##### The Theory Behind“Smith Chart”
Time : 2022-06-23 ### 1. What is “Smith Chart”?

The chart was invented by Phillip Smith in 1939 while working at RCA in the United States. Smith once said, “When I can use a slide rule, I’m interested in graphically representing mathematical relationships.”

The basics of the Smith chart lie in the following formula. where Γ represents the reflection coefficient of its line

That is, S11 and ZL in the S-parameter are the normalized load value ZL / Z0. Among them, ZL is the load value of the line itself, and Z0 is the characteristic impedance (intrinsic impedance) value of the transmission line, usually  50Ω  is used.

The circular line in the chart represents the real value of the electrical impedance, that is, the resistance value, and the horizontal line in the middle and the lines radiating upward and downward represent the imaginary value of the electrical impedance, which is generated by the capacitance or inductance at high frequencies. Resistance, where upwards are positive numbers and downwards are negative numbers.

The middle point of the graph (1+j0) represents a matched resistor value (ZL), and its reflection coefficient value will be zero.

The edge of the graph represents the length of the reflection coefficient of which is 1, which is 100% reflection.

The numbers on the sides of the graph represent the angle (0-180 degrees) and wavelength (from zero to half wavelength) of the reflectance.

Some graphs are expressed in admittance values, which can be rotated 180 degrees from the impedance version above.

Simply speaking, it is similar to a mathematical table, and the value of the reflection coefficient is known by searching.

### 2.How comes the“Smith Chart”?

First, you need to understand what the impedance of a resistor, capacitor, and inductor is.

In circuits with resistance, inductance, and capacitance, the resistance to current flow in the circuit is called impedance.

Impedance is usually represented by Z, which is a complex number, and called actual resistance while virtually called as reactance. The blocking effect of capacitance on alternating current in the circuit is called capacitive reactance, and the blocking effect of inductance on alternating current in the circuit is called inductance. The resistance of capacitance and inductance to alternating current in a circuit is collectively called reactance. The unit of impedance is ohm.

R, resistance: In the same circuit, the current through a conductor is proportional to the voltage across the conductor and inversely proportional to the resistance of the conductor, which is Ohm’s law.

Standard formula： (The ideal resistance is a real number, not involving the concept of complex numbers).

If the concept of complex numbers in mathematics is introduced, resistance, inductance, and capacitance can be represented by the same form of complex impedance. Which means: resistance is still a real number R (real part of complex impedance), capacitance and inductance are represented by virtual numbers, respectively:

Z= R+i( ωL–1/（ωC）)

Description: The load is a complex of three types of resistance, inductive reactance of inductance, and capacitive reactance of capacitor. After the combination, it is collectively referred to as “impedance”, which is written as a mathematical formula:

impedance Z= R+i(ωL–1/(ωC)).

Where R is the resistance, ωL is the inductive reactance, and 1/(ωC) is the capacitive reactance.

• If (ωL–1/ωC) > 0, it is called “inductive load”;
• On the contrary, if (ωL–1/ωC) < 0, it is called “capacitive load”.

Impedance formula, it is no longer a real number. It becomes a complex number because of the existence of capacitance and inductance.

In a complex plane, with the real part as the x-axis and the imaginary part as the y-axis, any complex number is represented. The impedance at this time, no matter how many resistors, capacitors, and inductors are connected in series or in parallel, can be represented in a complex plane.

Various impedance situations form this infinite plane and become the “Smith chart”.

As the signal travels along the transmission line, it experiences a transient impedance at every moment, which may be the transmission line itself or other components in the middle or at the end. For the signal, it doesn’t distinguish what it is, only the impedance is felt by the signal. If the perceived impedance of the signal is constant, then it will propagate forward normally, as long as the perceived impedance changes, no matter what causes it (maybe resistance, capacitance, inductance, via-holes, PCB corners, or connectors encountered halfway through), the signal will be reflected.

An important indicator to measure the amount of signal reflection is the reflection coefficient, which represents the ratio of the reflected voltage to the original transmission signal voltage.

To reduce the number of unknown parameters, it is possible to fix a parameter that occurs frequently and is often used in the application. Here Z0 (characteristic impedance) is usually a constant and a real number, and is a normalized standard value that is commonly used, such as 50Ω, 75Ω, 100Ω, and 600Ω.

3.The Usage of “Smith chart”:

The Smith chart is used for impedance matching between high frequency circuits.

Impedance represents the resistance ability of a circuit to electricity and is a vector. The impedance value is a complex number.

Impedance matching is one of the basic requirements for circuit connection. Impedance mismatch will cause the circuit to work inefficiently, work abnormally, or even burn directly.

For the impedance matching of high-frequency circuits, to be simply put, the input resistance and the output resistance are purely resistive, such as equal to 50 ohms or 75 ohms.

The main purpose of the Smith chart is to calculate the parameter values of the capacitors and inductors connected to the circuit. The second is to display the frequency-impedance characteristics of a circuit.

Impedance matching

Impedance matching is a part of microwave electronics. The load impedance and the internal impedance of the excitation source are matched with each other to obtain a working state of maximum power output.

It is mainly used on transmission lines to achieve all high-frequency microwave signals transmitted to the point of load, and no signal will be reflected to the source point, thereby improving energy efficiency.

For circuits with different characteristics, the matching conditions are different. In a pure resistance circuit, when the load resistance is equal to the internal resistance of the excitation source, the output power is the largest, and this working state is called matching, otherwise it is called mismatching.

When the internal impedance of the excitation source and the load impedance contain reactance components, in order to obtain the maximum power for the load, the load impedance and the internal resistance must satisfy the conjugate relationship, that is, the resistance components are equal, and the reactance components are only equal in value and opposite in sign. This matching condition is called conjugate matching.

### 4.How to use “Smith Chart”:

The Smith chart is equivalent to a map. Each point on it represents a impedance value in plural form. The center of the circle is called the matching point. It generally represents an ideal impedance of the real  50 ohms and the imaginary 0 ohms.

Impedance matching is to plan a line from the impedance point to the matching point.

In simple terms, it is to look at one line, two arcs and two circles.

Each point on the Smith chart has a real part value and an imaginary part value, and there are two types of lines on the chart too, the real part line and the imaginary part line. Looking at the picture is to find the value of the real part along the line of equal imaginary parts, and then find the value of the imaginary part along the line of equal real parts.

A line and two arcs are lines of equal imaginary parts; two circles are lines of equal real parts.

A line: resistance line. Divide the circle into upper and lower halves. The upper part is the inductance area, and the imaginary part of all points in this area is positive; the lower half is the capacitance area, and the imaginary part of all points in this area is negative. The impedance value of the imaginary part of the resistance line itself is neither positive nor negative, and the impedance value of each point above is 0 ohms.

The middle point of the resistance line is the center of the circle, which is the standard resistance value, usually  50 ohms .