**2. Problem formulation of SOMIs**

To realize second-order microwave integrator in microwave range, consider a two-port network, which is illustrated in **Figure 2**.

Its scattering matrix is defined as [4].

$$
\begin{bmatrix} b\_1 \\ b\_2 \end{bmatrix} = \begin{bmatrix} \mathbb{S}\_{11} & \mathbb{S}\_{12} \\ \mathbb{S}\_{21} & \mathbb{S}\_{22} \end{bmatrix} \begin{bmatrix} a\_1 \\ a\_2 \end{bmatrix} \tag{8}
$$

where *a*<sup>1</sup> and *a*<sup>2</sup> are incident waves at port 1 and port 2, respectively, and *b*<sup>1</sup> and *b*<sup>2</sup> are reflected waves at port 1 and port 2, respectively [4]. The chain scattering matrix of a two-port network can be established from the scattering matrix (S-matrix). The chain-scattering matrix of two-port network is defined as [5].

*a*1 *b*1 <sup>¼</sup> *<sup>T</sup>*<sup>11</sup> *<sup>T</sup>*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> T22 *<sup>b</sup>*<sup>2</sup> *a*2 (9)

**Figure 2.** *Two-port network.*

Multiplying throughout by C, we get

*Innovations in Ultra-WideBand Technologies*

as *RC* ≫ *t*, the term Ð*<sup>t</sup>*

**Figure 1.**

*R-C Integrator circuit.*

From Eq. (2),

**130**

*CVin* ¼ *iRC* þ

*CVin* ¼ *RC*

*<sup>i</sup>:dt* <sup>¼</sup> <sup>1</sup> *RC* ð*t* 0

*Vout* <sup>¼</sup> <sup>1</sup> *RC* ð*t* 0

Eq. (7) shows that the output of an integrator circuit is the integral of the input signal. These analog integrators are limited for low frequency application. Thus, the researcher moved to design digital integrators. Digital integrator is a system that performs mathematical operations on a sampled discrete time signal to reduce or enhance certain aspects of that signal. It is commonly used for applications such as waveform shaping, coherent detection, edge detection, and accumulator analysis in biomedical engineering and signal processing. It is widely utilized in biomedical engineering and signal processing applications, for example, as waveform shaping, coherent detection, edge detection, and accumulator analysis. It is also used in radar applications such as the allocation of mobile satellites, enterprise networks, commercial television services and digital services [1]. In order to design the wideband digital integrators, various methods were intended. Using the Newton-cotes integration rule and various digital integration techniques, the Recursive wideband digital integrators have been designed [2]. For low-speed applications up to barely a few hundred MHz, the integrators are primarily designed and implemented. Therefore, to cover wideband applications such as radar and wireless communication, the design and implementation of integrators for high-frequency applications is necessary. The microwave integrator is essentially used to measure the time integral of the input signal at microwave frequencies (0.3–300 GHz). Using wideband integrators, the high-frequency active filters can be introduced, and these wideband integrators can also be used for industrial and real-time applications for

<sup>0</sup>*i:dt* may be neglected

ð*t* 0

1 *C* ð*t* 0

Integrating with respect to *t* on both sides of Eq. (3)

ð*t* 0

ð*t* 0 *i:dt* (3)

*i:dt* (5)

*Vin:dt* (6)

*Vin:dt* (7)

*CVin* ¼ *iRC* (4)

For network computation the chain-scattering matrix from S-matrix (on solving Eq. (8) and Eq. (9))

$$T\_{11} = \frac{1}{S\_{21}}$$

$$T\_{12} = -\frac{S\_{22}}{S\_{21}}$$

$$T\_{21} = \frac{S\_{11}}{S\_{21}}$$

$$T\_{22} = \frac{S\_{12}S\_{21} - S\_{11}S\_{22}}{S\_{21}}$$

Then the chain-scattering matrix is

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} = \begin{bmatrix} \frac{1}{S\_{21}} & -\frac{S\_{22}}{S\_{21}} \\ \frac{S\_{11}}{S\_{21}} & \frac{S\_{12}S\_{21} - S\_{11}S\_{22}}{S\_{21}} \end{bmatrix} \tag{10}
$$

Subsequently, the chain scattering matrix of open-circuited stub is

purpose, which resembles a scaling by two on frequency axis. Then the chain

ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* ð Þ <sup>1</sup> � *<sup>k</sup>* <sup>D</sup>�<sup>2</sup> *<sup>k</sup>* � *kD*�<sup>2</sup> � �

ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* ð Þ <sup>1</sup> � *<sup>k</sup> <sup>z</sup>*�<sup>1</sup> *<sup>k</sup>* � *kz*�<sup>1</sup> ð Þ

. Likewise, the chain scattering matrix for the transmission line

ð Þ� <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>z</sup>*�<sup>1</sup> *<sup>δ</sup>* <sup>þ</sup> *<sup>δ</sup>z*�<sup>1</sup> ð Þ

" #

�*<sup>δ</sup>* � *<sup>δ</sup>z*�<sup>1</sup> ð Þ� <sup>1</sup> � *<sup>δ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup> <sup>z</sup>*�<sup>1</sup>

*R r*¼1

*αο* <sup>þ</sup> *<sup>α</sup>*1*z*�<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*2*z*�<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*3*z*�<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>α</sup>Nz*�*<sup>N</sup>* (21)

<sup>1</sup> � <sup>Γ</sup><sup>2</sup> *r* � �

" #

<sup>2</sup>*Zoc*. Now we substitute *<sup>z</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> in above Eq. (14) for designing

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup>

" #

�*<sup>k</sup>* <sup>þ</sup> *kD*�<sup>2</sup> ð Þþ <sup>1</sup> � *<sup>k</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup> <sup>D</sup>*�<sup>2</sup>

�*<sup>k</sup>* <sup>þ</sup> *kz*�<sup>1</sup> ð Þþ <sup>1</sup> � *<sup>k</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup> <sup>z</sup>*�<sup>1</sup>

ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* ð Þ <sup>1</sup> � *<sup>k</sup> <sup>z</sup>*�<sup>1</sup> (16)

<sup>1</sup> � <sup>Γ</sup><sup>2</sup>*z*�<sup>1</sup> �<sup>Γ</sup> � <sup>Γ</sup>*z*�<sup>1</sup> <sup>1</sup> � <sup>Γ</sup>*z*�<sup>1</sup> �Γ<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup>

" #

(14)

(15)

(17)

(18)

(19)

(20)

*T*<sup>11</sup> *T*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> *<sup>T</sup>*<sup>22</sup> " #

where *<sup>k</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup>

*T*<sup>11</sup> *T*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> *<sup>T</sup>*<sup>22</sup> " #

*open*

*open*

section in *Z* domain is given by [6].

*T*<sup>11</sup> *T*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> *<sup>T</sup>*<sup>22</sup> " #

and *S*<sup>21</sup> is given by

where *<sup>z</sup>* <sup>¼</sup> *<sup>e</sup><sup>j</sup>β<sup>l</sup>*

*T*<sup>11</sup> *T*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> *<sup>T</sup>*<sup>22</sup> " #

*Short*

<sup>¼</sup> <sup>1</sup> 1 � *z*�<sup>1</sup>

The overall *S*<sup>21</sup> in generalized form is given by

*<sup>S</sup>*<sup>21</sup> <sup>¼</sup> <sup>1</sup> *T*<sup>11</sup> ¼

**133**

where the coefficient *δ* is given by

<sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>D</sup>*�<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94843*

*Analysis of Wideband Second-Order Microwave Integrators*

scattering matrix of open-circuited stub is [6].

<sup>¼</sup> <sup>1</sup> 1 þ *z*�<sup>1</sup>

> *<sup>S</sup>*<sup>21</sup> <sup>¼</sup> <sup>1</sup> *T*<sup>11</sup>

*TLS*

where the reflection coefficient Γ is given by

<sup>¼</sup> <sup>1</sup> *z*�<sup>1</sup>

<sup>2</sup> <sup>1</sup> � <sup>Γ</sup><sup>2</sup> � �

<sup>Γ</sup> <sup>¼</sup> *ZTL* � *<sup>Z</sup>*<sup>0</sup> *ZTL* þ *Z*<sup>0</sup>

where *ZTL* represents the characteristic impedance of serial transmission line section. Similarly, the chain scattering matrix of a short-circuited stub is given by [8]

> *<sup>δ</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup> 2*ZSC*

where *ZSC* represents the characteristic impedance of the short-circuited stub. Serial line sections and short-circuited stubs shunted with open circuited stub should be employed in the transmission line configuration to design an integrator by cascading. A cascaded connection of two-port network is equivalent to a single two-port network containing a product of matrices. Assume the SOI is composed of *P* open-circuited stubs, *Q* short-circuited stubs and *R* transmission line sections.

<sup>1</sup> <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup> ð Þ*<sup>P</sup>* <sup>1</sup> � *<sup>z</sup>*�<sup>1</sup> ð Þ*<sup>Q</sup> <sup>z</sup>*�*R=*<sup>2</sup> <sup>Q</sup>

The formulation of SOMIs is employed with equal length line elements is cascading. The overall transfer function of a cascaded network can be established by multiplying the chain scattering matrices of the line elements. These line elements can be transmission line sections and stubs. Assume the length of all transmission line sections and stubs is *l* ¼ *λο=*4, where *λο* represents the wavelength of the lines at the normalizing angular frequency or we can say that the electrical length of each section (stubs and transmission lines) is set to be 90*<sup>ο</sup>* at the normalizing frequency. The electrical length of line element is *θ* ¼ *βl*.

The frequency response of an ideal second order integrator (SOI) is given by

$$H(j\alpha) = \frac{1}{\alpha^2} \tag{11}$$

where ω represents the angular frequency in radians per second. Since the magnitude response of an open-circuited stub decreases with frequency, an opencircuited stub can be chosen to design an integrator. Assume the impedance of an open-circuited stub to be *Zoc*. Its chain scattering is [5].

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix}\_{open} = \begin{bmatrix} \mathbf{1} + j \frac{Z\_0}{2Z\_{\text{oc}}} \tan\left(\beta l\right) & j \frac{Z\_0}{2Z\_{\text{oc}}} \tan\left(\beta l\right) \\\ -j \frac{Z\_0}{2Z\_{\text{oc}}} \tan\left(\beta l\right) & \mathbf{1} - j \frac{Z\_0}{2Z\_{\text{oc}}} \tan\left(\beta l\right) \end{bmatrix} \tag{12}
$$

where *Z*<sup>0</sup> represents the reference characteristic impedance that is 50 Ω, *Zoc* is the characteristic impedance of an open circuited stub and *β* represents the phase constant.

Assume *ω* is the angular frequency and *τ* is the propagation delay attributable to the length *<sup>l</sup>*. Therefore, the term *<sup>j</sup>*tan ð Þ¼ *<sup>β</sup><sup>l</sup> <sup>j</sup>*tan ð Þ *ωτ* can be expressed by *<sup>D</sup>*�<sup>1</sup> <sup>¼</sup> *e*�*jωτ*, which can be considered as a unit of delay, that is [5].

$$\text{jtan}(a\sigma) = \frac{e^{j\alpha\tau} - e^{-j\alpha\tau}}{e^{j\alpha\tau} + e^{-j\alpha\tau}} = \frac{D - D^{-1}}{D + D^{-1}} \tag{13}$$

*Analysis of Wideband Second-Order Microwave Integrators DOI: http://dx.doi.org/10.5772/intechopen.94843*

Subsequently, the chain scattering matrix of open-circuited stub is

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ \\ T\_{21} & T\_{22} \end{bmatrix}\_{open} = \frac{\mathbf{1}}{\mathbf{1} + D^{-2}} \begin{bmatrix} (\mathbf{1} + k) + (\mathbf{1} - k)\mathbf{D}^{-2} & (k - kD^{-2}) \\\\ -k + kD^{-2} & (\mathbf{1} - k) + (\mathbf{1} + k)D^{-2} \end{bmatrix} \tag{14}
$$

where *<sup>k</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup> <sup>2</sup>*Zoc*. Now we substitute *<sup>z</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> in above Eq. (14) for designing purpose, which resembles a scaling by two on frequency axis. Then the chain scattering matrix of open-circuited stub is [6].

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ \\ T\_{21} & T\_{22} \end{bmatrix}\_{open} = \frac{\mathbf{1}}{\mathbf{1} + \mathbf{z}^{-1}} \begin{bmatrix} (\mathbf{1} + k) + (\mathbf{1} - k)\mathbf{z}^{-1} & (k - k\mathbf{z}^{-1}) \\ & -k + k\mathbf{z}^{-1} & (\mathbf{1} - k) + (\mathbf{1} + k)\mathbf{z}^{-1} \end{bmatrix} \tag{15}$$

and *S*<sup>21</sup> is given by

For network computation the chain-scattering matrix from S-matrix (on solving

*<sup>T</sup>*<sup>11</sup> <sup>¼</sup> <sup>1</sup> *S*<sup>21</sup>

*<sup>T</sup>*<sup>12</sup> ¼ � *<sup>S</sup>*<sup>22</sup>

*<sup>T</sup>*<sup>21</sup> <sup>¼</sup> *<sup>S</sup>*<sup>11</sup> *S*<sup>21</sup>

*<sup>T</sup>*<sup>22</sup> <sup>¼</sup> *<sup>S</sup>*<sup>12</sup> *<sup>S</sup>*<sup>21</sup> � *<sup>S</sup>*11*S*<sup>22</sup> *S*<sup>21</sup>

> 1 *S*<sup>21</sup>

*S*<sup>11</sup> *S*<sup>21</sup>

The formulation of SOMIs is employed with equal length line elements is cascading. The overall transfer function of a cascaded network can be established by multiplying the chain scattering matrices of the line elements. These line elements can be transmission line sections and stubs. Assume the length of all transmission line sections and stubs is *l* ¼ *λο=*4, where *λο* represents the wavelength of the lines at the normalizing angular frequency or we can say that the electrical length of each section (stubs and transmission lines) is set to be 90*<sup>ο</sup>* at the normalizing frequency.

The frequency response of an ideal second order integrator (SOI) is given by

*H j*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

where ω represents the angular frequency in radians per second. Since the magnitude response of an open-circuited stub decreases with frequency, an opencircuited stub can be chosen to design an integrator. Assume the impedance of an

tan ð Þ *<sup>β</sup><sup>l</sup> <sup>j</sup> <sup>Z</sup>*<sup>0</sup>

tan ð Þ *<sup>β</sup><sup>l</sup>* <sup>1</sup> � *<sup>j</sup> <sup>Z</sup>*<sup>0</sup>

*ejωτ* <sup>þ</sup> *<sup>e</sup>*�*jωτ* <sup>¼</sup> *<sup>D</sup>* � *<sup>D</sup>*�<sup>1</sup>

where *Z*<sup>0</sup> represents the reference characteristic impedance that is 50 Ω, *Zoc* is the characteristic impedance of an open circuited stub and *β* represents the phase

Assume *ω* is the angular frequency and *τ* is the propagation delay attributable to the length *<sup>l</sup>*. Therefore, the term *<sup>j</sup>*tan ð Þ¼ *<sup>β</sup><sup>l</sup> <sup>j</sup>*tan ð Þ *ωτ* can be expressed by *<sup>D</sup>*�<sup>1</sup> <sup>¼</sup>

2*Zoc*

2*Zoc*

tan ð Þ *βl*

tan ð Þ *βl*

*<sup>D</sup>* <sup>þ</sup> *<sup>D</sup>*�<sup>1</sup> (13)

*S*<sup>21</sup>

� *<sup>S</sup>*<sup>22</sup> *S*<sup>21</sup>

*<sup>ω</sup>*<sup>2</sup> (11)

(10)

(12)

*S*<sup>12</sup> *S*<sup>21</sup> � *S*11*S*<sup>22</sup> *S*<sup>21</sup>

Eq. (8) and Eq. (9))

Then the chain-scattering matrix is

*Innovations in Ultra-WideBand Technologies*

The electrical length of line element is *θ* ¼ *βl*.

*T*<sup>11</sup> *T*<sup>12</sup> *T*<sup>21</sup> *T*<sup>22</sup> � �

constant.

**132**

open-circuited stub to be *Zoc*. Its chain scattering is [5].

*e*�*jωτ*, which can be considered as a unit of delay, that is [5].

jtanð Þ¼ *ωτ ejωτ* � *<sup>e</sup>*�*jωτ*

<sup>1</sup> <sup>þ</sup> *<sup>j</sup> <sup>Z</sup>*<sup>0</sup> 2*Zoc*

> �*<sup>j</sup> <sup>Z</sup>*<sup>0</sup> 2*Zoc*

*open* ¼

*T*<sup>11</sup> *T*<sup>12</sup> *T*<sup>21</sup> *T*<sup>22</sup> � �

¼

$$S\_{21} = \frac{\mathbf{1}}{T\_{11}} = \frac{\mathbf{1} + z^{-1}}{(\mathbf{1} + k) + (\mathbf{1} - k)z^{-1}} \tag{16}$$

where *<sup>z</sup>* <sup>¼</sup> *<sup>e</sup><sup>j</sup>β<sup>l</sup>* . Likewise, the chain scattering matrix for the transmission line section in *Z* domain is given by [6].

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ \hline \\ T\_{21} & T\_{22} \end{bmatrix}\_{TLS} = \frac{\mathbf{1}}{\mathbf{z}^{-\frac{1}{2}}(\mathbf{1} - \boldsymbol{\Gamma}^2)} \begin{bmatrix} \mathbf{1} - \boldsymbol{\Gamma}^2 \mathbf{z}^{-1} & -\boldsymbol{\Gamma} - \boldsymbol{\Gamma} \mathbf{z}^{-1} \\\\ \mathbf{1} - \boldsymbol{\Gamma} \mathbf{z}^{-1} & -\boldsymbol{\Gamma}^2 + \mathbf{z}^{-1} \end{bmatrix} \tag{17}$$

where the reflection coefficient Γ is given by

$$
\Gamma = \frac{Z\_{\rm TL} - Z\_0}{Z\_{\rm TL} + Z\_0} \tag{18}
$$

where *ZTL* represents the characteristic impedance of serial transmission line section. Similarly, the chain scattering matrix of a short-circuited stub is given by [8]

$$
\begin{bmatrix} T\_{11} & T\_{12} \\ \hline T\_{21} & T\_{22} \end{bmatrix}\_{\text{Short}} = \frac{\mathbf{1}}{\mathbf{1} - \mathbf{z}^{-1}} \begin{bmatrix} (\mathbf{1} + \delta) - (\mathbf{1} - \delta)\mathbf{z}^{-1} & (\delta + \delta \mathbf{z}^{-1}) \\\\ -\delta - \delta \mathbf{z}^{-1} & (\mathbf{1} - \delta) - (\mathbf{1} + \delta)\mathbf{z}^{-1} \end{bmatrix} \tag{19}
$$

where the coefficient *δ* is given by

$$\delta = \frac{Z\_0}{2Z\_{\text{SC}}} \tag{20}$$

where *ZSC* represents the characteristic impedance of the short-circuited stub. Serial line sections and short-circuited stubs shunted with open circuited stub should be employed in the transmission line configuration to design an integrator by cascading. A cascaded connection of two-port network is equivalent to a single two-port network containing a product of matrices. Assume the SOI is composed of *P* open-circuited stubs, *Q* short-circuited stubs and *R* transmission line sections. The overall *S*<sup>21</sup> in generalized form is given by

$$S\_{21} = \frac{1}{T\_{11}} = \frac{\left(\mathbf{1} + \mathbf{z}^{-1}\right)^{P} \left(\mathbf{1} - \mathbf{z}^{-1}\right)^{Q} \mathbf{z}^{-R/2} \prod\_{r=1}^{R} \left(\mathbf{1} - \Gamma\_{r}^{2}\right)}{a\_{o} + a\_{1}\mathbf{z}^{-1} + a\_{2}\mathbf{z}^{-2} + a\_{3}\mathbf{z}^{-3} + \dots + a\_{N}\mathbf{z}^{-N}} \tag{21}$$

where, *P* denotes the number of open-circuited stubs, *Q* denotes the number of short-circuited stubs, *R* denotes the number of transmission line section, *N* is the total number of line elements *N* ¼ *P* þ *Q* þ *R* and *αο*, *α*1, *α*2, ..., *α<sup>N</sup>* Coefficients are functions of the reflection coefficient of each section of the transmission line and stub characteristic impedances (in terms of open-circuited stub coefficient k and short-circuited stub coefficient δ). In addition, to design the SOMI, serial transmission lines cascaded with shunt circuited stubs can be employed. Then the frequency-domain response of the transfer function, which has an integrator characteristic, is thus obtained. In order to design a wideband SOMI, the following two methods are used.

Design-I (SOMI using two open stubs and three transmission line sections): In this design, we may select two open-circuited stubs and three transmission line sections as shown in **Figure 3**, where *P* ¼ 2, *R* ¼ 3. Overall *S*<sup>21</sup> of the design-1 SOMI is given by

$$S\_{21} = \frac{\left(\mathbf{1} + \mathbf{z}^{-1}\right)^2 \mathbf{z}^{-3/2} \prod\_{r=1}^3 \left(\mathbf{1} - \Gamma\_r^2\right)}{a\_o + a\_1 \mathbf{z}^{-1} + a\_2 \mathbf{z}^{-2} + a\_3 \mathbf{z}^{-3} + a\_4 \mathbf{z}^{-4} + a\_5 \mathbf{z}^{-5}} \tag{22}$$

Design-II (SOMI using one open stub, one short stub and three transmission line section): In this design, a short-circuited stub is cascaded with the transmission line sections and an open-circuited stub as shown in **Figure 5**, where *P* ¼ 1, *Q* ¼ 1, *R* ¼ 3. Overall *S*<sup>21</sup> of the design-2 SOMI is given by

$$S\_{21} = \frac{(\mathbf{1} - \mathbf{z}^{-2})\mathbf{z}^{-3/2} \prod\_{r=1}^{3} (\mathbf{1} - \Gamma\_r^2)}{a\_o + a\_1 \mathbf{z}^{-1} + a\_2 \mathbf{z}^{-2} + a\_3 \mathbf{z}^{-3} + a\_4 \mathbf{z}^{-4} + a\_5 \mathbf{z}^{-5}} \tag{23}$$

The term 1 <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup> ð Þ is due to open-circuited stub, the term 1 � *<sup>z</sup>*�<sup>1</sup> ð Þ is due to short-circuited stub, and *z*�3*=*<sup>2</sup> is the delay factor of the transmission line section. These line elements are transmission lines of equal length with a length of *l* ¼ *λ*0*=*4 at an operating frequency of 12.5 GHz.

The next task be to achieve the optimum value of characteristic impedances of design-1 and design-2 SOMI line elements. The optimization algorithms are used to obtain these characteristic impedances of the line elements. In order to lower the cost function, the design of SOMI is considered as an approximation problem. The cost function differs between the response in magnitude of an ideal SOI and the SOMI designed. The cost function is formulated in the sense of least squares and can be expressed as

$$\text{CF} = \min \int\_0^\pi \left| E(o) \right|^2 \, d\nu = \min \int\_0^\pi \left| H(o) - \text{S}\_{21} \right|^2 \, d\nu \tag{24}$$

where *E*ð Þ¼ *ω* ð Þ *H*ð Þ� *ω S*<sup>21</sup> is the cost function, in which *H*ð Þ *ω* is the frequency

*Profile of design-1 SOMI (a) Magnitude response, (b) Magnitude error response, (c) Phase response, (d) Pole-*

In infinite impulse response (IIR) systems, the error surface is generally nonquadratic and multimodal with respect to the system parameters. Minimization of such error fitness function using derivative-based search algorithm is difficult. This is due to the fact that the derivative-based search algorithm may not converge to the global minima and get stuck in local minima. Moreover, IIR systems are associated

response of an ideal SOI.

**Figure 4.**

**135**

**3. Employed optimization methods**

*zero plot, (e) Convergence profile, and (f) Improvement bae-graph.*

*Analysis of Wideband Second-Order Microwave Integrators*

*DOI: http://dx.doi.org/10.5772/intechopen.94843*

**Figure 3.** *Configuration of the design-1 SOMI.*

## *Analysis of Wideband Second-Order Microwave Integrators DOI: http://dx.doi.org/10.5772/intechopen.94843*

**Figure 4.**

where, *P* denotes the number of open-circuited stubs, *Q* denotes the number of short-circuited stubs, *R* denotes the number of transmission line section, *N* is the total number of line elements *N* ¼ *P* þ *Q* þ *R* and *αο*, *α*1, *α*2, ..., *α<sup>N</sup>* Coefficients are functions of the reflection coefficient of each section of the transmission line and stub characteristic impedances (in terms of open-circuited stub coefficient k and short-circuited stub coefficient δ). In addition, to design the SOMI, serial transmis-

sion lines cascaded with shunt circuited stubs can be employed. Then the

*<sup>S</sup>*<sup>21</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup> ð Þ<sup>2</sup>

*<sup>S</sup>*<sup>21</sup> <sup>¼</sup> <sup>1</sup> � *<sup>z</sup>*�<sup>2</sup> ð Þ*z*�3*=*<sup>2</sup>

*R* ¼ 3. Overall *S*<sup>21</sup> of the design-2 SOMI is given by

at an operating frequency of 12.5 GHz.

*CF* ¼ *min*

ð*π* 0

methods are used.

*Innovations in Ultra-WideBand Technologies*

is given by

be expressed as

**Figure 3.**

**134**

*Configuration of the design-1 SOMI.*

frequency-domain response of the transfer function, which has an integrator characteristic, is thus obtained. In order to design a wideband SOMI, the following two

Design-I (SOMI using two open stubs and three transmission line sections): In this design, we may select two open-circuited stubs and three transmission line sections as shown in **Figure 3**, where *P* ¼ 2, *R* ¼ 3. Overall *S*<sup>21</sup> of the design-1 SOMI

> *z*�3*=*<sup>2</sup> Q<sup>3</sup>

Design-II (SOMI using one open stub, one short stub and three transmission line section): In this design, a short-circuited stub is cascaded with the transmission line sections and an open-circuited stub as shown in **Figure 5**, where *P* ¼ 1, *Q* ¼ 1,

The term 1 <sup>þ</sup> *<sup>z</sup>*�<sup>1</sup> ð Þ is due to open-circuited stub, the term 1 � *<sup>z</sup>*�<sup>1</sup> ð Þ is due to short-circuited stub, and *z*�3*=*<sup>2</sup> is the delay factor of the transmission line section. These line elements are transmission lines of equal length with a length of *l* ¼ *λ*0*=*4

The next task be to achieve the optimum value of characteristic impedances of design-1 and design-2 SOMI line elements. The optimization algorithms are used to obtain these characteristic impedances of the line elements. In order to lower the cost function, the design of SOMI is considered as an approximation problem. The cost function differs between the response in magnitude of an ideal SOI and the SOMI designed. The cost function is formulated in the sense of least squares and can

> ð*π* 0

j j *H*ð Þ� *ω S*<sup>21</sup>

<sup>2</sup> *dω* (24)

j j *<sup>E</sup>*ð Þ *<sup>ω</sup>* <sup>2</sup> *<sup>d</sup><sup>ω</sup>* <sup>¼</sup> *min*

Q<sup>3</sup>

*<sup>r</sup>*¼<sup>1</sup> <sup>1</sup> � <sup>Γ</sup><sup>2</sup> *r* � � *αο* <sup>þ</sup> *<sup>α</sup>*1*z*�<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*2*z*�<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*3*z*�<sup>3</sup> <sup>þ</sup> *<sup>α</sup>*4*z*�<sup>4</sup> <sup>þ</sup> *<sup>α</sup>*5*z*�<sup>5</sup> (22)

*<sup>r</sup>*¼<sup>1</sup> <sup>1</sup> � <sup>Γ</sup><sup>2</sup> *r* � � *αο* <sup>þ</sup> *<sup>α</sup>*1*z*�<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*2*z*�<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*3*z*�<sup>3</sup> <sup>þ</sup> *<sup>α</sup>*4*z*�<sup>4</sup> <sup>þ</sup> *<sup>α</sup>*5*z*�<sup>5</sup> (23)

*Profile of design-1 SOMI (a) Magnitude response, (b) Magnitude error response, (c) Phase response, (d) Polezero plot, (e) Convergence profile, and (f) Improvement bae-graph.*

where *E*ð Þ¼ *ω* ð Þ *H*ð Þ� *ω S*<sup>21</sup> is the cost function, in which *H*ð Þ *ω* is the frequency response of an ideal SOI.
