Reprinted from reading this "Smith Chart" to say goodbye to bluffing RF!
Article source: http://www.mweda.com/hfss-cst-29038-1.html
what is this?
Answer three questions today:
1. What is it?
2. Why?
3. What for?
1. What is it?
The chart was invented by Phillip Smith in 1939 when he was working for RCA in the United States. Smith once said, "When I was able to use a slide rule, I was very interested in expressing mathematical connections in graphs."
The basis of the Smith chart lies in the following formula.
Representative among its line Γ reflectance (reflection coefficient)
Namely S11 in S-parameter, ZL is the normalized load value, namely 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 .
Simply put: Just like a mathematical table, you can find the value of the reflection coefficient through search.
2. Why?
We don’t know now how Mr. Smith came up with the inspiration for the “Smith Chart” representation method.
Many students looked at Smith’s original picture and memorized the shit to no point. In fact, they did not try to figure out the creative intention of Mr. Smith.
I personally speculate: Is it inspired by Riemannian geometry to "bend" a plane coordinate system.
The world map is actually a process of using a plane to represent a sphere. This process is a "straightening".
Smith’s original drawing is ingenious in that it uses a circle to represent an infinite plane.
2.1 First, we first understand the "infinity" plane
First of all, let's review the ideal resistance, capacitance, and inductor impedance.
In a circuit with resistance, inductance and capacitance, the hindrance to the current in the circuit is called impedance. Impedance is often represented by Z, which is a complex number. The real part is called resistance, and the imaginary part is called reactance. The obstructive effect of capacitance on alternating current in the circuit is called capacitive reactance, and the obstructive effect of inductance on alternating current in the circuit is called For inductive reactance, the obstructive effect of capacitance and inductance on 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 certain conductor is proportional to the voltage at both ends of the conductor and inversely proportional to the resistance of the conductor. This is Ohm's law.
(The ideal resistance is a real number and does not involve 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. That is, the resistance is still a real number R (the real part of the complex impedance), and the capacitance and inductance are represented by imaginary numbers, respectively:
Explanation: The load is a complex of three types: resistance, inductive reactance of inductance, and capacitive reactance of capacitors, which are collectively referred to as "impedance" after being combined, and the mathematical formula is: impedance Z= R+i(ωL–1/(ωC)) . Among them, 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) On the contrary, if (ωL–1/ωC)<0, it is called “capacitive load”;
We look carefully at the impedance formula, it is no longer a real number. It becomes a complex number because of the existence of capacitance and inductance.
If there is only resistance in the circuit, it only affects the amplitude change.
Through the above figure, we know that the amplitude of the sine wave has changed, at the same time, the phase has also changed, and the frequency characteristics will also change. Therefore, we need to consider the real part and the imaginary part in the calculation process.
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 all be expressed in a complex plane.
In the RLC series circuit, AC power supply voltage U = 220 V, frequency f = 50 Hz, R = 30 Ω, L = 445 mH, C = 32 mF.
In the above figure, we see that through the superposition of several vectors, the final impedance falls on the blue dot in the complex plane.
Therefore, the calculation result of any impedance can be placed in the corresponding position of the complex plane.
Various impedance conditions constitute this infinite plane.
2.2. Reflection formula
When the signal propagates forward along the transmission line, a transient impedance will be felt every moment. This impedance 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, all the signal feels is the impedance. If the impedance felt by the signal is constant, then it will propagate forward normally, as long as the sensed impedance changes, no matter what caused it (may be resistance, capacitance, inductance, vias, PCB corners encountered in the middle) , Connector), the signal will reflect.
The Qiantang River tide is the reflection caused by the change in the width of the river channel. This is analogous to the discontinuity of the impedance in the circuit and the signal reflection. The reflected energy is superimposed, causing overshoot. Maybe this analogy 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 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.
The reflection coefficient is defined as:
Among them: 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 and a 100 ohm chip resistance is encountered during transmission, the parasitic capacitance and inductance are not considered for the time being, and the resistance is regarded as an ideal pure resistance, then the reflection coefficient is:
1/3 of the signal is reflected back to the source.
If the voltage of the transmission signal is 3.3V, the reflected voltage is 1.1V. The reflection of a purely resistive load is the basis for the study of reflection phenomena. The change of resistive load is nothing more than the following four situations: impedance increases by a finite value, decreases by a finite value, open circuit (impedance becomes infinite), 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 of the source end is calculated as -0.33 based on the source end impedance (25 ohms) and the transmission line impedance (50 ohms) according to the reflection coefficient formula;
The reflectivity of the terminal is calculated as 1 based on the terminal impedance (infinity) and the transmission line impedance (50 ohm) according to the reflection coefficient formula;
We get this waveform by superimposing the initial pulse waveform according to the amplitude and delay of each reflection. This is why the impedance mismatch causes poor signal integrity .
Then we make an important assumption!
In order to reduce the number of unknown parameters, a parameter that often appears and is often used in the application can be cured. 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 constant and is 50 ohms. (Why is 50 ohms, not shown here for the time being; of course, other assumptions can also be made to facilitate understanding, we first set 50 ohms).
Then, according to the reflection formula, we get an important conclusion:
Each Zin corresponds to a unique "Γ", the reflection coefficient.
We describe the correspondence to the "complex plane" we just mentioned.
So we can define the normalized load impedance:
Accordingly, the formula of reflection coefficient is rewritten as:
Well, we are in the complex plane, forget Zin, only remember z (lowercase) and the reflection coefficient "Γ".
The preparations are done, and we are ready to "bend"
2.3 Bending
In the complex plane, there are three points with a reflection coefficient of 1, which is the infinity of the abscissa and the positive or negative infinity of the ordinate. One day in history, old Mr. Smith, if he had divine help, bend the black line and pinch the points marked by the three red circles in the picture above.
Perfect circle:
Although the plane of infinity becomes a circle, the red line is still red, and the black line is still black.
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 original Smith image, so the characteristics follow the color, and the characteristics remain unchanged.
The bottom half circle is the same as the work circle.
Because the Smith chart is a graph-based solution, the accuracy of the result directly depends on the accuracy of the graph. The following is an RF application example represented by a Smith chart:
Example: The characteristic impedance is known to be 50Ω, and the load impedance is as follows:
Normalize the above values and mark them in the circle chart (see Figure 5):
We can't see the picture above.
If it is "series", we can first determine the real part (search on the red line, the abscissa of the original complex plane) on the clear Smith chart, and then slide along the arc according to the sign of the imaginary part to find our corresponding的impedance. (Ignore the green line in the figure below)
Now the reflection coefficient Γ can be solved directly from the circle graph.
We can either directly read the value of reflection coefficient through rectangular coordinates, or read the value of reflection coefficient through polar coordinates.
Cartesian coordinates
Draw the impedance point (the intersection of the equal impedance circle and the equal reactance circle) , just read their projections on the horizontal and vertical axes of the rectangular coordinate, and get the real part Γr and imaginary part Γi of the reflection coefficient (see Figure 6) .
There may be eight situations in this example, and the corresponding reflection coefficient Γ can be directly obtained on the Smith chart shown in Figure 6:
Read the real and imaginary parts of the reflection coefficient Γ directly from the XY axis
What's the use of polar coordinates? Very useful, this is actually the purpose of Smith's original picture.
2.4 Red camp vs. green camp
We have just noticed that Smith’s original picture, except for the red curve, is a red world that is bent from the impedance complex plane. At the same time, in the figure, there are also green curves, which are generated from the admittance complex plane and bend. The process is the same as the process just now.
So what is the use of this admittance green?
Parallel circuit, we will use admittance calculation, we will be very convenient. At the same time, in the original Smith chart, we use the admittance green curve for query, which is also very convenient.
As shown in the figure, such a capacitor is connected in parallel, and the corresponding normalized impedance and reflection coefficient can be quickly queried through the green curve.
3. What are you doing?
I explained and introduced the long paragraph of the Smith chart, don’t forget, what we want to do. We actually hope that the closer the reflection coefficient of the circuit we design is to 0, the better .
But, 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 discovered the value of Smith's original picture. On the central axis, which is the red line mentioned earlier, are 25 ohms and 100 ohms respectively. Namely: Zin is between 1/2 Zo and 2 times Zo.
That is, when we hit the target in the blue area, we think the reflection coefficient is acceptable.
Reprinted from reading this "Smith Chart" to say goodbye to bluffing RF!