Smith Chart and Matching Circuit Basics

Smith Chart and Matching Circuit Basics The Smith Chart is a polar plot of complex reflection coefficient (gamma $\Gamma$). This chart plots the real and imaginary parts of complex impedance simultaneously. The real part R ranges from 0 to infinity ($\infty$), while the imaginary part X ranges from negative infinity to positive infinity, both of which can be plotted on the same chart. The parameters and functions that can be represented by the Smith Chart include, but are not limited to: Complex impedance Complex reflection coefficient Voltage standing wave ratio (VSWR) Transmission line effects Designing impedance matching networks Normalized Impedance We first need to convert complex impedance to normalized impedance before we can plot it on the Smith Chart. Normalized impedance is equal to the actual impedance divided by the system impedance: $$ Normalized Z=\frac{Actual Z}{System Z_0} $$ In most cases, the system impedance is 50Ω, so the actual impedance is basically divided by 50Ω. For example, if the actual impedance value is (37+j55)Ω, the value of the normalized impedance is approximately (0.74+j1.10)Ω. Smith Chart Details On the Smith Chart, the horizontal axis in the middle is the pure resistance axis (R). On this axis, the reactance is always 0. In the area above the main axis (R+jX), the reactance is inductive, while in the area below (R-jX), the negative reactance is capacitive. Key Points There are some key points on the Smith Chart. The middle point represents the system impedance (in a 50Ω impedance system, this point represents a 50Ω impedance value). Along the main axis, the rightmost point represents an open circuit (infinite impedance), while the leftmost point represents a short circuit (impedance is 0). Constant Resistance Circles On the constant resistance circles, the normalized resistance (real part) is a constant. For example, the circle passing through the center point has a resistance of 1 (50Ω), which means the real part of the normalized impedance is 1. The following are constant resistance circles with normalized resistance values of 0.4 and 3.0: Constant Reactance Arcs The constant reactance arcs extend from the open circuit point, and the points above the arc represent the reactance value, which is the imaginary part of the complex impedance. The larger the angle, the larger the absolute value of the reactance. As the angle decreases, the absolute value of the reactance will become smaller and smaller until it becomes 0 (on the pure resistance axis, neither inductive nor capacitive). The following are constant reactance arcs with reactance values of ±0.5 (imaginary part is ±j0.5) and ±1.0 (imaginary part is ±j1.0): Converting Complex Impedance to Complex Admittance Admittance (Y) is the reciprocal of impedance, conductance (G) is the reciprocal of resistance, and susceptance (B) is the reciprocal of reactance. If they are complex numbers, the calculation can be very complicated. However, on the Smith Chart, we only need to draw a circle passing through the complex impedance Z' point with the center point as the center to find the corresponding complex admittance value Y' on the other side. For example, if Z'=1+j1.1, then its corresponding Y'=0.45-j0.5. If it is a complex impedance curve, the Smith Chart can be rotated 180° to obtain the complex admittance curve. Constant Conductance Circles and Constant Susceptance Arcs The complete Smith Chart also shows constant conductance circles and constant susceptance arcs, marked in blue in the following figure: Marking General Values on Smith Chart To mark a general value, such as Z=25+j40, on the Smith Chart, we first normalize it to 50Ω impedance, and calculate Z'=0.5+j0.8. Then, we can find the circle with R'=0.5 and the arc with X'=0.8 on the chart, and their intersection is the desired Z': Radial Scaling Parameters of Smith Chart Generally, there is a radial scaling parameter axis underneath the Smith Chart. Assuming there is a complex impedance point on the chart, we can connect it to the center point, rotate it along the radius to the main axis, and project it onto the radial scaling parameter axis, as shown below: We can read some parameters on this axis: Voltage Standing Wave Ratio (VSWR): 2.3:1 Return Loss: 8.10dB Reflection Coefficient Voltage: 0.155 Power: 0.39 Moreover, we can find that the points on the circle have the same parameters. Relationship between VSWR and Transmission Line As mentioned earlier, for points with different impedances on the circle drawn with the system impedance (center point) as the center and the distance to the complex impedance point as the radius, the VSWR is the same. The length of one circle on the chart is equivalent to half a wavelength. Therefore, when the length of the transmission line from the source end to the antenna increases by half a wavelength, the complex impedance will rotate one more circle along the circle. Therefore, the length of the transmission line from the source to the antenna should be controlled to be an integer multiple of half a wavelength of the antenna to avoid deviation from the actual impedance of the antenna. If the length is a general value, the transmission line model can be used for compensation in the SimNEC simulation software. There is also an interesting phenomenon. If the length of the transmission line with an open end is 1/4 wavelength (half a circle), it looks like a short circuit; conversely, if the length of the transmission line with a short end is 1/4 wavelength, the observed characteristic looks like an open circuit: Design of Matching Circuit We usually connect inductors and capacitors in series or parallel in the circuit to move the complex impedance point and match the impedance. Series Inductor: Move clockwise along the constant resistance circle. Series Capacitor: Move counterclockwise along the constant resistance circle. Parallel Inductor: Move counterclockwise along the constant conductance circle. Parallel Capacitor: Move clockwise along the constant conductance circle. Qualitatively, whether in series or in parallel, adding an inductor will move the complex impedance point up, while adding a capacitor will move it down. Simple L-Matching Circuit Assuming the current measured complex impedance value of the circuit is $Z_L$, we need to adjust it to the ideal $Z_0$ (the best is the center point, followed by the 50Ω resistance circle) by using a simple L-shaped matching circuit composed of inductors or capacitors to move the complex impedance point on the Smith Chart. The combination of inductors or capacitors depends on the initial position of $Z_L$ on the Smith Chart. We can select a suitable combination to build an L-matching circuit based on the distribution of $Z_L$ shown in the figure below: For example, the initial complex impedance value $Z_L$ is shown in the figure below (black dot), and we can adjust it to the ideal $Z_0$ (red dot) based on the construction of the L-type matching circuit provided in the small figure: For convenience of debugging, we outline the standard constant conductance circle and constant resistance circle on the figure. The following steps can be performed in the SimNEC simulation software: Add a series inductor and move it clockwise along the constant resistance circle until it touches the constant conductance circle. Add a parallel capacitor and move it clockwise along the constant conductance circle until it reaches the center point. The method can be roughly summarized as: starting from the near load end, add the first component to move the complex impedance point to the standard constant conductance circle or constant resistance circle; add the second component to move it along the standard constant conductance circle or constant resistance circle to the center point. The values of the selected components can be read in the software. For the debugging of actual matching circuits, please refer to the article Design of General Antenna Matching Circuit. References and Acknowledgments Smith Chart: The History of Smith Chart and Why It's So Important for RF Designers Basics of the Smith Chart - Intro, impedance, VSWR, transmission lines, matching RF engineering basic concepts: the Smith chart Original: https://wiki-power.com/ This post is protected by CC BY-NC-SA 4.0 agreement, should be reproduced with attribution. This post is translated using ChatGPT, please feedback if any omissions.