*3.2.1 Phase shifter theoretical calculation*

In parallel feed network, the main role of phase shifter is to create the phase difference of waves coming to the antennas in order to scan a beam or to reconfigure a beam shape. Three characteristics are required for this phase shifter. Firstly, it must be capable of shifting phase of waves in full 360° range in order to meet all of the demands of the phase difference in phased array antenna system. Secondly, continuously shifting makes it possible to create any phase, which enables steering main beam with high resolution. Finally, with a uniform amplitude and spacing linear array, when power divider equally splits power to output ports, a phase shifter with low insertion loss variation is very important. Without this property, our linear array should be treated as a non-uniform amplitude linear array, which is more complex in beam steering principle.

Depending on the required output, phase shifters are classified into analog phase shifter and digital phase shifter. Digital phase shifters with semiconductor components such as PIN diode and FET switch predefined states to provide predetermined phase shift. The phase shift, generated by this type, has high accuracy. However, structure of this type will become cumbersome to meet high-resolution demands. With high resolution, the number of predefined states increases, resulting in the expansion of the number of switching elements as well as size of controller for them. Meanwhile, analog phase shifter with the use of varactors or Schottky diodes can continuously change the phase shift. These diodes have capacitance depending on the bias voltage, for this reason, they can be used as electrically variable capacitor in tuned circuit accordingly. By shifting the phase continuously, this type can provide high-resolution phase shift without changing its hardware structure. However, less accuracy and relatively narrow bandwidth are drawbacks of this type. On implementing a microwave phase shifter, these disadvantages must be enhanced for better performance.

There exist four types of phase shifters, as shown in **Figure 9**. One of the digital phase shifter, the switched-line one, adopts delay lines and switching elements to generate time delay differences.

The phase shift depends only on transmission line length; therefore, it is very stable over time and temperature. A basic schematic of switched line phase shifter is shown in **Figure 9a**. The second type, switched network phase shifter (**Figure 9b**), is similar to switched line phase shifter but delay lines are replaced by networks composed of inductors and capacitors. The dimension of this type does not change as much as switched line phase shifter, besides this type is suitable for low frequency design. Loaded line phase shifters (**Figure 9c**) are loaded with a shunt reactance that is electrically shortened or lengthen by PIN diode or FET in order to get the desired phase shift. This type has advantages of simplicity and low insertion loss for phase shift less than 45°. However, for larger values of phase shift, high sensitivity is required in order to increase insertion loss. Therefore, this type is only suitable for phase shift less than 45°. Reflection type phase shifter comprises a 3-dB hybrid couple and two tunable loads, as shown in **Figure 9d**. By selecting the appropriate load, this type can shift more than 360° continuously and has low insertion loss, like in [24–26]. From requirement of phase shifter, it is obvious that reflection type phase shifter with full 360° and continuous phase shift is the most appropriate choice for this phased array antenna system.

suppressed at port 2. The amplitude of waves at ports 3 and 4 equal to <sup>1</sup>ffiffi

*Beamforming Phased Array Antenna toward Indoor Positioning Applications*

compared with one at port 1, respectively (**Figure 10a**).

½ �¼ *<sup>S</sup>* <sup>1</sup> ffiffi 2 p

when there is only input wave from port j.

To be more specific, we have:

from (8), waves at ports 3 and 4 are:

**107**

*V*<sup>þ</sup> <sup>3</sup> ¼ � *<sup>j</sup>*

*V*� 1 *V*� 2 *V*� 3 *V*� 4

(V) waves at different ports.

*DOI: http://dx.doi.org/10.5772/intechopen.93133*

incoming wave at port 1 and phase of waves at port 3 and 4 shift 90° and 180°

In mathematical model, 3-dB hybrid coupler is characterized by a scattering matrix that represents voltage relationship between incoming (V+) and reflected

Here, *Sij* is ratio of the reflected wave at port i to the incoming wave at port j

� � � � � *V*þ *<sup>k</sup>* ¼0,*k*6¼*j*

> <sup>þ</sup> �<sup>1</sup> ffiffi 2 p *V*<sup>þ</sup> 4 � �

> <sup>þ</sup> �<sup>1</sup> ffiffi 2 p *V*<sup>þ</sup> 4 � �

> <sup>þ</sup> �<sup>1</sup> ffiffi 2 p *V*<sup>þ</sup> 3 � �

> <sup>þ</sup> �*<sup>j</sup>* ffiffi 2 p *V*<sup>þ</sup> 2 � �

*Sij* � � <sup>¼</sup> *<sup>V</sup>*� *i V*þ *j*

> �*j* ffiffi 2 p *V*<sup>þ</sup> 3 � �

�*j* ffiffi 2 p *V*<sup>þ</sup> 3 � �

�*j* ffiffi 2 p *V*<sup>þ</sup> 1 � �

�1 ffiffi 2 p *V*<sup>þ</sup> 1 � �

*<sup>Γ</sup>* <sup>¼</sup> *<sup>V</sup>*�

ffiffi 2 p *V*<sup>þ</sup>

Considering the reflection coefficient (*Γ*) generated by the load, the reflection coefficient is determined from the load impedance and the source impedance.

*<sup>V</sup>*<sup>þ</sup> <sup>¼</sup> j j *<sup>Γ</sup> <sup>e</sup>*

From the two components constituting the RTPS, the 3-dB hybrid coupler and the reflective load mentioned above, the model of RTPS is shown as in **Figure 10b**, with port 1 as incoming port, port 2 as outgoing port, and two loads ZL3 and ZL4 at port 3 and 4, respectively. The phase shift of RTPS is the phase difference between outgoing wave at port 2 compared with incoming wave at port 1. Suppose there is only an incoming wave at port 1. When passing through the 3-dB hybrid coupler,

<sup>1</sup> ;*V*<sup>þ</sup>

Because of appearance of ZL3 load at port 3, there is wave reflection with reflection coefficient Γ3. Therefore, from (9) and (10), the reflection wave at port 3 is:

<sup>4</sup> ¼ � <sup>1</sup>

ffiffi 2 p *V*<sup>þ</sup>

0 0 �*j* �1 0 0 �1 �*j* �*j* �10 0 �1 �*j* 0 0

2 <sup>p</sup> one of

(6)

(7)

(8)

*V*� ½ �¼ ½ � *S V*<sup>þ</sup> ½ � (5)

*<sup>j</sup><sup>ϕ</sup>* (9)

<sup>1</sup> (10)

*Types of phase shifter: (a) switched line; (b) switched network; (c) loaded line; (d) reflection type [24–26].*

The Reflection Type Phase Shifter (RTPS), an electrically adjustable phase shift, is made up of a 3-dB hybrid coupler combined with some components such as capacitors, resistors, transmission line, and varactors to eliminate the reflected wave at input port and shift phase of wave at output port. The 3-dB hybrid coupler is a 4-port network with all ports matched to the reference impedance of 50 Ω. When stimulating at port 1, incoming wave transmits to port 3, port 4, and is

**Figure 10.** *(a) 3-dB hybrid coupler; (b) structure of RTPS.*

*Beamforming Phased Array Antenna toward Indoor Positioning Applications DOI: http://dx.doi.org/10.5772/intechopen.93133*

suppressed at port 2. The amplitude of waves at ports 3 and 4 equal to <sup>1</sup>ffiffi 2 <sup>p</sup> one of incoming wave at port 1 and phase of waves at port 3 and 4 shift 90° and 180° compared with one at port 1, respectively (**Figure 10a**).

In mathematical model, 3-dB hybrid coupler is characterized by a scattering matrix that represents voltage relationship between incoming (V+) and reflected (V) waves at different ports.

$$[V^{-}] = [\mathbb{S}][V^{+}] \tag{5}$$

$$[\mathbf{S}] = \frac{\mathbf{1}}{\sqrt{2}} \begin{bmatrix} \mathbf{0} & \mathbf{0} & -j & -\mathbf{1} \\ \mathbf{0} & \mathbf{0} & -\mathbf{1} & -j \\ -j & -\mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & -j & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{6}$$

Here, *Sij* is ratio of the reflected wave at port i to the incoming wave at port j when there is only input wave from port j.

$$\left[\mathbf{S}\_{\vec{\eta}}\right] = \frac{V\_i^-}{\mathbf{V}\_j^+}\bigg|\_{V\_k^+ = \mathbf{0}, k \neq j} \tag{7}$$

To be more specific, we have:

$$\begin{aligned} \begin{bmatrix} V\_1^-\\ V\_2^-\\ V\_3^-\\ V\_4^-\\ V\_4^- \end{bmatrix} = \begin{bmatrix} \left(\frac{-j}{\sqrt{2}}V\_3^+\right) + \left(\frac{-1}{\sqrt{2}}V\_4^+\right) \\\\ \left(\frac{-j}{\sqrt{2}}V\_3^+\right) + \left(\frac{-1}{\sqrt{2}}V\_4^+\right) \\\\ \left(\frac{-j}{\sqrt{2}}V\_1^+\right) + \left(\frac{-1}{\sqrt{2}}V\_3^+\right) \\\\ \left(\frac{-1}{\sqrt{2}}V\_1^+\right) + \left(\frac{-j}{\sqrt{2}}V\_2^+\right) \end{bmatrix} \end{aligned} \tag{8}$$

Considering the reflection coefficient (*Γ*) generated by the load, the reflection coefficient is determined from the load impedance and the source impedance.

$$
\Gamma = \frac{V^-}{V^+} = |\Gamma|e^{j\phi} \tag{9}
$$

From the two components constituting the RTPS, the 3-dB hybrid coupler and the reflective load mentioned above, the model of RTPS is shown as in **Figure 10b**, with port 1 as incoming port, port 2 as outgoing port, and two loads ZL3 and ZL4 at port 3 and 4, respectively. The phase shift of RTPS is the phase difference between outgoing wave at port 2 compared with incoming wave at port 1. Suppose there is only an incoming wave at port 1. When passing through the 3-dB hybrid coupler, from (8), waves at ports 3 and 4 are:

$$V\_3^+ = -\frac{j}{\sqrt{2}} V\_1^+; V\_4^+ = -\frac{1}{\sqrt{2}} V\_1^+ \tag{10}$$

Because of appearance of ZL3 load at port 3, there is wave reflection with reflection coefficient Γ3. Therefore, from (9) and (10), the reflection wave at port 3 is:

The Reflection Type Phase Shifter (RTPS), an electrically adjustable phase shift,

is made up of a 3-dB hybrid coupler combined with some components such as capacitors, resistors, transmission line, and varactors to eliminate the reflected wave at input port and shift phase of wave at output port. The 3-dB hybrid coupler is a 4-port network with all ports matched to the reference impedance of 50 Ω. When stimulating at port 1, incoming wave transmits to port 3, port 4, and is

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

*Types of phase shifter: (a) switched line; (b) switched network; (c) loaded line; (d) reflection type [24–26].*

**Figure 9.**

**Figure 10.**

**106**

*(a) 3-dB hybrid coupler; (b) structure of RTPS.*

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

$$V\_3^- = \Gamma\_3 V\_3^+ = -\frac{j}{\sqrt{2}} \Gamma\_3 V\_1^{+\cdot} \tag{11}$$

When putting reactive loads to ports 3 and 4, the reflection coefficient is:

2

<sup>2</sup> � <sup>2</sup> *arctan*

From (16), we found that the range of phase shift values ∠S21 entirely depends on the capacitance range of varactors, which limits the phase shift range. To be more specific, in order to achieve the 360° phase shift, theoretically, the capacitance of varactors must reach infinity, which is impossible. Therefore, RTPS cannot get the full 360° with only one varactor on reflection loads. To overcome the above problem, the reflection load structure shown in **Figure 12** is employed. This endows us to choose the appropriate capacitance range, thus selecting the suitable varactors

Based on the transmission line impedance theory [27], because *ZT*<sup>2</sup> is a quarter-

*<sup>Z</sup>*<sup>2</sup> <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup> *T*2 *Zd*

Similarly, *ZT1* is also a quarter-wavelength, so the equivalent impedance is:

wavelength transmission line, the equivalent impedance of *ZT*<sup>2</sup> and *D*<sup>2</sup> is:

*ZL* <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup> *T*1 1 *Zd* þ 1 *Z*2

The equivalent circuit becomes (**Figure 13**):

*(a) Reflection load of RTPS; (b) the equivalent circuit of reflection load.*

<sup>¼</sup> <sup>1</sup> *Z*2 <sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup> *L*

> XL Z0

*<sup>j</sup>*<sup>2</sup> *arctan* � *<sup>X</sup> Z*0

¼ *e*

ð Þ *jXL* � *Z*<sup>0</sup>

� � (18)

� � (20)

2

(19)

� � (17)

<sup>¼</sup> ð Þ *jXL* � *<sup>Z</sup>*<sup>0</sup>

*e*

<sup>∠</sup>S21 <sup>¼</sup> <sup>π</sup>

! � � <sup>2</sup>

*Beamforming Phased Array Antenna toward Indoor Positioning Applications*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Z*2 <sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup> *L*

q

*DOI: http://dx.doi.org/10.5772/intechopen.93133*

ð Þ *jXL* þ *Z*<sup>0</sup> ð Þ *jXL* � *Z*<sup>0</sup>

*jarctan* � *<sup>X</sup> Z*0

From (14) and (15), the phase shift of RTPS with reactive loads is:

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

> <sup>¼</sup> <sup>1</sup> *Z*2 <sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>2</sup> *L*

and its voltage as follows:

**Figure 12.**

**Figure 13.**

**109**

*The equivalent circuit of final RTPS.*

Similarly, we have the reflected wave at port 4:

$$V\_4^- = \Gamma\_4 V\_4^+ = -\frac{j}{\sqrt{2}} \Gamma\_4 V\_1^+ \tag{12}$$

The reflected waves at ports 3 and 4 can be also regarded as incoming waves from ports 3, 4 transmitting to ports 1 and 2. We set *V*0þ <sup>3</sup> and *V*0þ <sup>4</sup> as incoming wave at ports 3 and 4, respectively. We have:

$$V\_3^{\prime +} = V\_3^{-} = -\frac{j}{\sqrt{2}} \Gamma\_3 V\_1^{+} \tag{13}$$

$$V\_4^{\prime +} = V\_4^- = -\frac{j}{\sqrt{2}} \Gamma\_4 V\_1^+ \tag{14}$$

These two waves transmit to ports 1 and 2 through 3 dB hybrid coupler, and outgoing waves from ports 1 and 2 are calculated from (6), (11), and (12) as follows:

$$V\_1^- = -\frac{j}{\sqrt{2}}V\_3^{\prime +} - \frac{1}{\sqrt{2}}V\_4^{\prime +} = -\frac{1}{2}\left(\Gamma\_3 V\_1^+ - \Gamma\_4 V\_1^+\right)$$

$$V\_2^- = -\frac{1}{\sqrt{2}}V\_3^{\prime +} - \frac{j}{\sqrt{2}}V\_4^{\prime +} = \frac{j}{2}\left(\Gamma\_3 V\_1^+ + \Gamma\_4 V\_1^+\right) \tag{15}$$

For the phase shifter, we wish that there is no reflected wave at port 1. Therefore, *Γ*<sup>3</sup> ¼ *Γ*<sup>4</sup> enables us to eliminate the reflected wave at port 1. From (9) and (13), choosing *ZL*<sup>3</sup> ¼ *ZL*<sup>4</sup> makes *Γ*<sup>3</sup> ¼ *Γ*<sup>4</sup> ¼ *Γ* and outgoing wave at port 2 as *V*� <sup>2</sup> ¼ *jΓV*<sup>þ</sup> <sup>1</sup> . Finally, the forward voltage gain *S*<sup>21</sup> of RTPS will be as follow:

$$\mathbf{S\_{21}} = \frac{V\_2^-}{V\_1^+} = j\boldsymbol{\varGamma} = j|\boldsymbol{\varGamma}|e^{j\phi} = |\boldsymbol{\varGamma}|e^{j\left(\frac{\pi}{2} + \phi\right)}$$

$$\boldsymbol{\varDelta S\_{21}} = \frac{\pi}{2} + \phi \tag{16}$$

#### *3.2.2 Design of reflection type phase shifter*

The reflection type phase shifter (RTPS) is a phase shifter, empowering continuous change of the phase of wave without the need to change hardware structure, based on the change in capacitance of varactors. **Figure 11** shows the principle diagram of the RTPS, which consists of a 3-dB hybrid coupled and reflection loads *XL* putting at ports 3 and 4. Waves enter port 1 and go out port 2.

**Figure 11.** *Schematic diagram of RTPS.*

*Beamforming Phased Array Antenna toward Indoor Positioning Applications DOI: http://dx.doi.org/10.5772/intechopen.93133*

When putting reactive loads to ports 3 and 4, the reflection coefficient is:

$$\begin{split} \boldsymbol{\Gamma} &= \frac{j\mathbf{X}\_L - \mathbf{Z}\_0}{j\mathbf{X}\_L + \mathbf{Z}\_0} = \frac{(j\mathbf{X}\_L - \mathbf{Z}\_0)^2}{(j\mathbf{X}\_L + \mathbf{Z}\_0)(j\mathbf{X}\_L - \mathbf{Z}\_0)} = \frac{\mathbf{1}}{\mathbf{Z}\_0^2 + \mathbf{X}\_L^2} (j\mathbf{X}\_L - \mathbf{Z}\_0)^2 \\ &= \frac{1}{\mathbf{Z}\_0^2 + \mathbf{X}\_L^2} \left( \sqrt{\mathbf{Z}\_0^2 + \mathbf{X}\_L^2} e^{j\arctan\left(\frac{-\mathbf{Z}}{Z\_0}\right)} \right)^2 = e^{j2\arctan\left(\frac{-\mathbf{Z}}{Z\_0}\right)} \end{split} \tag{17}$$

From (14) and (15), the phase shift of RTPS with reactive loads is:

$$
\Delta \mathbf{S}\_{21} = \frac{\pi}{2} - 2 \arctan \left( \frac{\mathbf{X}\_{\rm L}}{\mathbf{Z}\_{0}} \right) \tag{18}
$$

From (16), we found that the range of phase shift values ∠S21 entirely depends on the capacitance range of varactors, which limits the phase shift range. To be more specific, in order to achieve the 360° phase shift, theoretically, the capacitance of varactors must reach infinity, which is impossible. Therefore, RTPS cannot get the full 360° with only one varactor on reflection loads. To overcome the above problem, the reflection load structure shown in **Figure 12** is employed. This endows us to choose the appropriate capacitance range, thus selecting the suitable varactors and its voltage as follows:

Based on the transmission line impedance theory [27], because *ZT*<sup>2</sup> is a quarterwavelength transmission line, the equivalent impedance of *ZT*<sup>2</sup> and *D*<sup>2</sup> is:

$$Z\_2 = \frac{Z\_{T2}^2}{Z\_d} \tag{19}$$

The equivalent circuit becomes (**Figure 13**): Similarly, *ZT1* is also a quarter-wavelength, so the equivalent impedance is:

**Figure 12.** *(a) Reflection load of RTPS; (b) the equivalent circuit of reflection load.*

**Figure 13.** *The equivalent circuit of final RTPS.*

*V*�

*V*�

from ports 3, 4 transmitting to ports 1 and 2. We set *V*0þ

*V*0þ <sup>3</sup> ¼ *V*�

*V*0þ <sup>4</sup> ¼ *V*�

> <sup>3</sup> � <sup>1</sup> ffiffi 2 p *V*0þ

<sup>3</sup> � *<sup>j</sup>* ffiffi 2 p *V*0þ

<sup>1</sup> . Finally, the forward voltage gain *S*<sup>21</sup> of RTPS will be as follow:

¼ *jΓ* ¼ *j*j j *Γ e*

<sup>∠</sup>*S*<sup>21</sup> <sup>¼</sup> *<sup>π</sup>*

The reflection type phase shifter (RTPS) is a phase shifter, empowering continuous change of the phase of wave without the need to change hardware structure, based on the change in capacitance of varactors. **Figure 11** shows the principle diagram of the RTPS, which consists of a 3-dB hybrid coupled and reflection loads

ffiffi 2 p *V*0þ

> ffiffi 2 p *V*0þ

*<sup>S</sup>*<sup>21</sup> <sup>¼</sup> *<sup>V</sup>*� 2 *V*þ 1

*XL* putting at ports 3 and 4. Waves enter port 1 and go out port 2.

at ports 3 and 4, respectively. We have:

*V*� <sup>1</sup> ¼ � *<sup>j</sup>*

> *V*� <sup>2</sup> ¼ � <sup>1</sup>

*3.2.2 Design of reflection type phase shifter*

*jΓV*<sup>þ</sup>

**Figure 11.**

**108**

*Schematic diagram of RTPS.*

Similarly, we have the reflected wave at port 4:

<sup>3</sup> ¼ *Γ*3*V*<sup>þ</sup>

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

<sup>4</sup> ¼ *Γ*4*V*<sup>þ</sup>

<sup>3</sup> ¼ � *<sup>j</sup>*

<sup>4</sup> ¼ � *<sup>j</sup>*

The reflected waves at ports 3 and 4 can be also regarded as incoming waves

<sup>3</sup> ¼ � *<sup>j</sup>*

<sup>4</sup> ¼ � *<sup>j</sup>*

These two waves transmit to ports 1 and 2 through 3 dB hybrid coupler, and outgoing waves from ports 1 and 2 are calculated from (6), (11), and (12) as follows:

ffiffi 2 <sup>p</sup> *<sup>Γ</sup>*3*V*þ*:*

ffiffi 2 p *Γ*4*V*<sup>þ</sup>

ffiffi 2 p *Γ*3*V*<sup>þ</sup>

ffiffi 2 p *Γ*4*V*<sup>þ</sup>

<sup>4</sup> ¼ � <sup>1</sup> 2

> <sup>4</sup> <sup>¼</sup> *<sup>j</sup>* 2

For the phase shifter, we wish that there is no reflected wave at port 1. Therefore, *Γ*<sup>3</sup> ¼ *Γ*<sup>4</sup> enables us to eliminate the reflected wave at port 1. From (9) and (13), choosing *ZL*<sup>3</sup> ¼ *ZL*<sup>4</sup> makes *Γ*<sup>3</sup> ¼ *Γ*<sup>4</sup> ¼ *Γ* and outgoing wave at port 2 as *V*�

*Γ*3*V*<sup>þ</sup>

*Γ*3*V*<sup>þ</sup>

*<sup>j</sup><sup>ϕ</sup>* <sup>¼</sup> j j *<sup>Γ</sup> <sup>e</sup> <sup>j</sup> <sup>π</sup>* ð Þ <sup>2</sup>þ*<sup>ϕ</sup>*

<sup>1</sup> � *Γ*4*V*<sup>þ</sup> 1

<sup>1</sup> þ *Γ*4*V*<sup>þ</sup> 1 � � (15)

<sup>2</sup> <sup>þ</sup> *<sup>ϕ</sup>* (16)

� �

<sup>1</sup> (11)

<sup>1</sup> (12)

<sup>1</sup> (13)

<sup>1</sup> (14)

<sup>4</sup> as incoming wave

<sup>2</sup> ¼

<sup>3</sup> and *V*0þ

Substitute (17) to (18):

$$Z\_L = Z\_{T1}^2 \left(\frac{1}{Z\_d} + \frac{1}{Z\_2}\right) = Z\_{T1}^2 \left(\frac{1}{Z\_d} + \frac{Z\_d}{Z\_{T2}^2}\right) = \frac{Z\_{T1}^2}{Z\_{T2}^2} \frac{Z\_{T2}^2 + Z\_d^2}{Z\_d} \tag{21}$$

On the other hand, the impedance of the capacitance diodes is:

$$\mathbf{Z}\_d = j\alpha \mathbf{L}\_t + \frac{\mathbf{1}}{j\alpha \mathbf{C}\_d} = j\left(\alpha \mathbf{L}\_t - \frac{\mathbf{1}}{\alpha \mathbf{C}\_d}\right) = j\mathbf{X}\_d \tag{22}$$

At 2.45 GHz, for impedance matching, the varactor diode SMV1247, package type SC-79 of Skyworks is chosen with *Ls* = 0.7nH. To simplify the voltage controller of the varactors, the model capacitance range of SMV1247 from 8.5 pF to 0.7 pF is selected corresponding to the voltage range from 0 V to 5 V. From (23), *ZT*<sup>2</sup> = 15 Ω. For the impedance of *ZT*1, besides calculating the entire phase shift, the change of phase shift needs to be taken into account. It is desired that the relationship between the control voltage and the phase shift becomes as close to linear as possible, so that controlling phase shift with small resolution is easier. Thus, it is

*Beamforming Phased Array Antenna toward Indoor Positioning Applications*

and ∠*S*<sup>21</sup> is the phase shift. As for SMV1247, the relationship between bias voltage

<sup>1</sup> <sup>þ</sup> *VR VJ*

where CJ0 = 8.47 pF, VJ = 80 V, M = 70, CP = 0.54 pF are package parameters

*Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

*<sup>T</sup>*2*Z*0*MCJ*<sup>0</sup> *X*<sup>2</sup>

*Z*4 *T*2*Z*<sup>2</sup> 0*X*<sup>2</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>Z</sup>*<sup>4</sup>

where ω, VJ, Cd, ZT2, M, CJ0, Z0, Ls, CP are known, *ZT*<sup>1</sup> is unknown, and *VR* and

From expression (27), to see the effect of *ZT*<sup>1</sup> on the phase shift, different values of

*ZT*<sup>1</sup> is roughly experimented with different values from 10 to 80 *Ω* with the spacing of

Because the value of *ZT*<sup>1</sup> is found around the value of 20 Ω, the value of *ZT*<sup>1</sup> is examined at a narrower range from 15 to 22 Ω with the spacing of 1 Ω (**Figure 15**). Among the surveyed values of *ZT*1, the value of 19 Ω gives the best result and that is the best-chosen value. Thus, the RTPS will be implemented with the load topology, where the impedance of the two microstrip circuits is *ZT*<sup>1</sup> = 19 Ω and *ZT*<sup>2</sup> = 15 Ω, respectively. The varactors are Skyworks SMV1247 with capacitance from

As mentioned above, varactors are the elements in microwave circuit, and their capacitance value is controlled by a DC bias from controllers. With respect to the

10 Ω. From results in **Figure 14**, the value of *ZT*<sup>1</sup> can be found around 20 Ω.

*Z*2 *<sup>T</sup>*2*XdZ*<sup>0</sup>

*<sup>d</sup>* � *<sup>Z</sup>*<sup>2</sup> *T*2 � �

*<sup>d</sup>* <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *T*2 � �

*<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

*<sup>d</sup>* � *<sup>Z</sup>*<sup>2</sup> *T*2

*dVR* with smallest maximal point. Firstly,

� �<sup>2</sup> h i (29)

!

Substitute (25) to (22), the relational expression between ∠*S*<sup>21</sup> and VR is:

<sup>2</sup> � <sup>2</sup> *arctan*

*ωCd*

*T*1*Z*<sup>2</sup>

*Cd* <sup>¼</sup> *CJ*<sup>0</sup>

*dVR* where *VR* is the bias voltage of varactors

� �*<sup>M</sup>* <sup>þ</sup> *CP* (27)

(28)

important to examine the expression *<sup>d</sup>*∠*S*<sup>21</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93133*

and capacitive value as follows:

from varactor SMV1247's datasheet.

<sup>∠</sup>*S*<sup>21</sup> <sup>¼</sup> *<sup>π</sup>*

8

>>>>>>>>><

>>>>>>>>>:

Derivative ∠*S*<sup>21</sup> in VR:

∠*S*<sup>21</sup> are variables.

**111**

*d*∠*S*<sup>21</sup> *dVR*

*Xd* <sup>¼</sup> *<sup>ω</sup>Ls* � <sup>1</sup>

*Cd* <sup>¼</sup> *CJ*<sup>0</sup>

<sup>¼</sup> <sup>2</sup>*Z*<sup>2</sup>

8.5 to 0.7 pF corresponding to bias voltage from 0 to 5 V.

*3.2.3 Direct current (DC) feed and DC block*

*<sup>d</sup>* <sup>1</sup> <sup>þ</sup> *VR VJ* � �*<sup>M</sup>*þ<sup>1</sup>

*ωVJC*<sup>2</sup>

*ZT*<sup>1</sup> are investigated by MATLAB to find the *<sup>d</sup>*∠*S*<sup>21</sup>

<sup>1</sup> <sup>þ</sup> *VR VJ* � �*<sup>M</sup>* <sup>þ</sup> *CP*

Substitute (20) to (19):

$$\mathbf{Z}\_{L} = \mathbf{j}\mathbf{X}\_{L} = \frac{\mathbf{Z}\_{T1}^{2}(\mathbf{Z}\_{T2}^{2} - \mathbf{X}\_{d}^{2})}{\mathbf{Z}\_{T2}^{2}\mathbf{j}\mathbf{X}\_{d}} = \mathbf{j}\frac{\mathbf{Z}\_{T1}^{2}(\mathbf{X}\_{d}^{2} - \mathbf{Z}\_{T2}^{2})}{\mathbf{Z}\_{T2}^{2}\mathbf{X}\_{d}}\tag{23}$$

Substitute (21) to (16):

$$
\Delta S\_{21} = \frac{\pi}{2} - 2 \arctan \left( \frac{\mathbf{X}\_L}{Z\_0} \right) = \frac{\pi}{2} - 2 \arctan \left( \frac{Z\_{T1}^2 (\mathbf{X}\_d^2 - Z\_{T2}^2)}{Z\_{T2}^2 X\_d Z\_0} \right) \tag{24}
$$

In order to get 360° phase shift, we have:

$$\begin{aligned} \left| \mathcal{Z}\_{21} \right|\_{X\_{d\text{min}}} &= \left| \mathcal{Z}\_{21} \right| X\_{d\text{max}} \\ \Leftrightarrow \frac{\pi}{2} - 2 \arctan \left( \frac{Z\_{T1}^2 (X\_{d\text{min}}^2 - Z\_{T2}^2)}{Z\_{T2}^2 X\_{d\text{min}} Z\_0} \right) &= \frac{\pi}{2} - 2 \arctan \left( \frac{Z\_{T1}^2 (X\_{d\text{max}}^2 - Z\_{T2}^2)}{Z\_{T2}^2 X\_{d\text{max}} Z\_0} \right) \\ \Leftrightarrow \frac{Z\_{T1}^2 (X\_{d\text{min}}^2 - Z\_{T2}^2)}{Z\_{T2}^2 X\_{d\text{min}} Z\_0} &= \frac{Z\_{T1}^2 (X\_{d\text{max}}^2 - Z\_{T2}^2)}{Z\_{T2}^2 X\_{d\text{max}} Z\_0} \\ \Leftrightarrow \mathcal{X}\_{d\text{min}} &- \frac{Z\_{T2}^2}{X\_{d\text{min}}} = X\_{d\text{max}} - \frac{Z\_{T2}^2}{X\_{d\text{max}}} \\ \Leftrightarrow (\mathcal{X}\_{d\text{min}} - \mathcal{X}\_{d\text{max}}) \left( 1 + \frac{Z\_{T2}^2}{X\_{d\text{min}} X\_{d\text{max}}} \right) &= 0 \end{aligned}$$

Since *Xdmin* 6¼ *Xdmax*, we have:

$$Z\_{T2}^2 = -X\_{dmin} X\_{dmax} \tag{25}$$

Thus, by calculation of *ZT*<sup>2</sup> satisfying (23), we will have 360° phase shift. From (23), we have: *XdminXdmax* <0

$$\Leftrightarrow \left( \alpha L\_s - \frac{1}{\alpha \mathcal{C}\_{dmin}} \right) \left( \alpha L\_s - \frac{1}{\alpha \mathcal{C}\_{dmax}} \right) < 0$$

$$\Leftrightarrow \begin{cases} \mathcal{C}\_{dmin} < \frac{1}{\alpha^2 L\_s} \\\mathcal{C}\_{dmax} > \frac{1}{\alpha^2 L\_s} \end{cases}$$

$$\Rightarrow \frac{1}{\alpha^2 L\_s} \in \left[ \mathcal{C}\_{dmin}; \mathcal{C}\_{dmax} \right] \tag{26}$$

*Beamforming Phased Array Antenna toward Indoor Positioning Applications DOI: http://dx.doi.org/10.5772/intechopen.93133*

At 2.45 GHz, for impedance matching, the varactor diode SMV1247, package type SC-79 of Skyworks is chosen with *Ls* = 0.7nH. To simplify the voltage controller of the varactors, the model capacitance range of SMV1247 from 8.5 pF to 0.7 pF is selected corresponding to the voltage range from 0 V to 5 V. From (23), *ZT*<sup>2</sup> = 15 Ω.

For the impedance of *ZT*1, besides calculating the entire phase shift, the change of phase shift needs to be taken into account. It is desired that the relationship between the control voltage and the phase shift becomes as close to linear as possible, so that controlling phase shift with small resolution is easier. Thus, it is important to examine the expression *<sup>d</sup>*∠*S*<sup>21</sup> *dVR* where *VR* is the bias voltage of varactors and ∠*S*<sup>21</sup> is the phase shift. As for SMV1247, the relationship between bias voltage and capacitive value as follows:

$$\mathbf{C}\_d = \frac{\mathbf{C}\_{l0}}{\left(\mathbf{1} + \frac{V\_R}{V\_l}\right)^M} + \mathbf{C}\_P \tag{27}$$

where CJ0 = 8.47 pF, VJ = 80 V, M = 70, CP = 0.54 pF are package parameters from varactor SMV1247's datasheet.

Substitute (25) to (22), the relational expression between ∠*S*<sup>21</sup> and VR is:

$$\begin{cases} \mathcal{L}S\_{21} = \frac{\pi}{2} - 2 \arctan\left(\frac{Z\_{T1}^2 \left(X\_d^2 - Z\_{T2}^2\right)}{Z\_{T2}^2 X\_d Z\_0}\right) \\\\ X\_d = aL\_s - \frac{1}{a\alpha \mathcal{C}\_d} \\\ \mathcal{C}\_d = \frac{\mathcal{C}\_{f0}}{\left(1 + \frac{V\_P}{V\_I}\right)^M} + \mathcal{C}\_P \end{cases} \tag{28}$$

Derivative ∠*S*<sup>21</sup> in VR:

Substitute (17) to (18):

Substitute (20) to (19):

Substitute (21) to (16):

<sup>∠</sup>*S*<sup>21</sup> <sup>¼</sup> *<sup>π</sup>*

<sup>2</sup> � <sup>2</sup> *arctan*

<sup>⇔</sup> *<sup>π</sup>*

**110**

*ZL* <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup> *T*1 1 *Zd* þ 1 *Z*2 � � <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup> *T*1 1 *Zd* þ *Zd Z*2 *T*2

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

On the other hand, the impedance of the capacitance diodes is:

1 *jωCd*

*<sup>T</sup>*<sup>1</sup> *Z*<sup>2</sup>

*XL Z*0 � �

∠*S*21j

*dmin* � *<sup>Z</sup>*<sup>2</sup> *T*2

> *dmin* � *<sup>Z</sup>*<sup>2</sup> *T*2

> > *T*2 *Xdmin*

Thus, by calculation of *ZT*<sup>2</sup> satisfying (23), we will have 360° phase shift.

*Cdmin* <

*Cdmax* >

� �

*<sup>T</sup>*2*XdminZ*<sup>0</sup>

<sup>⇔</sup> *Xdmin* � *<sup>Z</sup>*<sup>2</sup>

⇔ ð Þ *Xdmin* � *Xdmax* 1 þ

*Z*2

� �

⇔

) 1 *ω*<sup>2</sup>*Ls*

*ωCdmin*

8 >><

>>:

<sup>⇔</sup> *<sup>ω</sup>Ls* � <sup>1</sup>

� �

*<sup>T</sup>*2*XdminZ*<sup>0</sup>

!

*Z*2 *<sup>T</sup>*2*jXd*

¼ *π*

*<sup>T</sup>*<sup>2</sup> � *<sup>X</sup>*<sup>2</sup> *d* � �

<sup>2</sup> � <sup>2</sup> *arctan*

*Xdmin* <sup>¼</sup> <sup>∠</sup>*S*21j*Xdmax*

¼ *π*

¼ *Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

<sup>2</sup> � <sup>2</sup> *arctan*

*Z*2

<sup>¼</sup> *Xdmax* � *<sup>Z</sup>*<sup>2</sup>

*Z*2 *T*2 *XdminXdmax* � �

*<sup>ω</sup>Ls* � <sup>1</sup>

1 *ω*<sup>2</sup>*Ls*

1 *ω*<sup>2</sup>*Ls*

� �

*ωCdmax*

*Zd* ¼ *jωLs* þ

*ZL* <sup>¼</sup> *jXL* <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup>

<sup>2</sup> � <sup>2</sup> *arctan*

In order to get 360° phase shift, we have:

*Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

⇔ *Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

Since *Xdmin* 6¼ *Xdmax*, we have:

From (23), we have: *XdminXdmax* <0

*Z*2

*Z*2

� �

<sup>¼</sup> *<sup>j</sup> <sup>ω</sup>Ls* � <sup>1</sup>

� �

¼ *j Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

*ωCd*

¼ *Z*2 *T*1 *Z*2 *T*2 *Z*2 *<sup>T</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *d Zd*

*<sup>d</sup>* � *<sup>Z</sup>*<sup>2</sup> *T*2 � �

> *<sup>d</sup>* � *<sup>Z</sup>*<sup>2</sup> *T*2 � �

!

*Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

*T*2

¼ 0

*<sup>T</sup>*<sup>2</sup> ¼ �*XdminXdmax* (25)

<0

∈ ½ � *Cdmin*;*Cdmax* (26)

*Z*2

*dmax* � *<sup>Z</sup>*<sup>2</sup>

� �

*<sup>T</sup>*2*XdmaxZ*<sup>0</sup>

!

*T*2

*Z*2 *<sup>T</sup>*2*Xd*

*Z*2 *<sup>T</sup>*2*XdZ*<sup>0</sup>

*dmax* � *<sup>Z</sup>*<sup>2</sup>

*T*2 *Xdmax*

� �

*<sup>T</sup>*2*XdmaxZ*<sup>0</sup>

*Z*2 *<sup>T</sup>*<sup>1</sup> *X*<sup>2</sup>

¼ *jXd* (22)

(21)

(23)

(24)

$$\frac{d\angle S\_{21}}{dV\_R} = \frac{2Z\_{T1}^2 Z\_{T2}^2 Z\_0 \text{MC}\_{\text{f0}} \left(X\_d^2 + Z\_{T2}^2\right)}{\alpha V\_f C\_d^2 \left(1 + \frac{V\_R}{V\_I}\right)^{M+1} \left[Z\_{T2}^4 Z\_0^2 X\_d^2 + Z\_{T1}^4 \left(X\_d^2 - Z\_{T2}^2\right)^2\right]} \tag{29}$$

where ω, VJ, Cd, ZT2, M, CJ0, Z0, Ls, CP are known, *ZT*<sup>1</sup> is unknown, and *VR* and ∠*S*<sup>21</sup> are variables.

From expression (27), to see the effect of *ZT*<sup>1</sup> on the phase shift, different values of *ZT*<sup>1</sup> are investigated by MATLAB to find the *<sup>d</sup>*∠*S*<sup>21</sup> *dVR* with smallest maximal point. Firstly, *ZT*<sup>1</sup> is roughly experimented with different values from 10 to 80 *Ω* with the spacing of 10 Ω. From results in **Figure 14**, the value of *ZT*<sup>1</sup> can be found around 20 Ω.

Because the value of *ZT*<sup>1</sup> is found around the value of 20 Ω, the value of *ZT*<sup>1</sup> is examined at a narrower range from 15 to 22 Ω with the spacing of 1 Ω (**Figure 15**).

Among the surveyed values of *ZT*1, the value of 19 Ω gives the best result and that is the best-chosen value. Thus, the RTPS will be implemented with the load topology, where the impedance of the two microstrip circuits is *ZT*<sup>1</sup> = 19 Ω and *ZT*<sup>2</sup> = 15 Ω, respectively. The varactors are Skyworks SMV1247 with capacitance from 8.5 to 0.7 pF corresponding to bias voltage from 0 to 5 V.

#### *3.2.3 Direct current (DC) feed and DC block*

As mentioned above, varactors are the elements in microwave circuit, and their capacitance value is controlled by a DC bias from controllers. With respect to the

**Figure 14.** *The results from the first survey of* ZT1 *value: (a) phase shift, (b)* <sup>d</sup>∠S21 dVR *.*

principle of microwave circuit, the varactors directly connect to transmission lines. From controller perspective, the DC bias is put to at two polar, anode and cathode, of varactors. Therefore, it is easily found that the microwave circuit and the controller are physically connected to each other, which leads to that high frequency waves can propagate to controller, and DC current can go to microwave circuit. When propagating to controller, controller can be considered as a new part in microwave circuit, so characteristics of our design might change unpredictably. For microwave circuit with reference impedance of 50 Ω, the maximal DC bias for varactors is 20 V, is corresponding to the power of approximately +36 dBm. This

value may exceed the limitation of microwave equipment and instrumentation and can damage them. From these problems, the demand for isolation between microwave and DC current appears. The common solution is to use DC Feed and DC

dVR *.*

*The results from the second survey of* ZT1 *value: (a) phase shift; (b)* <sup>d</sup>∠S21

*Beamforming Phased Array Antenna toward Indoor Positioning Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.93133*

DC Block is usually a capacitor, which prevents DC current, but permits microwave circuit to pass easily. The impedance of DC Block must be very low (<5 Ω) compared with reference impedance (50 Ω) at operating frequencies (2.45 GHz). To deploy capacitors on microstrip, gap and microstrip interdigital capacitors may be options. However, with impedance less than 5 Ω, the capacitance must be larger

Block components.

**Figure 15.**

**113**

*Beamforming Phased Array Antenna toward Indoor Positioning Applications DOI: http://dx.doi.org/10.5772/intechopen.93133*

value may exceed the limitation of microwave equipment and instrumentation and can damage them. From these problems, the demand for isolation between microwave and DC current appears. The common solution is to use DC Feed and DC Block components.

DC Block is usually a capacitor, which prevents DC current, but permits microwave circuit to pass easily. The impedance of DC Block must be very low (<5 Ω) compared with reference impedance (50 Ω) at operating frequencies (2.45 GHz). To deploy capacitors on microstrip, gap and microstrip interdigital capacitors may be options. However, with impedance less than 5 Ω, the capacitance must be larger

principle of microwave circuit, the varactors directly connect to transmission lines. From controller perspective, the DC bias is put to at two polar, anode and cathode, of varactors. Therefore, it is easily found that the microwave circuit and the controller are physically connected to each other, which leads to that high frequency waves can propagate to controller, and DC current can go to microwave circuit. When propagating to controller, controller can be considered as a new part in microwave circuit, so characteristics of our design might change unpredictably. For microwave circuit with reference impedance of 50 Ω, the maximal DC bias for varactors is 20 V, is corresponding to the power of approximately +36 dBm. This

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

dVR *.*

*The results from the first survey of* ZT1 *value: (a) phase shift, (b)* <sup>d</sup>∠S21

**Figure 14.**

**112**

than 13.2 pF at 2.45GHz. That capacitance value is hard to be met by gaps or microstrip interdigital capacitors. For capacitor components, in practice, at microwave frequency, a capacitor component is equivalent to a series of inductor, resistor, and capacitor, so the impedance of DC Block changes in frequency. Above a certain frequency threshold, that DC Block will act as an inductor. Thus, DC Blocks are usually capacitors with its self-resonant frequency near operating frequency, 2.45 GHz. Finally, the DC Block is VJ0603D8R2CXP capacitor of Vishay/Vitramon with impedance about 3.8 Ω at 2.45 GHz, as shown in **Figure 16**.

parameters calculated for the Roger4003c substrate radial stubs are: the outer radius

In phase array antennas, in order to be able to steer the antenna's main lobe in different directions, the phase-shift system must be placed in front of the antenna elements to generate the phase difference of the wave to each antenna so as to create the angle of beam. The phase shift entirely depends on the capacitance value of the varactors whereas every varactor always has its own C – V characteristic curve, representing the relationship between the input applied voltage and output capacitance. In other words, the phase shift is controlled by the voltage applied to the anode and cathode pins of varactors. For SMV1247 varactor, the voltage range of 0–5 V is applied. Thus, the controller is required to generate adjustable voltage in the range of 0–5 V, providing eight different voltages for eight RTPS and each voltage channel must meet the total power of four varactors in a RTPS.

To generate a DC voltage controlled by the microcontroller, popular methods are to modulate pulse width or control open angle of power semiconductor components such as Triac or Thysistor to convert AC to DC. These two methods are commonly used in power circuits, so we usually only care about the average power and voltage in a cycle, but in essence, the voltage generated is not flat over time but it is ripple. Obviously, these two methods will not be able to generate the control voltage for the varactors, which constantly changes the capacitance and results in undesired phase shifts. From the datasheet of SMV1247 varactor, the reverse current is very small, just a few nA to μA, so it is possible to use Digital to Analog Converter (DAC) units. Output current of these elements is about 20 mA. It is enough to satisfy the reverse currents of four varactors in a RTPS. In addition, the DACs have a variety of resolution options ranging from 8 bits to 24 bits. This facilitates flexible adjustment of the RTPS resolution. Therefore, the controller will be designed based on the eight DAC elements to generate bias voltages for eight RTPS. The controller model is described in **Figure 17**. The main components are DACs, and in order to increase the accuracy of the output voltage, the DACs are powered by a different source with a higher accuracy than the source for the microcontroller or can be adjusted for

backup case when higher voltage range of varactors is demanded.

Because the positioning system is executed indoor, the WiFi or Wireless LAN should be the most suitable protocol to use in order to radiate power to indoor mobile devices; therefore the 2.4 GHz to 2.484 bands is chosen as the operating frequency. On the other hand, in order to communicate with the mobile devices for localizing the position of the object, the angel of main beam of phased array antenna should change from 45 to 45° to scan the desired object, hence antenna element must have half power beam width [15] greater than 90°. These requirements demonstrate that the radio signal strength comparison completely depends on the array factor. Omnidirectional antennas can satisfy both requirements; however, some omnidirectional antennas such as dipoles, … [15] still have some undesired effects, that is, radiating to the back side of the antenna array, which results in some high side back lobes due to the reflection. The microstrip patch antenna can both meet the above condition and have small back lobes. In our array, the microstrip patch antenna was designed to operate at Wi-Fi band frequency, 2.4 to 2.484 GHz with an input impedance of 50 Ω using a low cost FR4 substrate with dielectric constant ε = 4.3, loss tangent tanδ = 0.02 and thickness h = 62 mil = 1.58 mm. The antenna

*3.2.5 Design of antenna element*

**115**

*Ro* = 540.621 mil and the inner radius *Ri* = 56.688 mil and angle α = 30°.

*Beamforming Phased Array Antenna toward Indoor Positioning Applications*

*3.2.4 Design of controller for reflection type phase shifter*

*DOI: http://dx.doi.org/10.5772/intechopen.93133*

In contrast, the DC feed is usually an inductive element, which blocks high frequency waves and passes DC current. The impedance of DC Feed element must be very high compared with the reference impedance (50 Ω) at operating frequency, 2.45 GHz. Similar to DC Block, the DC Feed may be an inductance element with a self-resonant frequency near 2.45 GHz operating frequency. However, in practice, due to cost constraint and the availability of inductors, these inductors are replaced by another option, taking advantage of some special case in length of transmission lines. To minimize the whole structure, the quarter-wavelength transmission line used is shorted at the end, of which the equivalent impedance is positive infinity at operating frequency. In other words, a quarter-wavelength transmission line with shorted at the end can be treated as an open circuit at the beginning of that transmission line. Consequently, the DC bias can be replaced at the end of quarter-wavelength transmission line, so that high frequency waves do not reach the DC circuit while the DC currents can flow directly to varactors.

There exist two ways to short a circuit. The first way is to use via holes to create physical connection from electric lines to the ground; however, this method also shorts the DC bias. The other way is to employ open quarter-wavelength straight stubs or radial stubs. These two kinds of stubs do not physically connect to the ground, so they do not short the DC bias. In [28], Gardner and Wickert mentioned that radial stubs, possibly realized as shunt stubs with low characteristic impedance, may avoid the problems of transverse resonance and poorly defined point of attachment associated with straight stubs. In our DC Feed, the radial stubs are the suitable choice. Some research groups have studied about the equations for the formulation of microstrip radial stubs [27, 29–31]. Based on these studies combined with optimization tool in Advanced Design System software, the optimized

**Figure 16.** *Impedance of DC block VJ0603D8R2CXP.*

parameters calculated for the Roger4003c substrate radial stubs are: the outer radius *Ro* = 540.621 mil and the inner radius *Ri* = 56.688 mil and angle α = 30°.
