Facing the "Smith Chart" calmly, no longer confused
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No matter how classic the RF tutorial is, why is it always in black and white? This makes students who want to understand Smith's original picture look confused.
What is this?Today we answer three questions:
1. What is it?
2. Why?
3. What are you doing?
1. What is it?
The diagram was invented in 1939 by Phillip Smith, who was working for RCA in the United States. Smith once said, "As early as I was able to use a slide rule, I became interested in expressing mathematical connections graphically."
The basis of the Smith chart is the following formula.
The Γ here represents the reflection coefficient of the line.
That is, S11 in the S-parameter, and ZL is the normalized load value, that is, 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.
Simply put: it is similar to using a mathematical table. By looking it up, you can find the value of the reflection coefficient.
2. Why?
We still don't know how Mr. Smith came up with the inspiration for the "Smith Chart" representation method.
Many students look at Smith's original drawings and memorize them by rote, but they don't get the point. In fact, they don't understand Mr. Smith's creative intention.
My personal guess is: maybe it was inspired by Riemannian geometry that the coordinate system of a plane was "bent" .
I am describing the process of "bending", and you will understand the meaning of this picture. (The coordinate system can be bent, but people should try not to bend; if you have already bent, I would like to express my blessing)
Now, I'll bend it for you to see.
The world map is actually a process of using a plane to represent a sphere, and this process is called "straightening".
The cleverness of Smith's original drawing lies in using a circle to represent an infinite plane.
2.1. First, let us understand the "infinite" plane.
First, let's review the impedance of ideal resistors, capacitors, and inductors.
In a circuit with resistance, inductance and capacitance, the resistance to the current in the circuit is called impedance. Impedance is often represented by Z, which is a complex number. The actual is called resistance, and the imaginary is called reactance. The resistance of the capacitor to the alternating current in the circuit is called capacitive reactance, and the resistance of the inductor to the alternating current in the circuit is called inductive reactance. The resistance caused by the capacitor and inductor to the alternating current in the circuit is collectively called reactance. The unit of impedance is ohm.
R, resistance: In the same circuit, the current passing through a conductor is directly proportional to the voltage across the conductor and inversely proportional to the resistance of the conductor. This is Ohm's law.
Standard formula:  . (The ideal resistor is a real number and does not involve the concept of complex numbers).
If we introduce the concept of complex numbers in mathematics, we can express resistance, inductance, and capacitance in the same form of complex impedance. That is, resistance is still a real number R (the real part of complex impedance), and capacitance and inductance are expressed in imaginary numbers, which are:
Z= R+i( ωL–1/（ωC）)
Note: Load is a complex of three types: resistance, inductive reactance of inductance, and capacitive reactance of capacitance. The composite is collectively called "impedance", which can be written as a mathematical formula: Impedance Z = R + i (ωL – 1/ (ωC)). R is resistance, ωL is inductive reactance, and 1/ (ωC) is capacitive reactance.
(1) If (ωL–1/ωC) > 0, it is called “inductive load”;
(2) Conversely, if (ωL–1/ωC) < 0, it is called “capacitive load”.
If we look closely at the impedance formula, we will find that it is no longer a real number. It becomes a complex number because of the presence of capacitance and inductance.
If there is only resistance in the circuit, it only affects the amplitude change.
From the above figure, we know that the amplitude of the sine wave has changed, and at the same time, the phase has changed, and the frequency characteristics will also change. Therefore, in the calculation process, we need to consider both the real part and the imaginary part.We can represent any complex number in a complex plane, with the real part as the x-axis and the imaginary part as the y-axis. Our impedance, no matter how many resistors, capacitors, and inductors are connected in series or in parallel, can be represented in a complex plane.
In the RLC series circuit, the AC power supply voltage U = 220 V, frequency f = 50 Hz, R = 30 Ω, L = 445 mH, C = 32 mF.
In the figure above, we can see that through the superposition of several vectors, the final impedance falls at the blue dot in the complex plane.Therefore, we can place the calculated result of any impedance at the corresponding position on this complex plane.
Various impedance conditions make up this infinite plane.
2.2 Reflection formula
When a signal propagates forward along a transmission line, it will feel a transient impedance at every moment. This impedance may be the transmission line itself, or it may be other components in the middle or at the end. For the signal, it will not distinguish what it is, and the signal will only feel the impedance. If the impedance felt by the signal is constant, then it will propagate forward normally. As long as the impedance felt changes, no matter what causes it (it may be resistance, capacitance, inductance, vias, PCB corners, connectors encountered in the middle), the signal will be reflected.
The Qiantang River tide is caused by the change in the width of the river, which causes reflection. This can be compared to the impedance discontinuity in the circuit, which causes signal reflection. The energy accumulated by the reflections is superimposed, causing overshoot. Maybe this metaphor is not appropriate, but it is quite vivid.
 So how much is reflected back to the starting point of the transmission line? An important indicator for measuring the amount of signal reflection is the reflection coefficient, which represents the ratio of the reflected voltage to the original transmitted signal voltage.The reflection coefficient is defined as:
Where: Z0 is the impedance before the change, and ZIN is the impedance after the change. Assuming that the characteristic impedance of the PCB line is 50 ohms, a 100 ohm chip resistor is encountered during the transmission process. The influence of parasitic capacitance and inductance is not considered for the time being, and the resistor is regarded as an ideal pure resistor, then the reflection coefficient is:
One third of the signal is reflected back to the source.If the voltage of the transmitted signal is 3.3 V, the reflected voltage is 1.1 V. The reflection of a pure resistive load is the basis for studying the reflection phenomenon. The changes in resistive loads are nothing more than the following four situations: impedance increases by a finite value, decreases by a finite value, open circuit (impedance becomes infinite), and short circuit (impedance suddenly becomes 0).
The initial voltage is the source voltage Vs (2V) divided by Zs (25 ohms) and the transmission line impedance (50 ohms).
Vinitial=1.33V
The subsequent reflectivity is calculated according to the reflection coefficient formula
The reflectivity at the source end is calculated as -0.33 based on the reflection coefficient formula based on the source impedance (25 ohms) and the transmission line impedance (50 ohms);
The reflectivity of the terminal is calculated as 1 based on the reflection coefficient formula based on the terminal impedance (infinity) and the transmission line impedance (50 ohms);
We superimpose the amplitude and delay of each reflection on the initial pulse waveform to obtain this waveform. This is why impedance mismatch causes poor signal integrity.
Let us make an important assumption!
In order to reduce the number of unknown parameters, a parameter that often appears and is often used in applications can be solidified. Here, Z0 (characteristic impedance) is usually a constant and a real number, which is a commonly used normalized standard value, such as 50Ω, 75Ω, 100Ω, and 600Ω.
Assume that Z0 is fixed at 50 ohms. (Why it is 50 ohms is not explained here for the time being; of course, other assumptions can be made, but for ease of understanding, we will first fix it at 50Ω).
Then, according to the reflection formula, we get an important conclusion:
Each Zin corresponds to a unique "Γ", the reflection coefficient.We depict the corresponding relationship on the "complex plane" we just mentioned.
We can then define the normalized load impedance:
Based on this, the formula for the reflection coefficient can be rewritten as:
Okay, now we are in the complex plane, forget about Zin, and just remember z (lowercase) and the reflection coefficient “Γ”.
All the preparations are done, now we are ready to "bend"
2.3 Bending
In the complex plane, there are three points where the reflection coefficient is 1, which means the horizontal coordinate is infinite and the vertical coordinate is positive and negative infinity. One day in history, Mr. Smith, as if with divine help, bent the black line and brought the three points marked with red circles in the above picture together.Bend, bend
It’s round, it’s round.
Perfect Circle:
Although the infinite plane becomes a circle, the red line is still the red line and the black line is still the black line.At the same time, we add three lines to the original complex plane, and they also bend as the plane closes.
The impedance on the black line has a characteristic: the real part is 0; (resistance is 0)The impedance on the red line has a characteristic: the imaginary part is 0; (inductance and capacitance are 0)
The impedance on the green line has a characteristic: the real part is 1; (the resistance is 50 ohms)
The impedance on the purple line has a characteristic: the imaginary part is -1;
The impedance on the blue line has a characteristic: the imaginary part is 1;
The impedance characteristics on the line are translated from the complex plane to the Smith original diagram, so the characteristics follow the color and remain unchanged.
The lower semicircle is divided in the same way as the working circle.
Because the Smith chart is a graphical solution, the accuracy of the result depends directly on the accuracy of the graph. The following is an RF application example represented by the Smith chart:
Example: The characteristic impedance is known to be 50Ω, and the load impedance is as follows:
| Z1 = 100 + j50Ω |
Z2 = 75 - j100Ω |
Z3 = j200Ω |
Z4 = 150Ω |
| Z5 = ∞ (an open circuit) |
Z6 = 0 (a short circuit) |
Z7 = 50Ω |
Z8 = 184 - j900Ω |
The above values are normalized and plotted in a circular graph (see Figure 5):
| z1 = 2 + j |
z2 = 1.5 - j2 |
z3 = j4 |
z4 = 3 |
| z5 = 8 |
z6 = 0 |
z7 = 1 |
z8 = 3.68 - j18 |
We can't see the picture above clearly.
If it is "series", we can first determine the real part on the clear Smith original diagram (search on the red line, the horizontal coordinate of the original complex plane), and then slide along the arc according to the positive and negative of the imaginary part to find our corresponding impedance. (Ignore the green line in the figure below for now)
Now we can solve the reflection coefficient Γ directly from the circle diagram.We can directly read the value of the reflection coefficient through rectangular coordinates or through polar coordinates.
Cartesian Coordinates
Draw the impedance points (intersection points of equal impedance circles and equal reactance circles), and read their projections on the horizontal and vertical axes of the rectangular coordinates to obtain the real part Γr and imaginary part Γi of the reflection coefficient (see Figure 6).
There may be eight cases in this example, and the corresponding reflection coefficient Γ can be directly obtained on the Smith chart shown in Figure 6:
| Γ1= 0.4 + 0.2j |
C 2 = 0.51 - 0.4j |
C 3 = 0.875 + 0.48j |
C 4 = 0.5 |
| C 5 = 1 |
C 6 = -1 |
C 7 = 0 |
Γ8= 0.96 - 0.1j |
Read the real and imaginary parts of the reflection coefficient Γ directly from the XY axis
Polar Coordinates
Polar coordinates, what is it useful for? Very useful, and this is actually the purpose of Smith's original diagram.2.4 Red Camp VS Green Camp
We have just noticed that in the Smith original graph, in addition to the red curve, which is the red world bent from the impedance complex plane, there are also green curves in the graph, which are bent from the admittance complex plane. The process is the same as before.So what is the use of this green color of admittance?
For parallel circuits, it is very convenient to use admittance calculation. At the same time, in the Smith original diagram, it is also very convenient to use the green curve of admittance for query.
As shown in the figure, by connecting a capacitor in parallel, the corresponding normalized impedance and reflection coefficient can be quickly found through the green curve.
3. What are you doing?
After explaining and introducing the Smith chart for such a long paragraph, don't forget what we want to do. We actually hope that the reflection coefficient of the circuit we design is as close to 0 as possible.
However, what kind of circuit is a qualified circuit? The reflection coefficient cannot be ideally 0, so what are our requirements for the reflection coefficient?
We hope that the absolute value of the reflection coefficient is less than 1/3, that is, the reflection coefficient falls into the blue area of the Smith chart (as shown below).
What are the characteristics of this blue ball? In fact, we have clearly found it through the values of Smith's original graph. On the central axis, that is, the red line mentioned earlier, there are two positions, 25 ohms and 100 ohms. That is: Zin is between 1/2 Zo and 2 times Zo.That is, if we hit the target in the blue area, we consider the reflection coefficient to be acceptable.
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