but the maximum value of tan *δ* is on the higher frequency side. Since 2.45 GHz is much smaller than these, it appears to be less efficient at heating water. However, when the amount of absorption is large, attenuation occurs rapidly in the surface and does not penetrate into the inner side. Therefore, it cannot be said that a larger loss is always effective for heating the inside to a wide area.

## **2.9 Changes in** *ε***r\* with temperature**

The complex dielectric constant according to Debye relaxation can be generalized by obtaining *ε*r(0), *ε*r(∞), and *τ* by measuring in a wideband and fitting to a semicircle. From this relationship, it can be seen that the temperature, frequency, and complex permittivity have the following relationship:

1.When the temperature rises, the molecule becomes disturbed. As a result, the external response amount *ε*<sup>r</sup> 0 (*ω*) decreases. This also applies to *ε*r(0).


As shown in **Figure 6**, the Bode diagrams of *ε*<sup>r</sup> 0 (*ω*) and *<sup>ε</sup>*r″(*ω*) shift from 1 ! <sup>2</sup> ! <sup>3</sup> with temperature rising [19]. Therefore, in the Nyquist diagram, the central angle *φ* indicated by the irradiation frequency *f*<sup>i</sup> can be classified into six types depending on the position [11–13].

Type I is the case where the relaxation frequency *f*<sup>c</sup> of the irradiated material before heating is much larger than *f*i, and the *φ* of the irradiated material before heating is in the vicinity of 0π/8 to 2π/8. As the temperature rises, *φ* approaches 0 and the amount of loss decreases.

Type II is the case where *f*<sup>c</sup> is slightly larger than *f*i, and *φ* indicates 2π/8 to 4π/8. Since *<sup>ε</sup>*r″(*ω*) is large, the temperature rises rapidly. However, as the temperature becomes high, *φ* decreases and the temperature rise rate greatly decreases.

Type III is when *f*<sup>i</sup> is close to or slightly smaller than *f*c, and *φ* is in the vicinity of <sup>4</sup>π/8 to 5π/8. Since *<sup>ε</sup>*r″(*ω*) is large as a whole, the temperature rise rate is high and *<sup>ε</sup>*r″ (*ω*) rises until *φ* becomes π/2 and then falls.

Type IV is when *f*<sup>c</sup> is smaller than *f*i, and *φ* indicates 5π/8 to 7π/8. The tan *δ* is a large region, and as temperature rises, *<sup>φ</sup>* approaches 4π/8, so *<sup>ε</sup>*r″(*ω*) further increases. Thermal runaway due to uneven heating may occur.

Type V is the case where *f*<sup>c</sup> is much smaller than *f*i, and *φ* indicates 7π/8 to 8π/8. Although it has the property that the temperature rises due to irradiation and the amount of absorption increases, there is a case where the irradiation frequency is too high, such that there is a lot of transmission and heating is not sufficient.

Type 0 is not shown in this figure. This corresponds to the case where contributions to the original dielectric loss, such as that from nonpolar molecules, are very small.

When the irradiated *f*<sup>i</sup> is considerably smaller than *f*c, the change in *ε*<sup>r</sup> <sup>0</sup> (*ω*) and *<sup>ε</sup>*r″ (*ω*) due to temperature rising is classified as type I, but when it irradiates a considerably larger *f*c, it becomes a V type. Therefore, this classification also means that the classification changes depending on the irradiation frequency, even for the same material. As an example, the physical property values of several liquids and the value of *τ* calculated from a semicircle are shown in **Table 1**.

**Figure 6.** *Area division of the bode diagram. Left: area division; Right: temperature rising.*

### **2.10 Liquid mixture**

For the relaxation time *τ* in the system to show only one value, it is necessary for the entire material to be in a uniform state. This occurs when there is only one kind of electric dipole that controls the dielectric constant, and the surrounding molecules that control the relaxation time are also uniform. This means that *τ* is distributed according to the *δ* function [20]. In the mixed liquid, there are various types of molecules exhibiting dielectric loss and various types of surrounding molecules. Therefore, the relaxation time is not always one. In the model, when the *ε*r\*(*ω*) of a system having the same *ε*r(0) and *ε*r(∞), different *τ* is calculated. **Figure 7** shows that wideband complex permittivity plot is separated into two semicircles when the difference between the two *τ* is large. On the other hand, when the difference in *τ* is not large, the distortion of the semicircle is small. In the case of simple two-component mixing, the trajectory shown in the Nyquist diagram is considered to be similar to any in **Figure 7**. However, there is little distortion in the measurement range as shown in **Table 1**, and there is almost a single semicircle. This can be regarded as a response in which *ε*r(0), *ε*r(∞), and *τ* show one average value. Furthermore, when *τ* is evaluated with respect to the mixing ratio, it continuously changes according to the composition, but it is not always shown on the straight line in terms of arithmetic mean. However, there is also a behavior that protrudes upward and downward.

**Figure 8** shows the Argan diagram as a complex relative permittivity at 2.45 GHz by the reflection probe method [21–23]. One line indicates the nine mixtures made with a volume ratio of 9:1 to 1:9 between two pure liquids. If the *ε*r\* obtained in the mixed sample obeys the additive property, the connection should be a straight line, but in many cases a curve is shown. This is because *ε*r(0), *ε*r(∞), and *τ* change due to the mixing of the two materials, such that *ε*<sup>r</sup> <sup>0</sup> and *<sup>ε</sup>*r″ do not always attain a single arithmetic average value.

A plot of the responses at other frequencies on the Argan diagram is also shown in **Figure 8**. Apparently, these figures have changed greatly. However, as shown in point A, ethanol:water = 4:6 mixed solution, acetonitrile:propylene carbonate = 2:8 mixed solution, and acetone:propylene carbonate = 2:8 mixed solution had close *ε*r\* values, regardless of frequency. Similar intersection points were also observed in Group B (acetone:propylene carbonate = 9:1, cyclopentyl methyl ether:acetonitrile = 3:7) and Group C (ethanol:propylene carbonate = 7:3, methanol = 10). This means that if the average *ε*r(0), *ε*r(∞), and *τ* obtained by mixing are close, *ε*r\*(*ω*) matches, and therefore the same *ε*r\* is shown at different frequencies. Thus, when the liquid mixture characteristic is shown as a "train map," the intersection corresponding to the "transfer station" does not change even if the frequency does.

**Figure 7.**

*Nyquist diagrams with two* τ*. Line 1: zero difference in* τ*, Line 2: small difference in* τ � *Line 6: large difference in* τ*.*

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

**Figure 8.**

*Argan diagrams of liquid mixtures. 1: Water; 2: Propylene carbonate; 3: Dimethyl sulfoxide; 7: Acetonitrile; 11: Methanol; 13: Acetone; 18: Ethanol; 27: 2-propanol; 28: Cyclopentyl methyl ether.*

## **2.11 Involvement of conductivity** *σ*

The conductive material is heated by the conductive loss [10, 24–27]. This property agrees with the dielectric loss in that it is proportional to the square of the electric field strength. These are losses to the electric field and not to the magnetic field. Since *ω* is not included in the conduction loss equation, it occurs even when the frequency is zero. Conduction loss is a phenomenon in which materials with a single charge, that is, positive and negative ion atoms (or molecules), are accelerated in opposite directions by application of an electric field. An increase in the distance between the counterions means that the electrostatic potential of the material is increased. Also, if ions that should linearly move with constant acceleration are decelerated to constant speed, this means that resistance has made ions motion isotropic, or the surrounding molecules have received the kinetic energy of ions. This means that the current has been converted to Joule heat. It is clear that such a conduction loss differs from dielectric loss in which the charge distance in the molecule is constant. Therefore, in the partial dielectric including conductivity, the conductive and dielectric losses appear separately. For example, the Nyquist diagrams of 0.1-mol/L = NaCl aqueous solution are shown in **Figure 9**. Comparing to **Figure 3**, it is shown that the locus is changed by adding NaCl due to conductive loss.

In many cases, only one parameter *σ* is used for the discussion of conductive loss. However, in the Nyquist diagram, *σ* must be a complex number with real and imaginary parts. If these parts have the same value, the low-frequency part shown in **Figure 9** should be a straight line with a 45° slope, but the measured value is not. Therefore, from this figure, when discussing losses in the microwave band, conductivity must also be considered to have a complex value.

The power density *w*\* [J/m<sup>3</sup> ] of one cycle can be calculated from the current density *i*(*t*) = *i*<sup>0</sup> sin*ω*t with amplitude *i*<sup>0</sup> [A/m] and the electric field intensity *E*(*t*)=E0 sin*ω*t with amplitude E0 [V/m]. From the definition of the complex conductivity *σ*\* [S/m] = *σ*<sup>0</sup> �*jσ*″, Eq. (38) is deformed as Eq. (59):

$$i(t) = \sigma^\* E(t). \tag{59}$$

Here, *w*\* is shown as Eq. (60):

$$
\omega^\* = \int\_0^{2\pi/\alpha} i(t)E(t)dt = \frac{1}{2f}\sigma^\* E\_0^2. \tag{60}
$$

Since *w*\* is repeated *f* times per second, the energy *W* received by the conductive material from the electric field per unit volume/unit time is expressed by Eq. (61):

$$W^\* = \frac{1}{2} \sigma^\* E\_0^2. \tag{61}$$

Here, *σ*\* is a complex value and describes mobility and loss at the same time. When the real part is mobility and the imaginary part is loss, the conduction loss can be described in the same way as Eq. (50), and Eq. (62) is obtained:

$$P\_{\sigma\_{-}loss} = \frac{1}{2} \sigma'' |E|^2. \tag{62}$$

Comparing Eqs. (50) and (64), it can be seen that *<sup>σ</sup>*″ and *ωε*0*ε*r″ have the same dimensions. Therefore, the loss equation of the combined electric field is Eq. (63):

**Figure 9.** *Nyquist and bode diagrams of 0.1-Mol/L NaCl aqueous solution.*

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

$$P\_{E\_\perp loss} = \frac{1}{2} (\sigma'' + o\varepsilon'') |E|^2. \tag{63}$$

The apparent relative permittivity measured in **Figure 9** is the sum of the permittivity term and the conductivity term. This response is represented by *ψ*<sup>r</sup> \* (*ω*). Here, since *σ*r\* has the same dimension as *ωε*r\*, *ψ*r\*(*ω*) is expressed by the following Eq. (64):

$$
\varphi\_r^\*\left(\boldsymbol{\alpha}\right) = \frac{\sigma^\*\left(\boldsymbol{\alpha}\right)}{\boldsymbol{\alpha}\varepsilon\_0} + \varepsilon\_r^\*\left(\boldsymbol{\alpha}\right) = \frac{\sigma\_r^\*\left(\boldsymbol{\alpha}\right)}{\boldsymbol{\alpha}} + \varepsilon\_r^\*\left(\boldsymbol{\alpha}\right). \tag{64}
$$

When the complex conductivity *σ*\* [S/m] = *σ*<sup>0</sup> �*jσ*″ is determined in the same manner as the complex conductivity *ε*\* [F/m] = *ε*<sup>0</sup> �*jε*″, the complex relative permittivity *ε*r\* [nd] = *ε*\*/*ε*0, the complex relative conductivity is derived as *σ*r\* = *σ*\*/ *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*r' � *<sup>j</sup>σ*r″. Here, *<sup>σ</sup>*r\* has the same dimensions as *ωε*r\*, [rad/s]. Although the international annealed copper standard (IACS) is defined as 58 MS/m as a standard for conductivity, *σ*r\* indicates a ratio to *ε*0, rather than IACS.

In the previous diagram of the NaCl aqueous solution, the Nyquist diagram shows a semicircle in the high-frequency band, so this region has dielectric properties. On the other hand, in the low-frequency region, a locus different from a semicircle based on the movement of ions is shown. This means the loss due to the phase delay is small and mainly due to the motion of the ionic molecules. Whether the heat generation behavior at the irradiated frequency is mainly caused by dielectric or conductive loss cannot be distinguished by measurement at one frequency. Unless it is a Nyquist or Bode diagram, the contribution ratio of the dielectric and conductive losses cannot be separated from the locus.

## **3. Extraordinary microwave effect**

In many discussions of microwave chemistry, temperature and heat are very important. It is shown that the loss of electromagnetic energy is based on the imaginary part, i.e. *<sup>ε</sup>*r″ or *<sup>σ</sup>*r″, as shown in Eq. (65). On the other hand, it is stated that energy is accumulated in a material with a large real part because propagation delay occurs, and substantial loss can be obtained by the tan *δ* term as shown in Eq. (53). Since microwave energy penetrates the material, the amount of energy on the spot can be increased without changing the amount of material. Therefore, it must be remembered that there is a high energy in the field, even if there is no conversion to heat. From this viewpoint, we considered what behavior should be taken if there is a nonthermal effect when microwave energy is applied. This section discusses deductions based on principles, not induction based on data [28, 29].

#### **3.1 Classification of microwave effects**

When the chemical reaction field under microwave irradiation is different from the non-irradiation condition, its form can be classified into four types.

1.Fast reaction rate (acceleration);

2. Slow reaction rate (deceleration);

3.Different products (selective production);

4.Different consumptions (selective consuming).

The first is an example in which the reaction speed increases when microwave energy is applied, and as a result, the reaction's end time is shortened. Since the reaction rate can be significantly accelerated by increasing the temperature, one can always discuss whether the temperature was accurately measured or whether the actual reaction field temperature was high. The second is an example opposite to the first. This phenomenon appears when the actual reaction field temperature is low, but this case has little detailed discussion. If phase transition is included in the category, supercooling and overheating phenomena correspond to this. The third is an example in which products differ depending on the presence or absence of microwave irradiation when multiple products are considered. This happens when there are multiple reaction paths, one of which is particularly accelerated. This includes cases where only intermediates are obtained in a multistep reaction. The fourth is an example of a case with multiple substrates, and a reaction of the specific substrates is prioritized. In any of these cases, the reaction rate is considered to have changed due to the application of microwave energy. That is,

1.The target reaction speed increased;


All four of these interpretations mean that the specific reaction rate was increased by the application of microwave energy. From the aforementioned, when the chemical reaction under microwave irradiation is different from that in the non-irradiation case, if it is not a thermal effect, an equation for changing the reaction rate must be derived even if the temperature *T* is constant. The following describes the possibility of such an expression.

#### **3.2 Real and imaginary parts**

Eq. (63) describes the attenuation of the energy of the electric field. Considering the propagation here, it is an expression in which the imaginary part is changed to a real part. This means that the propagation equation is derived as Eq. (65):

$$P\_{E\_{-}prop} = \frac{1}{2} (\sigma' + o\varepsilon') \left| E \right|^2. \tag{65}$$

Since the conductive material can be regarded as a special state of the dielectric, discussion of the material including *σ* will be omitted hereafter. Energy of electromagnetic waves vibrate between electric and magnetic fields as follows equations (Eqs. (66)–(69)). Herein, the magnetic field loss and the propagation energy PH are also expressed based on the complex permeability *μ*\* = *μ*<sup>0</sup> �*jμ*″ [H/m]:

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

$$P\_{E\_{-}prop} = \frac{1}{2} a \epsilon' |E|^2;\tag{66}$$

$$P\_{E\_\text{-loss}} = \frac{1}{2} \alpha \varepsilon^{\prime\prime} |E|^2;\tag{67}$$

$$P\_{H\_{\text{-}prop}} = \frac{1}{2}\alpha\mu'|H|^2;\tag{68}$$

$$P\_{H\_{\text{-}}\text{loss}} = \frac{1}{2}a\mu''|H|^2. \tag{69}$$

Therefore, propagation of electromagnetic waves occurs according to the following stages:


Whether *P*X\_prop is involved as a field that affects the reaction or *P*X\_loss is involved as a result of affecting the reaction is not clearly determined now. In any case, however, it should be energy terms which affect the chemical reaction.

#### **3.3 Interpretation of the Arrhenius equations**

In the reaction kinetics, the activation energy theory described by Arrhenius (Eq. (70)), which originated from gas molecular kinetics, is discussed:

$$k = A e^{-\frac{E\_t}{RT}}.\tag{70}$$

Strictly speaking, it cannot be applied theoretically except for the secondary reaction of two gas molecules, but it is useful as an empirical formula and can be used in a solution system. Here, *k*, *A*, *E*a, and *R* are the reaction rate constant, A-factor,

#### **Figure 10.**

*Relationship between molecular velocity v and existence probability* φ*E. Line 1: small distribution* � *Line 3: large distribution.*

activation energy [J/mol], and molar gas constant [J/Kmol]. *E*<sup>a</sup> is assumed to be a constant that does not change with the reaction temperature.

The Arrhenius equation can be interpreted as follows. The horizontal axis in **Figure 10** shows the molecular velocity v in an arbitrary reaction coordinate system. The vertical axis *φ*<sup>v</sup> represents the existence probability of the molecule with a normal distribution. In **Figure 10**, it is assumed that a molecule that thermally and isotropically exchanges kinetic energy at a temperature *T* has a normal velocity distribution according to Eq. (71):

$$\phi\_v = A e^{-\frac{mv\_x^2}{2kT}}.\tag{71}$$

Here, the mass per mol is *m* [kg/mol], the kinetic energy of the molecule is *E*<sup>v</sup> = *m*v2 /2 [J/ mol], and the horizontal axis is energy, *E*. If the range of *E* is 0 ≤ *E* < +∞, this frequency distribution *φ*<sup>E</sup> is canonical, as shown in **Figure 11**.

$$
\phi\_E = A e^{-\frac{E}{RT}} \tag{72}
$$

**Figure 11.** *Reaction coordinate and energy distribution at temperature* T*.* *Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

The progress of the reaction is assumed to occur when the motion of the original molecule coincides with the positive direction of the reaction coordinate system and its kinetic energy exceeds *E*a. The reaction coordinate is shown on the left-hand side of **Figure 11**. Here, the substrate, transition state, and product are Sb, Tr, and Pd, respectively. The integrals of the region *E*<sup>a</sup> < *E* < ∞ and of the whole region 0 ≤ *E* < ∞ are expressed by Eqs. (73) and (74):

$$\int\_{E\_a}^{\infty} A e^{-\frac{E}{RT}} dE = ARTe^{-\frac{E\_a}{RT}};\tag{73}$$

$$\int\_0^\infty Ae^{-\frac{E}{RT}}dE =ART.\tag{74}$$

Therefore, the ratio of the region exceeding *E*<sup>a</sup> to the entire region is represented by Eq. (75), indicating that A is irrelevant:

$$\frac{\int\_{E\_x}^{\infty} A e^{-\frac{E}{kT}} dE}{\int\_0^{\infty} A e^{-\frac{E}{kT}} dE} = e^{-\frac{E\_0}{kT}}.\tag{75}$$

The value obtained by Eq. (75) is the Boltzmann factor. In the Arrhenius equation, the reaction rate is proportional to the Boltzmann factor because of the occupation ratio of the high energy state.

Here, the temperature *T* of the heat bath is increased to *T* + *ΔT*. In gas molecule kinetics, the kinetic energy of a molecule increases with temperature, and this is expressed by (*m*B*v*<sup>2</sup> )/2 = 3*k*B*T*/2. Here, *m*<sup>B</sup> and *k*<sup>B</sup> are the mass of one molecule and the Boltzmann constant, respectively. Therefore, a temperature increase of *ΔT* is the same as one molecule receiving *k*B*ΔT*/2 of energy as *m*B*v*<sup>x</sup> 2 /2, *m*B*v*<sup>y</sup> 2 /2, and *m*B*v*<sup>z</sup> 2 /2, respectively. Considering this at 1 mol, since *R* = *N*A*k*<sup>B</sup> (where *N*<sup>A</sup> is Avogadro's number), the kinetic energy of the system is distributed to the *x*, *y*, and *z* components by *RΔT*/2, respectively.

When any one direction is taken as the reaction coordinate, the energy distribution is the same in any state from Sb to Pd in the reaction coordinate system. Therefore, as expected, *E*<sup>a</sup> does not change, even if the temperature is raised. This is shown on the left-hand side of **Figure 12**. On the other hand, an increase in *T* corresponds to a decrease in the kurtosis of the Boltzmann distribution in the original molecule and an increase in the tail. As a result, the occupancy rate of molecules having an energy exceeding *E*<sup>a</sup> increases, and the reaction rate *k* increases, as shown at right in **Figure 12**.

**Figure 12.** *Reaction coordinate and energy distribution at temperature* T *+* ΔT*.*

Next, consider the supply of microwave energy instead of *ΔT* as an external energy supply. The Arrhenius equation presupposes that it is in thermal equilibrium, which indicates that energy can be exchanged quickly with an external heat bath. Therefore, if the rate at which the system releases heat to the outside matches the heating rate by microwaves, a constant temperature state can be maintained, and *T* can be regarded as constant. This means that there can be reaction systems with the same *T* and different microwave application intensities. Microwave energy is calculated as energy per unit time. On the other hand, the reaction kinetics discussed earlier calculate the change in the molar concentration of molecules as the reaction rate. Therefore, in order to align the units, the electromagnetic wave energy supplied to the volume per mole with respect to the concentration *M* [mol/m<sup>3</sup> ] of the target reaction molecule is *E*add [J/mol]. *E*add is proportional to the oscillated energy *E*MW [J/mol], which is the product of power per mole and irradiation time, but not all energy works effectively.

Herein, the effective efficiency is defined by *α* and *γ*, and *E*add = *αRT* = *γE*MW. This means that *α*RT is added to the thermal potential RT that the field already has. Assuming that this added amount of external energy is incorporated into the canonical distribution in the same manner as R*ΔT*, the potential distribution is derived in Eq. (76) by replacing RT in Eq. (72) with (1 + *α*)*RT*:

$$\phi\_E = A\_1 e^{-\frac{E}{(1+a)kT}} = A\_1 e^{-\frac{E}{RT + E\_{\rm add}}}.\tag{76}$$

The coefficient *A*<sup>1</sup> in Eq. (76) was introduced to consider the possibility that the A-factor changes depending on the presence or absence of external energy.

From the right-hand side of **Figure 13**, integration of the region *E*<sup>a</sup> < *E* < ∞, integration of the whole region 0 ≤ *E* < ∞, and the ratio between both are obtained as Eqs. (77), (78), and (79):

$$\int\_{E\_4}^{\infty} A\_1 e^{-\frac{E}{(1+a)RT}} dE = (1+a)A\_1 RTe^{-\frac{E\_4}{(1+a)RT}};\tag{77}$$

$$\int\_0^\infty A\_1 e^{-\frac{E}{(1+a)RT}} dE = (1+a)A\_1RT;\tag{78}$$

$$\begin{split} \int\_{E\_{a}}^{\infty} A\_{1} e^{-\frac{E}{(1+a)RT}} dE\\ \int\_{0}^{\infty} A\_{1} e^{-\frac{E}{(1+a)RT}} dE \end{split} \tag{79}$$

**Figure 13.** *Reaction coordinate and energy distribution at temperature* T *under microwave irradiation.*

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

From the aforementioned, it is apparent that the exponential form of the Boltzmann factor is maintained, and this expression is irrelevant to *A*1, just as was Eq. (75). Here, when the reaction rate constant under microwave non-irradiation is *k*<sup>0</sup> and that under irradiation is *k*1, the Arrhenius equation is described as follows:

$$k\_0 = A\_0 e^{-\frac{E\_t}{kT}};\tag{80}$$

$$k\_1 = A\_1 e^{-\frac{E\_a}{RT + E\_{add}}}.\tag{81}$$

Comparing both equations, *k*<sup>0</sup> = *k*<sup>1</sup> at *E*add = 0. Therefore, *A*<sup>0</sup> = *A*<sup>1</sup> is derived. Eq. (81) shows that the A-factor does not change under microwave irradiation.

### **3.4 Introduction of a nonthermal constant**

Generalizing Eq. (81) under an applied microwave irradiation yields Eq. (82):

$$k = A e^{-\frac{E\_a}{RT + \gamma E\_{MW}}}.\tag{82}$$

*E*MW is an intensive property that can change the applied amount from the outside. Therefore, changing *k* by a variable independent of temperature *T* is not thermal, that is, a nonthermal effect. Therefore, the coefficient *γ* in Eq. (82) is a nonthermal constant. Here, if *γ* is a positive value, the group has the same *T*, but the variance of the canonical distribution widens, and the occupation ratio of *E*<sup>a</sup> or higher is increased and *k*<sup>0</sup> < *k*1.

Eq. (82) does not change *E*<sup>a</sup> and *A*. Therefore, the reaction route is not changed. This is a reaction rate equation that can be applied, even when the microwave irradiation is 0, and the irradiation intensity of the microwave is variable. In this case, the population maintains a canonical distribution. If there is a nonthermal effect, it must be expressed by a reaction equation with a microwave intensity independent of temperature as a variable. Eq. (82) obtained by deduction meets this condition.

#### **3.5 Working principle of a nonthermal constant**

Eqs. (81) or (82) can be transformed into Eq. (83). *E*<sup>a</sup> and *A* can be obtained without microwave irradiation. If *E*<sup>a</sup> and A are not changed by microwave irradiation, *E*add, that is, *γ*EMW can be obtained from *k* at *T*. Assuming that the volume of the irradiation target is same and EMW is proportional to the irradiation power, *γ* can be calculated from Eq. (83):

$$a\mathbb{R}T = E\_{add} = \gamma E\_{MW} = \frac{E\_a}{\ln\left(A\right) - \ln\left(k\right)} - RT.\tag{83}$$

If the effect of microwave irradiation is expressed by Eq. (81), the ratio *r* between Eqs. (81) and (80) indicates the efficiency of increase by microwave irradiation (Eq. (84)):

$$r = \frac{Ae^{-\frac{E\_a}{(1+a)kT}}}{Ae^{-\frac{E\_a}{RT}}} = e^{\left(\frac{a}{1+a}\right)\frac{E\_a}{RT}}.\tag{84}$$

When *α* is 0, there is no irradiation, so *r* = 1. The upper limit of *r* is considered when *α* is infinite. Therefore, the upper limit *r*max of *r* is expressed by Eq. (85):

$$r\_{\max} = \lim\_{a \to \infty} e^{\left(\frac{a}{1+a}\right)\frac{E\_a}{RT}} = e^{\frac{E\_a}{RT}}.\tag{85}$$

This equation suggests that in a system having an effective *α*, the effect of microwave irradiation appears more markedly as the reaction temperature *T* is lower and *E*<sup>a</sup> is higher.

## **4. Conclusions**

Microwave chemistry was described with a focus on the behavior of dielectrics. We discussed the relationship between physical properties such as dielectric constant, energy propagation in the material, and energy loss in the material. By discussing energy, if there is a special effect other than heat in microwave chemical reactions, we have derived *a priori* by mathematical formulas factors that are required. The "Arrhenius equation under microwave irradiation" proposed in this section is one model. In this equation, a microwave energy term is added.

In order to confirm the results of this deduction inductively, correct measurements are required. A simple method is a surface temperature measurement with a radiation thermometer, but the fact that there is a temperature difference from the inside has been investigated in various experiments, and unless this problem is solved, it will lead to incorrect measurement. The temperature and its distribution should be validated with multiple methods, such as internal radiation measurements using an optical fiber, a fiber-grating method that measures the local volume changes of the optical fiber, and temperature dependence of the lifetime of the fluorescent material at the tip [30–32].

As presented in this paper, understanding microwave effects algebraically in terms of energy theory is different from considering energy distribution geometrically. For example, the existence of nonequilibrium local heating, which is influenced by the structure of the irradiated object, has been reported [33]. This phenomenon, which can be explained as a peculiar heating method occurring in microwave heating, is due to the geometric intensity of the electromagnetic field distribution. In this example, the unique heat distribution structure gives "geometric" microwave effects rather than "algebraic" microwave effects.

Microwaves in the GHz band have a decimator wavelength. If the size of the reaction vessel is smaller than the wavelength, the result may vary greatly depending on the place of installation and the shape of the vessel, based on the microwave interference. Therefore, in chemical synthesis assisted by microwave heating, it is necessary to consider the shape of the electromagnetic field distribution in the apparatus. Otherwise, other conditions may change at the same time that the temperature, substrate, solvent, and scale are changed. If this unintended influence is ignored, it becomes difficult to clarify general trends in the condition search, or the result will lead to excessively good or bad evaluations. Thus, microwave chemistry seems to be complicated in terms of chemistry and electromagnetic field analysis. However, it can be expected that more efficient reaction control will be possible by fully utilizing the control as an external field and utilizing it highly. In addition to the algebraic interpretation based on energetics, if the structure of the

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

irradiated object is geometrically controlled as a metamaterial, further microwave effects will be manifested. We believe that microwave chemistry will be a useful technique that can be used to manipulate chemical synthesis by applying external energy to a simple heating reaction.

## **Author details**

Jun-ichi Sugiyama\*, Hayato Sugiyama, Chika Sato and Maki Morizumi National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan

\*Address all correspondence to: sugiyama-j@aist.go.jp

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Ritter SK. Microwave chemistry remains hot, fast, and a tad mystical. Chemical & Engineering News. 2014; **92**(4):26-28

[2] Kappe A, Pieber B, Dallinger D. Effects in organic synthesis: Myth or reality? Angewandte Chemical International Edition. 2013;**52**: 1088-1094. DOI: 10.1002/ anie.201204103

[3] Rosana MR, Tao Y, Stiegman AE, Dudley GB. On the rational design of microwave-actuated organic reactions. Chemical Science. 2012;**3**:1240-1244. DOI: 10.1039/c2sc01003h

[4] Dudley GB, Stiegman AE, Rosana MR. Correspondence on microwave effects in organic synthesis. Angewandte Chemie, International Edition. 2013;**52**:7918-7923. DOI: 10.1002/anie.201301539

[5] Kappe CO. Reply to the correspondence on microwave effects in organic synthesis. Angewandte Chemie, International Edition. 2013;**52**: 7924-7928. DOI: 10.1002/ anie.201304368

[6] Nushiro K, Kikuchi S, Yamada T. Microwave effect on catalytic enantioselective Claisen rearrangement. Chemical Communications. 2013;**49**: 8371-8373. DOI: 10.1039/c3cc44610g

[7] Sugiyama J. Heating principle of microwave from a viewpoint of the mathematical formula. In: Proceedings of the 4th JEMEA Symposium. Fukuoka. Tokyo: JEMEA; 2010. pp. 26-27

[8] The International System of Units. 9th ed. The International Bureau of Weights and Measures; 2019 ISBN 978-92-822-2272-0

[9] Sugiyama J. Change of complex permittivity and electromagnetic field accompanying with temperature rising. In: Proceedings of the JEMEA Safety/ technology Seminar. Fukuoka. Tokyo: JEMEA; 2010. pp. 40-50

[10] Sugiyama J, Morizumi M, Sato C. Separation of conductivity sigma and permittivity epsilon in electrolyte solution. IEICE Technical Report. 2014; **MW2014**:1-6

[11] Sugiyama J. Development of the resonator to measure a complex permittivity at the wide temperature. IEICE Technical Report. 2009; **MW2009-79**:31-36

[12] Sugiyama J. Evaluation of the phase displacement delta by the complex permittivity measuring with resonators. IEICE Technical Report. 2010;**MW2009**: 11-16

[13] Sugiyama J. Calculation of the relaxation time tau by resonators and thermal behavior in heating device. IEICE Technical Report. 2010;**MW2010**: 13-18

[14] Sugiyama J. What is the microwave heating from a viewpoint of a material? JEMEA-Bulletin. 2017;**3**(1):15-18

[15] Sugiyama J. Heating principle of microwave from a viewpoint of the mathematical formula. In: Proceedings of the 7th JEMEA Symposium. Fukuoka. Tokyo: JEMEA; 2013. pp. 148-149

[16] Sugiyama J, Morizumi M, Sato C. Frequency dependence of the dielectricrelaxation of single liquid. In: Proceedings of the 7th JEMEA Symposium. Tokyo: JEMEA; 2013. pp. 188-189

*Electromagnetic Relations between Materials and Fields for Microwave Chemistry DOI: http://dx.doi.org/10.5772/intechopen.106257*

[17] Sugiyama J. Microwave heating from the viewpoint of materials. In: Proceedings of the 1st JEMEA Summer School. Takayama. Tokyo: JEMEA; 2016. pp. 35-80

[18] Sugiyama J. Microwave Course I, II, III. In: Proceedings of the 13th JEMEA Symposium. Tsukuba. Tokyo: JEMEA; 2019. p. 28

[19] Gabriel C, Gabriel S, Grant EH, Halstead BSJ, Mingos DMP. Dielectric parameters relevant to microwave dielectric heating. Chemical Society Reviews. 1998;**27**(3):213-223. DOI: 10.1039/a827213z

[20] Sugiyama J, Morizumi M, Sato C. Frequency dependence of the dielectricrelaxation of blended liquid. In: Proceedings of the 7th JEMEA Symposium. Tokyo: JEMEA; 2013. pp. 190-191

[21] Sugiyama J, Sato C. Heating of small amount solution by a solid oscillator. In: Proceedings of the 10th JEMEA Symposium. Tokyo: JEMEA; 2016. pp. 160-161

[22] Sugiyama H, Sugiyama J, Sato C. Study of characteristic correlation for dielectric liquid mixture. In: Proceedings of the 13th JEMEA Symposium. Tokyo: JEMEA; 2019. pp. 162-163

[23] Sugiyama J, Morizumi M, Sato C. Mathematical analysis of the empirical relationship of the dielectric-relaxation. In: Proceedings of the 8th JEMEA Symposium. Kochi, Tokyo: JEMEA; 2014. pp. 74-75

[24] Miyamoto S, Sugiyama J. Microwave characteristics measurement of the liquid which has electrical conductivity sigma. IEICE Technical Report. 2011; **MW2011**:13-18

[25] Sugiyama J, Yamazaki T, Moriike T, Suzuki M, Segawa T, Kato Y, et al. Prototyping of large-scale circular microwave oven and its actual heating efficiency at complete evaporation of water. IEICE Technical Report. 2011; **MW2010**:51-56

[26] Nagashima I, Sugiyama J, Sakuta T, Sasaki M, Shimizu H. Efficiency of 2.45 and 5.80GHz microwave irradiation for a hydrolysis reaction by thermostable βglucosidase HT1. Bioscience, Biotechnology, and Biochemistry. 2014; **78**(5):758-760. DOI: 10.1080/ 09168451.2014.891931

[27] Sugiyama J, Morizumi M, Sato C. Evaluation of the real and imaginary part of complex conductivity. In: Proceedings of the 8th JEMEA Symposium. Kochi, Tokyo: JEMEA; 2014. pp. 198-199

[28] Sugiyama J. Proposition of theory on the quantitativity of non-thermal effects. In: Proceedings of the 13th JEMEA Symposium. Tsukuba, Tokyo: JEMEA; 2019. pp. 41-42

[29] Sugiyama J, Yoneya M. Simulation of molecular motion in the application of alternating electromagnetic fields in the microwave band. Computation of Chemical Japan. 2021;**20**(3):123-125. DOI: 10.2477/jccj.2021-0035

[30] Sugiyama J. Electromagnetic relationship between microwaves and flow reactor systems. Chemical Record. 2019;**19**:146-156. DOI: 10.1002/ tcr.201800120

[31] Wada D, Sugiyama J, Zushi H, Murayama H. An optical fiber sensing technique for temperature distribution measurements in microwave heating. Measurement Science and Technology. 2015;**26**(085105):1-7. DOI: 10.1088/ 0957-0233/26/8/085105

[32] Wada D, Sugiyama J, Zushi H, Murayama H. Temperature distribution monitoring of a coiled flow channel in microwave heating using an optical fiber sensing technique. Sensors and Actuators B: Chemical. 2016;**232**: 434-441. DOI: 10.1016/j.snb.2016.03.156

[33] Wada Y, Tsubaki S, Maitani MM, Fujii S, Kishimoto F, Haneishi N. Physical insight to microwave special effects: Nonequilibrium local heating and acceleration of Electron transfer. Journal of the Japan Petroleum Institute. 2018;**61**(2):98-105. DOI: 10.1627/ jpi.61.98

## **Chapter 5**

## Power Consumption in CMOS Circuits

*Len Luet Ng, Kim Ho Yeap, Magdalene Wan Ching Goh and Veerendra Dakulagi*

## **Abstract**

In this chapter, we explain the two types of power consumption found in a complementary metal-oxide-semiconductor (CMOS) circuit. In general, a CMOS circuit tends to dissipate power at all times—be it active or inactive. The power consumed by the circuit when it is performing computational tasks is known as dynamic power. On the contrary, the power lost due to current leakage during which the circuit is dormant is referred to as static power. By carefully and properly designing the circuit, current leakage can be suppressed to its minimum. Hence, dynamic power consumption is usually significantly higher than its static counterpart. Some of the techniques that could be adopted to save dynamic power consumption include reducing the supply voltage, clock frequency, clock power, and dynamic effective capacitance. By probing into the activity factors of the design modules, the techniques can be applied to those with high power consumption.

**Keywords:** dynamic power, static power, switching power, short-circuit power, leakage power, supply voltage, clock frequency, dynamic effective capacitance, switching activity

## **1. Introduction**

More than half a century has elapsed since the three physicists from the AT&T Bell Laboratories—Brattain, Bardeen, and Shockley—invented the first solid-state transistor in December 1947 [1–3]. In comparison with the thermionic triode (which is colloquially known as the vacuum tube), the solid-state transistor is much smaller in size, consumes much lower power, operates at a relatively lower temperature, and exhibits significantly faster response time. Hence, the solid-state transistor swiftly replaced its predecessor as the predominant building block for electronic devices. The inexorable widespread application of solid-state transistors in electronic circuits has triggered a dramatic revolution in the electronic industries.

Today, microchips are built from the solid-state metal-oxide-semiconductor fieldeffect transistors (MOSFETs). A typical microchip consists of arrays of negative and positive MOSFETs, which are commonly denoted as the NMOS and PMOS transistors, respectively. **Figures 1** and **2** illustrate the symbols and cross-sections of the NMOS and PMOS transistors. As can be seen from the figures, the source and drain terminals

#### **Figure 1.**

*The (a) symbol and (b) cross-section of a NMOS transistor.*

#### **Figure 2.**

*The (a) symbol and (b) cross-section of a PMOS transistor.*

of the NMOS transistor are heavily doped with donator ions, such as phosphorous (P) or arsenic (As), whereas, its body is moderately doped with boron (B) acceptor ions. The PMOS transistor, on the other hand, consists of a high density of B ions at its source and drain and a moderate density of P ions at its body. Since the combination of these two transistors dissipates lower static power and offers higher noise immunity than implementing either the NMOS or PMOS transistor alone, they are both applied concurrently when designing electronic circuits. An electronic circuit that constitutes both the NMOS and PMOS transistors is referred to as the complementary metaloxide-semiconductor or CMOS.

The insatiable desire to incorporate more functionalities into a microchip has issued a clarion call for a higher number of transistors to be fabricated within it. A state-of-the-art electronic device today, for instance, may be equipped with the fifthgeneration (5G) telecommunication, neural engine (NE), augmented reality (AR), cloud computing, facial and speech recognition, and wireless power transmission technologies. These features could only be supported by millions, if not billions, of transistors in the chip. In order to build more transistors into the chip, the size of a transistor has undergone significant reductions over the years [4–6]. By shrinking the size of the transistor, the switching speed of the logic components can also be enhanced, while the operating power can be saved [7]. An advanced microprocessor today possesses more than 50 billion transistors, with technology nodes as small as 5 nm, clock rates of about 5 GHz [3], and an area less than 500 mm<sup>2</sup> . Microchips are

now interwoven seamlessly with the fabric of mankind, and, in many aspects, they have become an indispensable necessity to mankind.

## **2. Power consumption in a CMOS circuit**

The total power consumption *Ptotal* in a CMOS circuit comprises two major components, namely, the dynamic power *Pdynamic* and static power *Pstatic*, that is,

$$P\_{\text{total}} = P\_{\text{dynamic}} + P\_{\text{static}} \tag{1}$$

*Pdynamic* refers to the power consumed by the circuit when it is performing useful work during the active mode, whereas *Pstatic* is the power lost due to the leakage current that flows through the transistors when the circuit is inactive [8]. An overview of the different types of power consumption is displayed in **Figure 3**.

## **2.1 Dynamic power consumption**

A CMOS circuit dissipates dynamic power *Pdynamic* in either of the following conditions:


#### **Figure 3.**

*Different types of power consumption in a CMOS circuit.*

Dynamic power dissipation due to the transient short-circuit current path is considerably lower than that caused by the circuits with high switching activities. Hence, emphasis is usually given on finding ways to reduce *Pswitch*. Nevertheless, the noise created by the short-circuit current may sometimes be disturbing since it could cause errors in the output logic.

### *2.1.1 Switching power*

In general, the energy delivered to a CMOS circuit can be classified into two parts, namely, the charging and discharging of the load capacitance *CL*. To understand how energy delivery takes place, a simple CMOS inverter is shown in **Figure 4**, which is used for illustration.

During the charging phase, the input gate signal switches from logic 1 to 0, and, as a result, the PMOS transistor is switched on, while its NMOS counterpart is switched off. As can be seen in **Figure 4(a)**, the load capacitance *CL* is connected to the supply voltage via the PMOS transistor, thereby allowing current *I*(*t*) to charge *CL* to the supply voltage *VDD*. The energy *Ed* delivered to *CL* is derived in Eq. (2) given below:

$$E\_d = \int\_0^\infty I(t) V\_{DD} \, dt. \tag{2}$$

*The (a) charging path (VDD to CL) and (b) discharging path (CL to* GND*) of the capacitive load in the CMOS circuit.*

Since the current to charge a capacitor *C* to voltage *V* can be obtained from

$$I(t) = \mathcal{C} \frac{dV}{dt},\tag{3}$$

*Ed* in Eq. (2) can, therefore, be expressed as [8].

$$E\_d = \mathbf{C}\_L \mathbf{V}\_{\rm DD} \mathbf{2} \,\tag{4}$$

Likewise, the energy stored *Ec* in *CL* during the charging phase for each transition is derived in Eq. (5) given below:

$$E\_c = \int\_0^\infty I(t)V(t)dt = \frac{1}{2}C\_L V\_{DD}^{-2} \tag{5}$$

It can be observed between Eqs. (4) and (5) that only half of the delivered energy *Ed* is stored in *CL*, while the remaining half is dissipated in the PMOS transistor. In other words, a CMOS circuit encounters power loss for each logic transition when the current passes the transistors. The changing of the logic state is known as the switching activity. Each time a switching activity occurs for a particular node, the transistors will consume energy. Hence, Eq. (4) rather than Eq. (5) is to be used when determining the switching power consumption of a CMOS circuit. This is because both the energy stored in the load and that dissipated in the transistors have to be taken into account.

When the gate signal changes from logic 0 back to 1, the opposite scenario as that of the charging phase occurs this time, the PMOS transistor is switched off and NMOS switched on. During this discharging phase, the energy stored previously in the load capacitance *Ec* <sup>¼</sup> <sup>1</sup> <sup>2</sup>*CLVDD* <sup>2</sup> is drained completely to the ground via the NMOS transistor, as seen in **Figure 4(b)**.

When deriving Eq. (4), only a single state transition is considered. In reality, however, the scenario of the signal change at the circuit node may be more complicated than that. Since the feeding signal may have more than one transition within a time interval, Eq. (4) has to be multiplied by *N* times of transitions to obtain the total delivered power *Edt*, that is,

$$E\_{\rm dt} = \mathbf{N} \cdot \mathbf{C}\_{L} \mathbf{V}\_{\rm DD} \,^2. \tag{6}$$

Assuming that a circuit node toggles at frequency *fswitch* over a time interval *T*, *N* can be written as

$$N = T \cdot f\_{switch}.\tag{7}$$

Substituting Eq. (7) into (6), the total energy delivered can be expressed as

$$E\_{\rm dt} = T \cdot f\_{\rm switch} \mathbf{C}\_L \mathbf{V}\_{\rm DD} \,^2. \tag{8}$$

Since power is defined as the rate at which energy is used, the relationship between the switching power *Pswitch* and the total delivered power *Edt* can be described as

$$P\_{switch} = \frac{E\_{dt}}{T}.\tag{9}$$

Substituting Eq. (8) into (9), the switching power *Pswitch* can be written as

$$P\_{switch} = f\_{switch} C\_L V\_{DD} \,^2. \tag{10}$$

Eq. (10) can only be used to accurately calculate *Pswitch* as long as the assumption in Eq. (7) holds valid. In most CMOS circuits, however, logic does not really switch at a constant frequency *fswitch*. It is, therefore, more persuasive to express *fswitch* in terms of the product of *AF* and the clock frequency *fclk*, that is,

$$f\_{switch} = AF \cdot f\_{clk} \tag{11}$$

Doing so, *Pswitch* in Eq. (10) becomes

$$P\_{switch} = AF \cdot f\_{clk} \cdot C\_L V\_{DD} \,^2 \tag{12}$$

The product of *CL*�*AF* is typically represented by the variable *Cdyn*, which is called the dynamic effective capacitance. Hence, Eq. (12) can be rewritten as

$$P\_{switch} = f\_{clk} \cdot C\_{dyn} V\_{DD} \,^2. \tag{13}$$

According to Weste and Harris [9], activity factor (*AF*) is defined as the probability that the circuit node changes from logic 0 to logic 1, and this is the only time that the circuit consumes switching power. Therefore, *AF* is an important element to estimate the power consumption of a circuit. In order to gain a better understanding of *AF*, assume that the clock is triggered at every single cycle, that is, *fswitch* = *fclk*. From Eq. (11), it can be seen that *AF* = 1. Likewise, *AF* is found to be 2 for a data signal, which toggles once every two clock cycles. This phenomenon is graphically depicted in **Figure 5**. In most cases, the least significant bit (LSB) and the most significant bit (MSB) contain the highest and lowest *AF*, respectively.

For a circuit node that switches its logic states in an irregular manner, *AF* can be determined by multiplying the probability the node switches to logic 0, *P*<sup>0</sup> *<sup>f</sup>* , with the probability it switches to logic 1, *P*<sup>1</sup> *<sup>f</sup>* , that is,

$$AF = P\_f^0 \times P\_f^1 \tag{14}$$

For instance, a switching function expression is given as *F* ¼ *A* þ *BC* þ *BC*. The truth table of *F* is shown in **Table 1**. Out of the eight combinations of input values in

**Figure 5.** *Clock signal (*AF *= 1) and data signal (*AF *= 0.5).* *Power Consumption in CMOS Circuits DOI: http://dx.doi.org/10.5772/intechopen.105717*


#### **Table 1.** *Truth table of F* ¼ *A* þ *BC* þ *B C.*

the truth table, two produce logic 1 at output *F,* while the remaining ones produce logic 0. Hence, *P*<sup>0</sup> *<sup>f</sup>* and *P*<sup>1</sup> *<sup>f</sup>* can be obtained as <sup>6</sup> <sup>8</sup> and <sup>2</sup> <sup>8</sup>, respectively. Substituting these values into Eq. (14), *AF* is then found to be 0.1875. This is to say that, the probability that the circuit is active is only 0.1875, which is clearly low. Comparing this value with the frequency of occurrence for logic 1 found in the truth table, it can be seen that *AF* gives a good indication of the active rate of the circuit.

## *2.1.2 Short-circuit power*

Since it takes time for the parasitic capacitance to charge and discharge, the signal fed to a circuit does not change its logic state instantly. The input signal consists of the finite rise and fall times by this means. Unlike the switching power dissipation, where only one of the transistors is switched on at a time, short-circuit power dissipation *Pshort* is induced from the concurrent activation of both the NMOS and PMOS transistors. When the logic changes its state, there is a short window of time where the PMOS and NMOS transistors are switched on simultaneously. A direct current path connecting *VDD* to the ground is produced within this interval, resulting in shortcircuit power dissipation *Pshort*. As can be seen in **Figure 6**, the short-circuit current that passes both the PMOS and NMOS transistors during the transition state does not contribute to the charging and discharging of the load capacitance. The power

**Figure 6.** *Short-circuit path (VDD to* GND*) in a CMOS circuit.*

produced does not deliver any meaningful activities at the output and is therefore wasted. The short-circuit power *Pshort* can be mathematically expressed as

$$P\_{short} = T\_{\kappa} \cdot V\_{DD} \cdot I\_{peak} \tag{15}$$

where *Tsc* is the rising or falling time of the input signal and *Ipeak* is the peak current, which could be estimated from the transistor size and technology process.

### **2.2 Static power**

Static power *Pstatic* refers to the power lost when the CMOS circuit is dormant. The main culprit of *Pstatic* is the leakage current, which exists mainly because of the shortchannel effects [10]. As the technology node continues to reduce toward the subnanometer range, leakage current has become a major problem. In 2011, the severity of the leakage reached the brink, which prompted Intel Corporation to introduce the 22 nm tri-gate transistor. The tri-gate transistor is more popularly referred to as the FinFET, owing to its protruding drain and source structures, which resemble the fin of a fish. In comparison with the planar MOSFETs, the FinFET has better control of the current flow, thereby reducing leakage [11].

Among the short-channel effects that contribute predominantly to the *Pstatic* dissipation are the sub-threshold leakage current and the gate leakage current [12]. Sub-threshold leakage current is the weak current that exists between the source and drain terminals during the off-state of the transistor, as a result of the weak inversion layer at the oxide–substrate interface. This phenomenon occurs when the gate voltage is lower than the threshold voltage *VTH*. The sub-threshold leakage increases exponentially as the feature size continues to shrink [13]. This is mainly caused by the reduction of the *VTH*, which is also scaled down accordingly.

When the size of the transistor decreases, the oxide layer is thinned as well. The oxide thickness is continuously thinned until a certain extent, an undesired electric field is induced at the oxide–substrate interface whenever voltage is applied at the gate terminal. The electric field increases the probability of electrons to tunnel through the oxide layer from the channel region into the gate and vice versa [14], leading to gate leakage current. Although its impact is less severe compared to the sub-threshold leakage, current leakage by virtue of this mechanism has gradually exacerbated when the technology node penetrates the nanometric regime.

The equation that describes static power dissipation can be expressed as,

$$P\_{\text{static}} = V\_{\text{DD}} \cdot I\_{\text{leakage}} \tag{16}$$

where *Ileakage* denotes the total leakage current.

## **3. Dynamic power optimization**

With an increasing number of features being installed into a microchip today, higher computing power is drawn by electronic devices. Power consumption has, therefore, become a key concern in a CMOS circuit designs. Clearly, it is imperative to employ effective approaches to cut down the usage of power in microchips.

The first step for power optimization is to analyze the overall power consumed by a CMOS circuit. By substituting Eqs. (13), (15), and (16) into Eq. (1), the total power consumption *Ptotal* can be obtained as

$$P\_{\text{total}} = \left(\mathcal{f}\_{\text{clk}} \cdot \mathbb{C}\_{\text{dyn}} \cdot V\_{\text{DD}}\,^2\right) + \left(T\_{\text{sc}} \cdot V\_{\text{DD}} \cdot I\_{\text{peak}}\right) + \left(V\_{\text{DD}} \cdot I\_{\text{leakage}}\right). \tag{17}$$

Upon inspection, it can be observed that *Ptotal* is dictated by six power components, namely, *fclk*, *Cdyn*, *VDD*,*Tsc*, *Ipeak*, and *Ileakage*. By carefully and properly designing the logic circuit, *Pshort* and *Pstatic* can usually be minimized, if not completely suppressed. Hence, only clock frequency *fclk*, dynamic effective capacitance *Cdyn*, and supply voltage *VDD* are typically adjusted to optimize the power consumption of the circuit. Although modifying any of these three components may result in power saving, precaution is to be heeded since some approaches may also impose adverse effects on the circuits. The reduction of *VDD*, for example, yields a quadratic effect on diminishing the usage of power. Doing so, however, may also impair the performance of the circuit. This approach is generally not recommended unless there is a need to switch the device from high performing to high power saving. Similarly, reducing the clock frequency *fclk* may not necessarily be effective in curtailing power consumption because the system has to operate for a longer period of time when fed with the same load. Tuning *Cdyn* is perhaps one of the most popular options since it involves decreasing either the parasitic capacitance or the activity factor *AF* for idle nodes. A summary of some of the existing methods adopted to optimize power usage is presented in the subsequent sections. The first three methods discussed in the subsequent sections are related to the adjustments of the clock frequency *fclk* and supply voltage *VDD*, while the remaining ones deal with the dynamic effective capacitance *Cdyn*.

## **3.1 Dynamic frequency scaling**

Weissel and Bellosa [15] proposed an event-driven clock scaling approach for dynamic power management. In their work, schedulers were utilized to determine the appropriate clock frequency for each thread. Finally, the frequency of the dedicated applications can be adjusted based on the recurrent analysis of the thread-specific performance profile. Their analysis showed that at least 37% of energy can be saved with a performance loss of less than 10% when tested on the Intel XScale architecture.

Although this approach shows positive results in power saving, it may suffer from longer application execution time, since the frequency is reduced. As a consequence of this, applications may be corrupted if certain deadlines were missed.

### **3.2 Multiple supply voltages**

Chabini et al. [16] in their study proposed to minimize the dynamic power consumption under performance constraints by scaling down the supply voltage of computational elements off critical paths. In their work, the mixed-integer linear programming (MILP) method was first used to determine a schedule of the computational blocks that will lead to the maximum reduction of dynamic power consumption with designed performance constraints. Once the valid periodic schedule was computed, registers were inserted into the circuit to preserve the behavior of the original circuit. When compared to the design using the highest supply voltages, power reduction factors as high as 69.75% were obtained from their work.

Since multiple voltage domains are involved, multiple power grid structures are required. Power and floor plannings must, therefore, be implemented with care when employing this approach to optimize power consumption.

### **3.3 Clock gating for clock tree**

According to Donno et al. [17], the power drawn by the clock tree in advanced microchips tends to dominate. Hence, they introduced a clock-gating approach that could be adopted at the register transfer level (RTL) to reduce clock power. In the algorithm that they developed, a clock tree topology, which balanced the reduction in clock switching against clock and activation function capacitive loading estimates, was first built. Clock-gating logic was then incorporated into the tree, achieving a balance between its power consumption and the power on the gated clock sub-tree. The physical and functional information was taken as inputs to generate a clock netlist at the output. The netlist was then used to update the structural description of the design. The results show that the capability of their approach in power saving is 75% better than conventional clock-gating methods.

The work proposed in ref. [17] focused on the calculation of the active and idle time frames of different registers and inserting the clock gating logics into the netlist. Since the algorithm implemented in the work did not account for an optimized location for the gating logic, the registers may end up being placed far apart after the netlist is updated. A larger clock network size and higher power consumed by the clock tree may, therefore, ensue.

#### **3.4 Downsizing gates**

Gate sizes are proportional to the parasitic capacitances. By this means, dynamic effective capacitance *Cdyn* could be reduced when gates are downsized. By selectively downsizing the gates of a circuit, Aizik and Kolodny [18] demonstrated that dynamic and leakage energy dissipations can be reduced. Doing so, however, may lead to an increase in speed delay. This is to say that the operating speed of a circuit can be traded off in exchange for power reduction. The findings reveal that 25% of dynamic power can be saved for circuits in 32-nm technology when the delay constraint is relaxed by 5%.

#### **3.5 Interconnect-power reduction**

A circuit consumes switching power *Pswitch* when the interconnection capacitances are charged and discharged. The analysis in ref. [19] showed that the interconnect power occupied more than half of the total dynamic power consumption *Pdynamic*, with 90% of it contributed from 10% of the interconnections. To reduce *Cdyn*, larger wire spacing and minimal length routing were implemented for the high-power consuming interconnects. The researchers re-configured the interconnects without sacrificing the area and timing degradation and the results that they obtained showed that *Pdynamic* was saved by 14%.

#### **3.6 Clock network optimization**

Lu et al. [20] demonstrated that power consumption is affected by the size of the clock network. They developed an algorithm that navigated the registers' locations during cell placement. When performing the navigation process, the Manhattan ringbased register guidance, the center of gravity constraints for registers, pseudo pin and

## *Power Consumption in CMOS Circuits DOI: http://dx.doi.org/10.5772/intechopen.105717*

net, and register cluster contraction were observed. The clock net wirelength was decreased by 16–33%, with no more than 0.5% increase in signal net wirelength.

However, precautions must be taken when adopting this approach, particularly for circuits with high densities. This is because, reducing the clock network size may increase the risk of routing congestion and this may lead to a poor signal net to wirelength.

## **3.7 Net ordering and wire space optimization**

To optimize power consumption, Moiseev et al. [21] endeavored to reduce the capacitance for the most active nodes within parallel bundles. This can be achieved by finding the best arrangement of adjacent signals in the bundle and rearranging the positions of the wires so that the most active signal shares the smallest crosscapacitance. The approaches that they adopted are net ordering and wire space optimization. In their work, signals with high switching activity (*SA*) share a relatively larger space than those with lower *SA* (as seen in **Figure 7**). Further, the LSB is proposed to be

#### **Figure 8.**

*Net ordering. The LSB (highest switching activity signal) is positioned at the center, while the MSB (lowest switching activity signal) is positioned near the sidewall.*

placed at the center of the bundle and the MSBs at the ends, as depicted in **Figure 8**. This spacing and ordering optimization method was applied on industrial layouts of 65 nm process technology and the power saved ranged from 9 to 37%.

#### **3.8 Leaving unused routing conductors floating**

To reduce the effective coupling capacitance, Huda, Anderson, and Tamura [22] proposed to leave routing conductors adjacent to those used by timing critical or high activity nets floating. The purpose of doing so is to ensure that the original coupling capacitance among the conductors stays in series with other capacitances in the circuit. It is to be noted that, the equivalent capacitance of a chain of series capacitances is lower. As can be seen in **Figure 9**, tri-state buffers were used to disconnect the unused conductors. During high switching conditions, the tri-state buffer is allowed to be used as a normal buffer, without the loss or delay of data. The results show that the interconnect dynamic power was reduced up to approximately 15.5%, with a critical path degradation of about 1% and a total area overhead of about 2.1%.

## **4. Switching activity**

The power optimization techniques discussed in the previous section involve redesigning the circuit topology. Most of these techniques are conducted based on the assessment of the switching activity. Switching activity *SA* comprises two basic

#### **Figure 9.**

*The unused adjacent routing conductors are left floating, by connecting them to tri-state buffers.* C*<sup>1</sup> and* C*<sup>2</sup> denote the coupling capacitance, while* Cp *denotes the plate capacitance (i.e., the parasitic capacitance formed between the substrate and the metal layers).*

elements, namely, (i) the toggle rate and (ii) static probability. The toggle rate gives an indication of the frequency the node toggles for a specified interval, and it is usually measured in millions of transistors per second. The static probability, on the other hand, predicts the logic state of the signal. An *SA* that shows 0.4 static probability, for example, suggests that the signal gives a logic 1 for 40% of the time and a logic 0 for the remaining 60%. **Figures 10** and **11** illustrate two sets of signals with opposite scenarios—the signals in **Figure 10** consist of identical toggle rates but different static probabilities, whereas those in **Figure 11** show different toggle rates, but the same static probability.

Switching activity is an important reference tool when performing power analysis. By reading the *SA*, critical design blocks that consume high power can be identified. The details of the circuit, such as where and when it has the lowest and highest toggle rate, can also be ascertained. This information is important in deciding the appropriate approach to save power.

#### **Figure 10.**

*The pulse trains in (a) and (b) consist of identical toggle rate but different static probabilities. For the pulse train in (a), toggle rate = 6 and static probability = 0.5; and that in (b), toggle rate = 6 and static probability = 0.25.*

#### **Figure 11.**

*The pulse trains in (a) and (b) consist of identical static probability but different toggle rates. For the pulse train in (a), toggle rate = 4 and static probability = 0.5; and that in (b), toggle rate = 2 and static probability = 0.5.*

## **5. Conclusion**

In this chapter, the power consumed by CMOS circuits is expounded. The total power consumption *Ptotal* in a CMOS circuit can be classified into two types—viz, the dynamic power *Pdynamic* and static power *Pstatic*. Dynamic power refers to the power dissipated by the circuit when it is operating. It is induced by the switching activities, which take place at the nodes, and the short-circuit current formed at the transition state of the logic switch. Static power, on the other hand, occurs when the circuit is idle. It is caused mainly by the subthreshold and gate leakage currents. Since dynamic power takes up a significant fraction of the overall power consumption, different approaches have been developed to minimize it. The approaches focus on the reduction of the supply voltages, clock frequencies, or dynamic effective capacitance. By studying the activity factors of the design modules, the approaches can be applied to those with high power consumption.

*Power Consumption in CMOS Circuits DOI: http://dx.doi.org/10.5772/intechopen.105717*

## **Author details**

Len Luet Ng<sup>1</sup> , Kim Ho Yeap<sup>2</sup> \*, Magdalene Wan Ching Goh<sup>3</sup> and Veerendra Dakulagi<sup>4</sup>


© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Yeap KH, Nisar H. Introductory chapter: VLSI. In: Yeap KH, Nisar H, editors. Very-Large-Scale-Integration. London, UK: InTechOpen; 2018. pp. 3-11

[2] Yeap KH, Nisar H. Introductory chapter: Complementary metal oxide semiconductor (CMOS). In: Yeap KH, Nisar H, editors. Complementary Metal Oxide Semiconductor. London, UK: InTechOpen; 2018. pp. 3-7

[3] Yeap KH, Isa MM, Loh SH. Introductory chapter: Integrated circuit Chip. In: Yeap KH, Hoyos JJS, editors. Integrated Circuits/Microchips. London, UK: InTechOpen; 2020. pp. 1-13

[4] Ho YK, Ahmad I, Sulong MS. Characterization of a 0.14 μm submicron NMOS with Silvaco TCAD simulator. Journal of Science and Technology. 2009;**1**:1-10

[5] Sulong MS, Ahmad I, Foo LT, Ho YK. Characterization of a 130 nm CMOS device using CVIV and focused ion beam. Jurnal Sains dan Teknologi. 2007; **4**:37-46

[6] Ho YK, Meng MK, Chun LK, Chiong TP, Nisar H, Rizman ZI. Design and analysis of 15 nm MOSFETs. Journal of Telecommunication, Electronic and Computer Engineering. 2016;**8**:1-4

[7] Yeap KH, Thee KW, Lai KC, Nisar H, Krishnan KC. VLSI circuit optimization for 8051 MCU. International Journal of Technology. 2018;**9**:142-149

[8] Rabaey JM, Chandrakasan A, Nikolic B. Digital Integrated Circuits: A Design Perspective. UK: Pearson Education; 2003

[9] Weste NH, Harris D. CMOS VLSI Design: A Circuits and Systems

Perspective. India: Pearson Education; 2015

[10] Yeap KH. Fundamentals of Digital Integrated Circuit Design. 1st ed. UK: Author House; 2011

[11] Yeap KH, Lee JY, Yeo WL, Nisar H, Loh SH. Design and characterization of a 10 nm FinFET. Malaysian Journal of Fundamental and Applied Sciences. 2019;**15**:609-612

[12] Jacob B, Wang D, Ng S. Memory Systems: Cache, DRAM, Disk. US: Morgan Kaufmann; 2010

[13] Sharroush SM. An MTCMOS subthreshold-leakage reduction algorithm. In: 2nd Novel Intelligent and Leading Emerging Sciences Conference. Giza, Egypt: IEEE; 2020. pp. 7-14

[14] Chinta V. Subthreshold and Gate Leakage Current Analysis and Reduction in VLSI Circuits. Master of Science Thesis. Rochester, NY: Kate Gleason College of Engineering; 2007

[15] Weissel A, Bellosa F. Process cruise control: Event-driven clock scaling for dynamic power management. In: Proceedings of the International Conference on Compilers, Architecture, and Synthesis for Embedded Systems. Grenoble, France: Association for Computing Machinery (ACM); 2002. pp. 238-246

[16] Chabini N, Chabini I, Aboulhamid EM, Savaria Y. Methods for minimizing dynamic power consumption in synchronous designs with multiple supply voltages. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 2003;**22**:346-351

*Power Consumption in CMOS Circuits DOI: http://dx.doi.org/10.5772/intechopen.105717*

[17] Donno M, Ivaldi A, Benini L, Macii E. Clock-tree power optimization based on RTL clock-gating. In: Proceedings of the 40th Annual Design Automation Conference. Anaheim CA USA: Association for Computing Machinery (ACM); 2003. pp. 622-627

[18] Aizik Y, Kolodny A. Finding the energy efficient curve: Gate sizing for minimum power under delay constraints. VLSI Design. 2011;**2011**: 845957

[19] Magen N, Kolodny A, Weiser U, Shamir N. Interconnect-power dissipation in a microprocessor. In: Proceedings of the International Workshop on System Level Interconnect Prediction. NY USA: Association for Computing Machinery (ACM); 2004. pp. 7-13

[20] Lu Y, Sze CN, Hong X, Zhou Q, Cai Y, Huang L, et al. Navigating registers in placement for clock network minimization. In: Proceedings of the 42nd Annual Design Automation Conference. NY USA: Association for Computing Machinery (ACM); 2005. pp. 176-181

[21] Moiseev K, Kolodny A, Wimer S. Timing-aware power-optimal ordering of signals. ACM Transactions on Design Automation of Electronic Systems. 2008; **13**:1-17

[22] Huda S, Anderson J, Tamura H. Optimizing effective interconnect capacitance for FPGA power reduction. In: Proceedings of the ACM/SIGDA International Symposium on Field-Programmable Gate Arrays; NY USA: Association for Computing Machinery (ACM); 2014. pp. 11-20

## **Chapter 6**
