Automatic Procedure and the Use of the Smith Chart in Impedance Matching in Analog Circuits

Abstract

This paper presents a comprehensive methodology for impedance matching in analog circuits, integrating analytical methods with computer-aided design techniques. It focuses on maximizing power transfer through impedance adaptation and emphasizes the practical utility of the Smith chart for identifying optimal matching configurations. This study examines various impedance matching topologies—including L, T, and Pi networks—with an emphasis on using reactive components such as capacitors and inductors. A MATLAB-based tool is developed to automate the synthesis of matching networks, providing four equivalent circuit solutions for each scenario. Illustrative examples and simulations confirm the method’s efficiency, flexibility, and applicability to a broad range of radiofrequency (RF), microwave, and wireless power transfer systems.

1. Introduction

In today’s context of increasing energy efficiency demands, maximizing the transfer of active power from source to load is a critical design objective [1]. Achieving this requires effective impedance matching, which is essential for optimizing both the input impedance of an electrical load and the output impedance of a source [2,3].

The calculation of matching impedances is particularly useful in determining the load impedance parameters of analog circuits and wireless power transfer systems so that the active powers transmitted to the loads are maximum. Matching impedances in signal transmission lines is of particular importance for eliminating noise signals and reflections [4,5,6,7,8,9].

The maximum active power transfer theorem states that the active power delivered from the source to the load is maximized when the load resistance matches the source resistance and the load reactance is equal in magnitude but opposite in sign to the source reactance. This translates into the language of complex numbers with the condition that the load impedance is equal to the complex conjugate of the source impedance [10,11,12,13].

For a complex source impedance Z_S and a complex load impedance Z_L, one obtains the maximum active power transfer using the following condition:

$${\underset{\_}{Z}}_S={\underset{\_}{Z}}_L^{*}$$

(1)

If ${\underset{\_}{Z}}_S$ is the transmission line characteristic impedance, the condition for having minimal signal reflection on the transmission line is obtained when

$${\underset{¯}{Z}}_S={\underset{¯}{Z}}_L$$

(2)

Fulfilling the condition in Equation (2) leads to the non-existence of the wave reflected by the load. Another measure of impedance matching is represented by the return loss parameter (RL). RL is used to measure the attenuation of the signal reflected by the load:

$$RL\left(\mathrm{d}\mathrm{B}\right)=10\mathit{lg}{\displaystyle \frac{P_r}{P_i}}$$

(3)

where

• P_i is the power of the incident wave;
• P_r is the power of the wave reflected by the load.

For a perfectly matched electrical load (ideal), the RL parameter tends to infinity. Considering that the powers of the incident and reflected waves are directly related to the squared value of the signal amplitudes, the RL can also be defined as a function of the reflection coefficient $\underset{\_}{\Gamma }$.

$$RL\left(\mathrm{d}\mathrm{B}\right)=20\mathit{lg}{\displaystyle \frac{U_r}{U_i}}=20\mathit{lg}\underset{\_}{\Gamma }$$

(4)

Insertion loss is defined by the ratio between the input signal power value and the transmitted signal power value

$$IL\left(\mathrm{d}\mathrm{B}\right)=10\mathit{lg}{\displaystyle \frac{P_i}{P_t}}$$

(5)

Using S parameters, the performance of a radio frequency circuit can be completely characterized without the need to know the internal composition, being considered a black box. The internal components can be active or passive, with only the linearity condition being imposed (the circuit output is proportional to the input). Figure 1 is a representation of a simple circuit with an input and an output, called ports. Each port has, in harmonic mode, an incident signal (a) and a reflected signal (b). We denote the incident wave at port 1 as a₁ and at port 2 as a₂. They have as consequences the reflected waves b₁ and b₂ [4,9,10].

The incident and reflected waves are used to define S parameters for a two-port network described by linear equations:

$$\left\{\begin{array}{l}{\underset{¯}{b}}_1={\underset{¯}{S}}_{11}{\underset{¯}{a}}_1+{\underset{¯}{S}}_{12}{\underset{¯}{a}}_2\\ {\underset{¯}{b}}_2={\underset{¯}{S}}_{21}{\underset{¯}{a}}_1+{\underset{¯}{S}}_{22}{\underset{¯}{a}}_2\end{array}\right.$$

(6)

These waves are referred to as power waves, and the corresponding S parameters are known as generalized S parameters. We obtain the S parameters as follows:

$${\underset{¯}{S}}_{11}={\left.{\displaystyle \frac{{\underset{¯}{b}}_1}{{\underset{¯}{a}}_1}}\right|}_{{\underset{¯}{a}}_2=0}\hspace{1em}\hspace{1em}{\underset{¯}{S}}_{12}={\left.{\displaystyle \frac{{\underset{¯}{b}}_1}{{\underset{¯}{a}}_2}}\right|}_{{\underset{¯}{a}}_1=0}\hspace{1em} {\underset{¯}{S}}_{21}={\left.{\displaystyle \frac{{\underset{¯}{b}}_2}{{\underset{¯}{a}}_1}}\right|}_{{\underset{¯}{a}}_2=0}\hspace{1em} {\underset{¯}{S}}_{22}={\left.{\displaystyle \frac{{\underset{¯}{b}}_2}{{\underset{¯}{a}}_2}}\right|}_{{\underset{¯}{a}}_1=0}$$

(7)

Furthermore, RL and IL can be expressed as functions of S parameters:

$$R{L}_{dB}=10\mathit{log}{\displaystyle \frac{{\left|{\underset{¯}{a}}_1\right|}^2}{{\left|{\underset{¯}{b}}_1\right|}^2}}=20\mathit{log}{\displaystyle \frac{\left|{\underset{¯}{a}}_1\right|}{\left|{\underset{¯}{b}}_1\right|}}=20\mathit{log}{\displaystyle \frac{1}{\left|S_{11}\right|}}=-20\mathit{log}\left|S_{11}\right|$$

(8)

$$I{L}_{dB}=10\mathit{log}{\displaystyle \frac{{\left|{\underset{¯}{a}}_1\right|}^2}{{\left|{\underset{¯}{a}}_2\right|}^2}}=20\mathit{log}{\displaystyle \frac{\left|{\underset{¯}{a}}_1\right|}{\left|{\underset{¯}{a}}_2\right|}}=20\mathit{log}{\displaystyle \frac{1}{\left|{\underset{¯}{S}}_{21}\right|}}=-20\mathit{log}\left|{\underset{¯}{S}}_{21}\right|$$

(9)

This representation can be extended to circuits with an arbitrary number of ports. The number of S parameters increases with the square of the number of ports, so the mathematics becomes more complicated, but it is simplified using matrix algebra.

While L-sections are quite practical circuits, they do not give us the flexibility to choose the bandwidth. The bandwidth of these circuits is constant for a given input and output impedance. When a narrower bandwidth than that of an L-section is required, more complicated arrangements, such as T or Pi networks, can be employed. Keep in mind that these types of networks can only increase the quality factor (or equivalently reduce the bandwidth) of the circuit. Further investigations will focus on matching circuits that can provide a wider bandwidth than that of a simple L-section.

Although L-sections are simple and practical, their bandwidth is inherently limited by fixed impedance ratios. To overcome this constraint, more advanced configurations such as T and Pi networks are introduced, enabling greater flexibility in bandwidth control through the adjustable quality factor (Q). These networks allow fine-tuning of bandwidth, making them ideal for applications requiring specific performance characteristics.

This paper used the Matlab R2023b environment to develop a computer-aided procedure for generating matching networks. This procedure identifies the values for the circuit elements, capacitors or inductors (series and/or parallel), based on the Smith chart and provides four equivalent schemas for the impedance matching process. This procedure simplifies the work and offers a reliable tool for obtaining the target matching impedance.

The procedures for matching impedances and transmission lines are based on the use of matching circuits composed of reactive elements (inductors and capacitors) that can be easily implemented in a program in the MAPLE 2015 or MATLAB programming environments.

In this paper, we study impedance matching with series and parallel components. Combining the Smith chart for impedance with that for admittance appears to facilitate this process without difficulty.

Obtaining the precise values of the inductors and capacitors is not critical. The main objective of impedance matching is to understand how these components affect movement on the Smith chart and to select approximate component values that achieve the desired matching. Exact values for capacitors and inductors do not exist (i.e., you can find a 0.8 pF capacitor, but you cannot find a 0.832 pF capacitor), and the components are non-ideal (resistance not always negligible, non-ideal frequency characteristics, etc.). As a consequence, impedance matching reduces to understanding how the components move an impedance on the Smith chart.

Section 2 presents the use of the Smith chart to compute the impedance matching circuit elements. Section 3 presents the analytical calculation procedures for circuit elements in the L, T, and Pi impedance matching structures. Automatic calculation procedures based on the Smith chart, for circuit elements in the L, T, and Pi structures, are exposed in Section 4.

Significant examples are presented that confirm the validity of the procedures presented in this paper.

2. Smith Chart in Adapting Analog Circuits

The Smith chart is a visual depiction of the Möbius Transformation with [14,15,16,17,18,19,20]

$$\underset{¯}{\mathsf{\Gamma }}={\displaystyle \frac{\underset{¯}{Z}-Z_0}{\underset{¯}{Z}+Z_0}}$$

(10)

With this transformation, the positive real values of impedance Z are transferred onto the unit circle of plane Γ (Figure 2).

The main properties of this transformation are as follows:

(1)

The lines with complex values are transformed into circles;

(2)

Angles are preserved locally.

Usually, the Smith chart works with normalized characteristic impedance (Z₀):

$$\underset{¯}{z}={\displaystyle \frac{\underset{¯}{Z}}{Z_0}}$$

(11)

and the transformation is simplified as follows:

$$\underset{¯}{\mathsf{\Gamma }}={\displaystyle \frac{\underset{¯}{z}-1}{\underset{¯}{z}+1}}\iff \underset{¯}{z}={\displaystyle \frac{1+\underset{¯}{\mathsf{\Gamma }}}{1-\underset{¯}{\mathsf{\Gamma }}}}$$

(12)

Considering the real and imaginary parts of Γ and z, relation Equation (12) becomes [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]

$$\begin{array}{l}\underset{¯}{z}={\displaystyle \frac{1+{\mathsf{\Gamma }}_r+j{\mathsf{\Gamma }}_i}{1-{\mathsf{\Gamma }}_r-j{\mathsf{\Gamma }}_i}}={\displaystyle \frac{1-{\mathsf{\Gamma }}_r^2-{\mathsf{\Gamma }}_i^2}{{\left(1-{\mathsf{\Gamma }}_r\right)}^2+{\mathsf{\Gamma }}_i^2}}+j{\displaystyle \frac{2{\mathsf{\Gamma }}_i}{{\left(1-{\mathsf{\Gamma }}_r\right)}^2+{\mathsf{\Gamma }}_i^2}}=r+jx\\ r={\displaystyle \frac{1-{\mathsf{\Gamma }}_r^2-{\mathsf{\Gamma }}_i^2}{{\left(1-{\mathsf{\Gamma }}_r\right)}^2+{\mathsf{\Gamma }}_i^2}}\leftrightarrow {\left[{\mathsf{\Gamma }}_r-{\displaystyle \frac{r}{r+1}}\right]}^2+{\left[{\mathsf{\Gamma }}_i-0\right]}^2={\left[{\displaystyle \frac{1}{r+1}}\right]}^2\\ x={\displaystyle \frac{2{\mathsf{\Gamma }}_i}{{\left(1-{\mathsf{\Gamma }}_r\right)}^2+{\mathsf{\Gamma }}_i^2}}\leftrightarrow {\left[{\mathsf{\Gamma }}_r-1\right]}^2+{\left[{\mathsf{\Gamma }}_i-{\displaystyle \frac{1}{x}}\right]}^2={\left[{\displaystyle \frac{1}{x}}\right]}^2\end{array}$$

(13)

For the real part of complex impedance, a circle with center coordinates $\left({\displaystyle \frac{r}{r+1}},0\right)$ and radius $\left({\displaystyle \frac{1}{r+1}}\right)$ is plotted.

For the imaginary part of complex impedance, sectors of circles with center coordinates $\left(1,{\displaystyle \frac{1}{x}}\right)$ and radius $\left({\displaystyle \frac{1}{x}}\right)$ are plotted.

Each point on the Smith chart simultaneously represents both a value of z and the corresponding value of Γ, as presented in Figure 3.

Γ—the reflection factor—represents a measure that describes how much of a wave is reflected back at two different media boundaries; it is the ratio of the electrical field strength E of the reflected wave over the forward traveling wave [1]:

$$\mathsf{\Gamma }={\displaystyle \frac{E_{of reflected wave}}{E_{of forward traveling wave}}}$$

(14)

In the Smith chart we find three important points:

(1)

$\mathsf{\Gamma }=1, z\to \infty$—for an open circuit;

(2)

$\mathsf{\Gamma }=-1, z=0$—for a short circuit;

(3)

$\mathsf{\Gamma }=0, z=1$—for a matched load;

The reflection factors are represented as concentric circles around the center. The radius of these circles is proportional to the magnitude of the reflection factor (e.g., a radius of 0.5 is assigned to a 3dB reflection (demi-reflected signal) and a radius of 1 corresponds to full reflection). Since the mismatch impedance problems correspond to values of reflection coefficient always greater than 0, the Smith chart provides a better understanding.

The Möbius Transformation provides a Smith chart with the values of complex admittance on the same plane:

$$\underset{¯}{\mathsf{\Gamma }}={\displaystyle \frac{1-\underset{¯}{y}}{1+\underset{¯}{y}}}={\displaystyle \frac{Y_0-\underset{¯}{Y}}{Y_0+\underset{¯}{Y}}}={\displaystyle \frac{\frac{1}{Z_0}-\frac{1}{\underset{¯}{Y}}}{\frac{1}{Z_0}-\frac{1}{\underset{¯}{Y}}}}={\displaystyle \frac{\underset{¯}{Z}-Z_0}{\underset{¯}{Z}+Z_0}}={\displaystyle \frac{\underset{¯}{z}-1}{\underset{¯}{z}+1}}$$

(15)

For complex admittance, we have a similar chart, rotated 180 degrees from the center of the Smith chart (Figure 4).

As mentioned, on the Smith chart it is possible to represent any value of z and easy to visualize the trajectory of this point for series- or parallel-connected variable inductance or a variable capacitor (Figure 5).

For the series-connected elements one uses the circles in the impedance plane. These are moved clockwise for an added inductance and anticlockwise for an added capacitor. For elements connected in parallel one uses the circles but in the admittance plane. For a clockwise displacement a capacitor is added and for an anticlockwise one an inductance is added.

The advantages of the Smith chart can be summarized as follows [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]:

(1)

The Smith chart is a compact and useful image of passive impedances from 0 to ∞.

(2)

Impedance mismatch problems are solved easily in the chart. There are eight possible impedance matching networks (Figure 6):

(3)

The Smith chart is especially useful for radio frequency, microwave applications, and wireless power transfer systems because it translates impedances and admittances into reflection factors and vice versa. In these higher frequency ranges, it is more convenient to describe electrical quantities using forward and backward waves.

3. Analytic Computation and Finding L- and T-Section Component Values

3.1. Finding L-Section Component Values

Assume that the normalizing impedance is Z₀ = 50 Ω and the frequency of interest is 1 GHz. The component values of the L-section can be found as follows. The transformation along the Smith chart path from zLoad to point A corresponds to a series inductor with a normalized reactance of xA − xLoad = j0.4 − j0 = j0.4. This requires a 3.18 nH series inductor at 1 GHz (Figure 7).

The transformation along the Smith chart path from point A to zSource in Figure 7 requires a parallel capacitor with a susceptance of bSource − bA = 0j − (−j2) = j2. This can be produced by a 6.37 pF parallel capacitor. The final L-section is shown in the figure.

The values of the parameters L1 and C1 can also be determined analytically as follows:

$$\begin{array}{l}f=1 \mathrm{GHz}; {\underset{¯}{Z}}_{C1}=-j/\left(2\pi \ast f\ast C_1\right); {\underset{¯}{Z}}_{L1}=j\ast \left(2\pi \ast f\ast L_1\right); R_{Load}=10 \mathsf{\Omega }\Rightarrow \\ \Rightarrow {\underset{¯}{Z}}_{e\_Lser}=\left({\underset{¯}{Z}}_{C1}\ast \left({\underset{¯}{Z}}_{L1}+R_{Load}\right)\right)/\left({\underset{¯}{Z}}_{C1}+{\underset{¯}{Z}}_{L1}+R_{Load}\right) \mathrm{and} \mathrm{setting} \mathrm{the} \mathrm{conditions}:\\ \left\{\begin{array}{l}\mathrm{Re}\left({\underset{¯}{\mathrm{Z}}}_{e\_Lser}\right)=50.0\\ \mathrm{Re}\left({\underset{¯}{\mathrm{Z}}}_{e\_Lser}\right)=0\end{array}\right. \Rightarrow \left\{\begin{array}{l}{C}_1=6.3694 \mathrm{pF} \cong 6.37 \mathrm{pF}\\ L_1=3.184 \mathrm{nH} \cong 3.18 \mathrm{nH}\end{array}\right.\Rightarrow \\ \Rightarrow {\underset{¯}{Z}}_{e\_Lser}=50.0-0.314\ast {10}^{-06}j\approx 50.0 \mathsf{\Omega }.\end{array}$$

(16)

3.2. Finding T Network Component Values

Next, we can find the component values of the T network in a similar manner. In this case, the transformation along the Smith chart path from zLoad to point B corresponds to a series inductor with a normalized reactance of j0.8, which can be produced by a 6.37 nH inductor.

The next transformation along the Smith chart path requires a capacitive susceptance of bC − bB = −j0.47 − (−j1.2) = j0.73, which can be obtained by a 2.2954 pF capacitor. Finally, we need a series capacitor with reactance −j1.53 to go to the center of the chart, which can be achieved by a 2.0555 pF capacitor. The final T network is depicted in Figure 8.

The matching (adaptation) circuit in the case of T-connection requires that the values of three elements be determined analytically, although there are only two equations: Re(Ze_Tnet) = Re(ZS) and Im(Ze_Tnet) = Im(ZS). To analytically compute the values of the parameters of the elements of the matching (adaptation) circuit with T-connection, the value of the parameter of one of the three circuit elements must be arbitrarily chosen through experiments (

Table 1. The three case values of the parameters of the matching circuit elements.

Arbitrarily Choose Parameter Value	Overall Parameter Values Two Elements and Complex Matching Impedance
L2 = 9.0 nH	C2 = 5.914 pF, C3 = 4.397 pF, Ze_Tnet_L1 = 49.9999 + 0.808 × 10−06 j
C2 = 4.0 pF	L2 = 12.505 nH, C3 = 2.568 pF, Ze_Tnet_C2 = 49.9999 + 0.3746 × 10−05 j
C3 = 1.0 pF	L2 = 16.261 nH, C2 = 2.658 pF, Ze_Tnet_C3 = 50.0 + 0.125 × 10−05 j

). The experiments must take into account the value of the operating frequency of the circuit (signal transmission network) so that the resonant frequencies are close to the operating frequency.

The bandwidth (BW) and quality factor (Q) in impedance matching circuits are interconnected, defined as follows:

$$Q={\displaystyle \frac{f_0}{BW}}$$

where $f_0$ is the operating frequency. The benefit of T and Pi networks is that Q can be directly controlled by altering the component values. For instance, consider a T network at $f_0=1 Gz$ with an obtained quality factor of Q = 5. The corresponding bandwidth is calculated explicitly as follows:

$$BW={\displaystyle \frac{f_0}{Q}}={\displaystyle \frac{1 GhZ}{5}}=200 MhZ$$

Adjusting the reactances and susceptances allows the designer to achieve wider or narrower bandwidths, a flexibility absent in simpler L-section circuits.

3.3. Designing a Pi Matching Network

Another type of three-element matching network is the Pi network depicted below (Figure 9).

Let us design a Pi network to translate zLoad = 3.33 to the Smith chart center at 1 GHz. Assume that the maximum Qn is set to be 4. With a Pi circuit, the elements next to ZLoad and ZSource are parallel components, and thus we are allowed to move along constant-conductance circles going through the source and load impedances. An intersection of these constant-conductance circles and the Qn = 4 curve can be used as an intermediate point, as shown in Figure 10.

In this example, the g = 0.3 circle is the constant-conductance circle that goes through zLoad. The intersection of this circle with the Qn = 4 curve (point A above) is used as the intermediate point for impedance transformation. The next move should be along the constant-resistance circle going through point A (this corresponds to the series component of the Pi network). In our example, the r = 0.2 circle is the constant-resistance circle that goes through point A. The intersection of the r = 0.2 Ω circle and the g = 1 S constant-conductance circle is our next intermediate impedance (point B). Finally, we move along the g = 1 S circle to reach the center of the Smith chart. Using a more complete Smith chart, we can find the reactance (x) and susceptance (b) of points A and B, as provided in

Table 2. The reactance and susceptance for points A and B.

Intermediate Point	Reactance (x)	Susceptance (b)
A	0.8	−1.2
B	0.4	−2

.

Using the above information, the component values cand be found. The transformation along the Smith chart path from zLoad to point A implies a −j1.2 normalized susceptance. This can obtained by a 6.67014 nH parallel inductor at 1 GHz (assuming Z0 = 50 Ω). Furthermore, the transformation along the Smith chart path A to B implies a j0.4−j0.8 = −j0.4 normalized reactance. This can be obtained by an 8.1415 pF series capacitor. Finally, the transformation along the Smith chart path from B to zSource needs a normalized susceptance of j2, which calls for a 6.37 pF parallel capacitor. The final circuit is shown in Figure 11.

$$\begin{array}{l}f=1 \mathrm{GHz}; {\underset{¯}{Z}}_{C2}=-j/\left(2\pi \ast f\ast C_2\right); {\underset{¯}{Z}}_{C3}=-j/\left(2\pi \ast f\ast C_3\right); {\underset{¯}{Z}}_{L1}=j\ast \left(2\pi \ast f\ast L_1\right); \\ R_{Load}=166.5 \mathsf{\Omega }\Rightarrow {\underset{¯}{Z}}_{e\_C2L1Rload}={\underset{¯}{Z}}_{C2}+\left({\underset{¯}{Z}}_{L1}\ast R_{Load}\right)/\left({\underset{¯}{Z}}_{L1}+R_{Load}\right); \\ {\underset{¯}{Z}}_{e\_Pi}=\left({\underset{¯}{Z}}_{C3}\ast {\underset{¯}{Z}}_{1\_C2L1RLoad}\right)/\left({\underset{¯}{Z}}_{C3}+{\underset{¯}{Z}}_{e\_C2L1RLoad}\right)=50.00017-j0.000505\approx 50 \mathsf{\Omega }.\end{array}$$

(17)

For the circuit given in Figure 11, one computes the input impedance. It results in a Z_in that is reasonably close to the target 50 Ω impedance.

In a Pi-connection matching circuit, three element values must be determined analytically, though only two equations are available. To solve this, one component’s value is chosen experimentally (

Table 3. The three case values of the parameters for the matching circuit elements.

Arbitrarily Choose Parameter Value	Overall Parameter Values Two Elements and Complex Matching Impedance
L1 = 9.0 nH	C2 = 5.914 pF, C3 = 4.397 pF, Ze_Tnet_L1 = 49.9999 + 0.808 × 10−06 j
C2 = 4.0 pF	L1 = 12.505 nH, C3 = 2.568 pF, Ze_Tnet_C2 = 49.9999 + 0.3746 × 10−05 j
C3 = 1.0 pF	L1 = 16.261 nH, C2 = 2.658 pF, Ze_Tnet_C3 = 50.0 + 0.125 × 10−05 j

). The experiments must take into account the value of the operating frequency of the circuit (signal transmission network) so that the resonant frequencies are close to the operating frequency [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35].

4. Automatic Procedure for Adapting Analog Circuits: Four Different L–Sections for One Impedance Matching Problem

4.1. The Automatic Procedure for Adapting Analog Circuits

Starting from the equations of the circles for resistances, reactances, conductances, and susceptances (Equation (13)), a Matlab procedure was developed to compute the values for capacitance and/or inductance in four different scenarios. Figure 12 presents the diagram of the developed procedure.

Let us examine this through an example. Assume that the load impedance Z₁ = 10 + j30 Ω is to be matched to a source impedance of Z₂ = 50 Ω at a frequency of 1 MHz.

With a normalizing impedance of Z₀ = 50 Ω, the normalized impedance is z₁ = 0.2 + j0.6 (point A).

This is represented by point (A) both on the Z Smith chart impedance diagram (Figure 13a—resistance circle with constant value of 0.2 and reactance arc with value 0.6) and on the Y Smith chart admittance diagram (Figure 13b—conductance circle of 0.5 and susceptance arc of −1.5) [35,36,37,38,39,40,41,42,43].

To get from point (A), which defines the position of the load impedance, to point (O), which meets the impedance matching condition, we have four options:

-

In the impedance diagram, the movement from point (A) to point (B), on the 0.2 resistance circle, is equivalent to connecting in series a capacitor with the value C_s = 15.915 nF.

$$C_S={\displaystyle \frac{1}{\left(x_A-x_B\right)2\pi f{Z}_0}}$$

(18)

The translation from point (B) to point (O), on the conductance circle of value 1, results in the parallel connection of a capacitor with the value C_p = 6.366 nF.

$$C_p={\displaystyle \frac{b_B}{2\pi f{Z}_0}}$$

(19)

resulting Z_in = 49.998 + j*0.004 (equivalent to Z₀ = 50).

The Matlab procedure automatically identifies the circles corresponding to the points on the Smith chart and calculates the values of the circuit elements resulting from movements along the circles. The electrical diagram and circuit element values for version 1 is given in Figure 14.

-

Also in the impedance diagram, the movement from A to C, on the resistance circle with value of 0.2, is equivalent to connecting in series a capacitor with the value C_s = 3.183 nF

$$C_S={\displaystyle \frac{1}{\left(x_A-x_C\right)2\pi f{Z}_0}}$$

(20)

The translation from point (C) to point (O), on the conductance circle of 1, results in the parallel connection of a coil with the value L_p = 3.979 uH.

$$L_p={\displaystyle \frac{Z_0}{b_{C}2\pi f}}$$

(21)

resulting Zin = 50.006 + j × 0.001 (equivalent to Z₀ = 50). The values obtained for this variant by our procedure are the following (Figure 15):

-

In the admittance diagram, the movement from A to B, on the conductance circle with value of 0.5, is equivalent to connecting in parallel a capacitor with the value C_p = 3.183 nF

$$C_p={\displaystyle \frac{\left(b_A-b_{B\prime }\right)}{2\pi f{Z}_0}}$$

(22)

The translation from point (B’) to point (O), on the resistance circle of value 1, results in the series connection of a capacitor with the value C_s = 3.183 nF.

$$C_S={\displaystyle \frac{1}{x_{B\prime }2\pi f{Z}_0}}$$

(23)

resulting Z_in = 49.997 – j × 0.002 (Figure 16).

Also in the admittance diagram, the movement from point (A) to point (C′), on the conductance circle with value of 0.5, is equivalent to connecting in parallel a capacitor with the value Cp = 6.366 nF

$$C_b={\displaystyle \frac{\left(b_B-b_{C\prime }\right)}{2\pi f{Z}_0}}$$

(24)

The translation from point (C′) to point (O), on resistance circle 1, results in the series connection of a coil with the value L_s = 7.957 mH

$$L_S={\displaystyle \frac{\left(x_A-x_C\right)Z_0}{2\pi f}}$$

(25)

resulting Z_in = 50.006 – j × 0.004 (Figure 17).

The simulations clearly demonstrate that while L-section networks have a fixed bandwidth (determined by the impedance ratio), T and Pi networks exhibit tunable bandwidth characteristics. Adjustments to component values lead to substantial changes in the Q-factor, clearly observable as variations in bandwidth. This explicit bandwidth control is essential in RF systems requiring tailored performance profiles.

4.2. Frequency-Sweep Analysis and Bandwidth Evaluation of Matching Networks

To illustrate the practical advantages in bandwidth control provided by T and Pi networks over the simpler L-section topology, frequency-sweep simulations were performed. Figure 18 compares the magnitude of reflection coefficient S11 for L-section, T network, and Pi network configurations across a relevant frequency range.

Figure 18 illustrates how the magnitude of the reflection coefficient S11 varies with frequency for L-section, T network, and Pi network configurations. The following aspect can be deduced:

(1)

L-section (Q = 5, BW = 200 MHz): Offers a moderate fixed bandwidth, illustrating typical bandwidth limitations;

(2)

T network (Q = 3, BW = 333 MHz): Demonstrates a broader bandwidth, reflecting its advantage in applications requiring wider frequency coverage;

(3)

Pi network (Q = 8, BW = 125 MHz): Shows significantly narrower bandwidth, ideal for selective applications demanding higher quality factors.

4.3. Practical Considerations: Component Tolerances, Parasitic Effects, and PCB Implementation

While the previous analyses assumed ideal components, practical RF and microwave designs are significantly affected by non-idealities such as component tolerances, parasitic inductances, capacitances, and resistances, as well as PCB layout effects [35,36,37,38,39,40]. Such practical factors often shift operational characteristics away from their theoretically predicted behavior.

4.3.1. Component Tolerances and Availability

Standard capacitors and inductors typically exhibit manufacturing tolerances ranging from ±1% to ±10%, directly influencing resonance frequencies and impedance matching performance [37,40]. Variations caused by component tolerances may degrade return loss and alter bandwidth, necessitating sensitivity analysis, such as Monte Carlo simulations, within MATLAB or equivalent software environments [14,15,34].

4.3.2. Parasitic Effects

Real inductors and capacitors at RF and microwave frequencies demonstrate substantial parasitic elements (Equivalent Series Resistance—ESR; Equivalent Series Inductance—ESL; Equivalent Parallel Capacitance—EPC), dramatically affecting frequency response, insertion loss, and impedance matching accuracy [36,37,38,39]. For instance, a nominal 10 nH inductor may possess significant ESR at GHz frequencies, substantially impacting circuit efficiency and the accuracy of impedance matching [36,37,39]. Accurate component models from datasheets or measured data should thus be integrated into simulations.

4.3.3. PCB Layout Considerations

The PCB layout introduces additional parasitic inductances and capacitances, influencing resonance frequencies, Q-factors, and impedance characteristics [35,38,39,40]. High-frequency design principles—controlled impedance guidelines, minimal trace lengths, microstrip or coplanar waveguide structures, and robust grounding—must be employed to ensure stable and consistent performance in practical circuit implementations [38,40].

4.3.4. Experimental Validation and Measurements

Although analytical and simulation methods dominate the current methodology, practical RF/microwave circuit designs require empirical validation via Vector Network Analyzer (VNA) measurements [34,35,36]. These experimental validations identify deviations due to real-world effects and component non-idealities, significantly enhancing circuit reliability. Future research should incorporate extensive experimental studies, validating matching network implementations on PCB prototypes and comparing empirical results with theoretical predictions [34,36,40].

5. Conclusions

The study presented in this paper demonstrated a comprehensive and effective methodology for impedance matching in analog circuits, combining analytical calculations, graphical insights via the Smith chart, and computer-aided automation using MATLAB. The developed MATLAB-based tool efficiently generates multiple equivalent impedance matching solutions, significantly simplifying the design process and enabling rapid evaluation and prototyping for RF and microwave engineers.

Through explicit frequency-sweep simulations, the practical advantages of T and Pi matching networks over simpler L-section circuits were clearly demonstrated, particularly regarding flexible bandwidth control via adjustments to the quality factor (Q). This capability is essential for tailoring performance to specific application requirements.

Furthermore, this paper addressed crucial practical considerations—including component tolerances, parasitic effects, and PCB layout implications—highlighting the necessity of incorporating these factors into simulation and design practices to ensure accurate and reliable performance in real-world applications.

Future work should focus on optimizing impedance matching networks to achieve broader bandwidth capabilities, exploring multisection or distributed element configurations. Additionally, integrating parasitic-aware optimization techniques and conducting comprehensive experimental validations using network analyzer measurements will substantially enhance the practical applicability and robustness of impedance matching solutions in next-generation RF, microwave, and wireless power transfer systems.