Circuit synthesis-super cool-directly synthesize the microstrip circuit diagram based on S11 parameters
1. Comprehensive principles and practice of circuits-comprehensive principles of reactance function
2. Circuit synthesis principle and practice-positive real function and instigated electrical impedance function
3. Circuit synthesis principle and practice-single and double-port ideal microstrip line (pseudo) hand calculation of S parameters and time domain waveforms
We have previously introduced how to obtain the S parameters by solving the microstrip line structure of the circuit (3. Comprehensive circuit principles and practices-single and double-port ideal microstrip line (pseudo) hand calculation of S parameters and time domain waveforms). Here is an introduction Something particularly cool is that the S11 parameters of the microstrip line give you a structure that can be used to implement it.
Directly obtaining the microstrip line circuit diagram from the S11 parameters is a technology called simplified real frequency, which can be used in the design of matching circuits. For example, if I need to match 10 ohms to 50 ohms at 1GHz, I only need to control the S11 parameters of 1GHz at 10 ohms. Just do it (you need to convert S parameters into Z parameters first).
1. Expression of S11 parameter of Richard domain
The parameter expression of S11 here is in the Richard domain, which may not be consistent with the same format as the S11 image we usually see, such as:
So what exactly does this formula represent? For a lossless transmission line, it can be expressed in the following form:
Among them, beta is the phase shift constant, and l is the actual length of the microstrip line. In the following analysis, I use the indirect expression method of electrical length to illustrate. For example, the actual length of a microstrip line with an electrical length of 60° at 1GHz isl=ele_l/360*c/f. Where ele_l is the electrical length (60, unit degree), c is the speed of light, and f is the frequency at which the electrical length is located (1GHz).
In addition, beta=beta=2* pi *freq_solve/c, where freq_solve is the frequency of solution (that is, which frequency S parameter needs to be solved). In this way, corresponding to a fixed expression, its S11 parameter will change with frequency. In addition, because lambda is a periodic function, the final S11 parameter must also be periodic.
For the above S11 expression, use the following program to draw its corresponding S11 parameters (The microstrip line uses a microstrip line with an electrical length of 60 degrees at 1GHz, and the corresponding actual length is 0.05m, ignoring the dielectric constants etc.):
clear
close all
clc
%Microstrip line electrical length
ele_l=60;
%The frequency at which the electrical length of the microstrip line lies
f=1e9;
%Solution frequency range, unit GHz
f_start=0.01;
f_stop=10;
f_step=0.01;
%speed of light
c=299792458;
%Solution range
freq_solve=[f_start:f_step:f_stop]*1e9;
% Calculate the physical length, unit m
l=ele_l/360*c/f;
%Calculate the phase shift constant beta at different frequencies
beta=2*pi*freq_solve/c;
%Convert to lambda domain
lamda=1j*tan(beta*l);
S11=(lamda.^2 + 19*lamda)./(lamda.^2 + 21*lamda + 8);
figure
plot(freq_solve/1e9,20*log10(abs(S11)))
xlabel('Frequency(GHz)')
ylabel('dB(S11)')
title('S11')
The resulting picture looks like this:
2. Solving the theoretical process
STEP0:
Take the above case as an example:
Combine:
and then:
k=1 means that the number of cascaded microstrip lines is 1.
STEP1: Put lamda=1 into S11 expression:
STEP2: Calculate Zi:
STEP3: Calculate K:
STEP4: Calculate S:
STEP5: Calculate the next level S:
STEP6: Calculate impedance:
3. Circuit Generation Case 1-Open Microstrip Line Synthesis
In a more general case, we design the matching circuit based on the S11 parameters in the last picture of the first part. We can use the S11 parameter to reverse the expression. Here is an example (using the expression from the first part as a case):
clear
clc
syms lambda
S11=(lamda.^2 + 19*lamda)./(lamda.^2 + 21*lamda + 8);
Zin=(1 + S11)/(1-S11);
S11_Z0=S11;
Z0=1;
ind=1;
s11_tmp(ind)=subs (S11_Z0,Z0);
for cnt=1:1:2
if ind==1
Z(ind)=Z0*(1 + s11_tmp(ind))/(1-s11_tmp(ind));
K(ind)=(Z0-Z(ind))/(Z0 + Z(ind));
S11_Zi(ind)=(K(ind) + S11_Z0)/(1 + K(ind)*S11_Z0);
S111_Zi(ind)=simplify(S11_Zi(ind)*(1 + lamda)/(1-lamda));
ind=ind + 1;
else
Z(ind)=simplify(Z(ind-1)*(1 + S111_Zi(ind-1))/(1-S111_Zi(ind-1)))
Z(ind)=subs (Z(ind),Z0);
K(ind)=(Z(ind-1)-Z(ind))/(Z(ind-1) + Z(ind));
S11_Zi(ind)=(K(ind) + S111_Zi(ind-1))/(1 + K(ind)*S111_Zi(ind-1));
S111_Zi(ind)=simplify(S11_Zi(ind)*(1 + lamda)/(1-lamda));
ind=ind + 1;
end
end
Z
The running results are as follows:
It means that the impedance can be composed of two parts. The first part is a microstrip line with an impedance of 5 ohms, and the second part is a parallel connection of capacitor and resistor:
Based on the analysis results, the following circuit schematic diagram is constructed:
Run the simulation and the simulation results are consistent with the theoretical analysis results:
4. Circuit Generation Case 2-Synthesizing Multi-Microstrip Circuit Based on Actuation Point Impedance Function
The input impedance function to be synthesized is as follows (here t is lambda):
% 100*t^3 + 50*t^2 + 300*t + 30 % Z(t) = ---------------------------------- % 9*t^3 + 170*t^2 + 31*t + 30
Draw the corresponding S11 parameter and Zin parameter curves (assuming that a microstrip line with a power length of 60 degrees at a frequency of 1GHz is used for implementation), use the following code:
clear
close all
clc
%Microstrip line electrical length
ele_l=60;
%The frequency at which the electrical length of the microstrip line lies
f=1e9;
%Solution frequency range, unit GHz
f_start=0.01;
f_stop=10;
f_step=0.01;
%speed of light
c=299792458;
%Solution range
freq_solve=[f_start:f_step:f_stop]*1e9;
% Calculate the physical length, unit m
l=ele_l/360*c/f;
%Calculate the phase shift constant beta at different frequencies
beta=2*pi*freq_solve/c;
%Convert to lambda domain
lamda=1j*tan(beta*l);
Zin=(100*lamda.^3 + 50*lamda.^2 + 300*lamda + 30)./(9*lamda.^3 + 170*lamda.^2 + 31*lamda + 30);
S11=(Zin-1)./(Zin + 1);
figure
plot(freq_solve/1e9,(abs(Zin)))
xlabel('Frequency(GHz)')
ylabel('Zin')
title('Zin')
figure
plot(freq_solve/1e9,20*log10(abs(S11)))
xlabel('Frequency(GHz)')
ylabel('dB(S11)')
title('S11')
The obtained S11 parameter and Zin parameter results are as follows:
Use the following code for synthesis:
clear
clc
syms lambda
Zin=(100*lamda.^3 + 50*lamda.^2 + 300*lamda + 30)./(9*lamda.^3 + 170*lamda.^2 + 31*lamda + 30);
S11=(Zin-1)./(Zin + 1);
S11_Z0=S11;
Z0=1;
ind=1;
s11_tmp(ind)=subs (S11_Z0,Z0);
for cnt=1:1:4
if ind==1
Z(ind)=Z0*(1 + s11_tmp(ind))/(1-s11_tmp(ind));
K(ind)=(Z0-Z(ind))/(Z0 + Z(ind));
S11_Zi(ind)=(K(ind) + S11_Z0)/(1 + K(ind)*S11_Z0);
S111_Zi(ind)=simplify(S11_Zi(ind)*(1 + lamda)/(1-lamda));
ind=ind + 1;
else
Z(ind)=simplify(Z(ind-1)*(1 + S111_Zi(ind-1))/(1-S111_Zi(ind-1)))
Z(ind)=subs (Z(ind),Z0);
K(ind)=(Z(ind-1)-Z(ind))/(Z(ind-1) + Z(ind));
S11_Zi(ind)=(K(ind) + S111_Zi(ind-1))/(1 + K(ind)*S111_Zi(ind-1));
S111_Zi(ind)=simplify(S11_Zi(ind)*(1 + lamda)/(1-lamda));
ind=ind + 1;
end
end
Z
The running results are as follows:
Build ADS simulation verification:
The results are consistent:
