9. Circuit synthesis-microstrip circuit design based on arbitrary amplitude-frequency response of simplified real frequency

Chapters 1-8 in the Overview of Network Synthesis and Simplified Real Frequency Theory Learning introduced some basic concepts and experimental methods of SRFT, and finally reached the ultimate use of SRFT, which is to directly synthesize a microstrip circuit given any response.

1. Microstrip circuit design with arbitrary amplitude-frequency response

We demonstrate the design process of a classical microwave filter using corresponding transmission lines. The first step in the design method is to choose an appropriate transfer function. Obviously, the form of the transfer function must reflect the design requirements. For many practical situations it may be sufficient to use off-the-shelf low-pass type prototype functions constructed from Butterworth or Chebyshev polynomials.

However, for some applications, the desired shape of the transducer power gain (TPG) can be determined in any way by the problem under consideration. For example, the gain distortion of a communication channel can be equalized using a lossless filter that compensates for channel distortion.

Of course, this design method can also be used for design analysis of matching circuits (generating corresponding microstrip line circuits based on matching targets).

2. Some previous reviews and summaries

Some cases of circuit synthesis have been given before, but these cases must be based on S parameters in the form of analytical functions. In more general cases, it is difficult for us to complete the design based on this. We often need to base the design on the amplitude of the circuit to be designed. Frequency response and phase-frequency response to design the actual circuit:

5. Circuit synthesis – super cool – directly synthesize the microstrip circuit diagram based on S11 parameters
Based on the expression of the given S11 parameter, the corresponding microstrip circuit diagram is synthesized. Note that the S11 parameter expression here needs to be in analytical form (that is, it must be in the form of a function expression)

6. Circuit synthesis-design of SRFT microstrip line-cut Byshev low-pass filter based on simplified real frequency
Circuit synthesis is performed based on the Chebyshev function, and the circuit synthesis is performed directly based on the target parameters to obtain the corresponding microstrip circuit. The corresponding theory and operation steps are given (with Matlab code)

7. Circuit synthesis – SRFT microstrip line Butterworth low-pass filter design based on simplified real frequency
Conduct circuit synthesis based on the Butterworth function and directly conduct circuit synthesis based on the target parameters to obtain the corresponding microstrip circuit. The corresponding theory and operation steps are given (with Matlab code)

8. Circuit synthesis-bandpass filter design based on simplified real frequency SRFT microstrip line
Conduct circuit synthesis based on Butterworth and Chebyshev functions, directly conduct circuit synthesis based on target parameters to obtain the corresponding bandpass microstrip circuit, and provide the corresponding theory and operation steps (with Matlab code)

3. Microstrip circuit design steps with arbitrary amplitude-frequency response

Suppose I need to design a microstrip line circuit with the following amplitude-frequency characteristics. The frequency f1 is 0GHz. The modulus value of S21 at this frequency is 1; the frequency f2 is 1GHz , the modulus value of S21 at this frequency is 0, and the modulus value of S21 is required to be 0 within 1-3GHz (this characteristic is used to determine the electrical length of the microstrip line).

STEP1: Realize the design goal and convert the design goal into the form of an array. The frequency range covered by the array is 0-3GHz. Here, an array with a length of 58 is used to store this goal. It is worth noting that this array exceeds Large optimizations will also be slower:

STEP2: Determine the circuit structure required to implement this structure. The main thing that needs to be determined is the order n to achieve this amplitude-frequency response. Through the study of the previous tutorials, we know that the circuit-based structure can directly determine F (lamda), You only need to determine the coefficient of H (lamda) to perform circuit synthesis.

This means that we need to obtain the coefficients of H (lamda) that meet the above amplitude-frequency response, and then we can use SRFT for circuit synthesis. This step requires the use of an optimization algorithm to solve the coefficients. The defined error function is as follows (simply put, it is the actual value minus the target value. The optimization goal is to minimize the final error to 0):

In MATLAB, this optimization process can be completed using lsqnonlin, and its usage tutorials such as nonlinear least squares solver

STEP3: After a certain number of iterative solutions, a corresponding set of solutions can be obtained. This solution is the coefficient of h, and circuit synthesis is carried out. For the specific synthesis process, please refer to the previous tutorial.

4. Matlab programming implementation

clear
clc

% Use 1GHZ microstrip line, control up to 3GHz
f=1;
fe=3;
we=2*pi*fe;
tau=pi/2/we;
ele_l=360*tau*f;
%Use 6 cascaded microstrip lines for design
k=6;
%There is no zero point at DC
q=0;
%Initialize the coefficient of H
h=[1 1 1 1 1 1 1];


disp(['Here we use a microstrip line with a power-off length of',num2str(f/1e9),'GHz of',num2str(ele_l),'° for implementation']);


%0-3GHz points
Nopt=57;
fr_opt=linspace(0,3,Nopt + 1);
% Define the modulus value of S21 parameter
mag=[1,0.952631578947368,0.905263157894737,0.857894736842105,0.810526315789474,0.763157894736842,0.715789473684211...
    ,0.668421052631579,0.621052631578947,0.573684210526316,0.526315789473684,0.478947368421053,0.431578947368421...
    ,0.384210526315789,0.336842105263158,0.289473684210526,0.242105263157895,0.194736842105263,0.147368421052631...
    ...
    ...
    ...
    ...
    ...
    ...
    ,0.0100000000000000,0.0100000000000000,0.0100000000000000];
% Define the phase, only the amplitude is designed here, the phase is arbitrary
phase=linspace(0,0,Nopt + 1);
% Draw the amplitude of the target circuit
figure (1)
plot(fr_opt,mag)
xlabel('Frequency')
ylabel('MAG-S21')
title('Target S21 parameters of the microstrip line to be designed')


% Run optimization algorithm
n1=length(h);
n=n1-1;
h(n + 1)=0;
for i=1:n
    x0(i)=h(i);
end
% Call optimization with no transformer
OPTIONS=optimset('MaxFunEvals',20000,'MaxIter',50000);
x=lsqnonlin('objective',x0,[],[],OPTIONS,fe,q,k,fr_opt,mag, phase);
h(n + 1)=0;
for i=1:n
    h(i)=x(i);
end

% Calculate other parameters based on h obtained from optimization
[G,H,F,g]=SRFT_htoG(h,q,k);
tau=1/4/fe;
N=length(mag);
j=sqrt(-1);

% Draw the obtained analytical form
for i=1:N
    teta=2*pi*tau*fr_opt(i);
    omega=tan(teta);
    lmbda=j*tan(teta);
    fval=(-1)^q*(lmbda)^q*(1-lmbda^2)^(k/2);
    freal=real(fval);
    fimag=imag(fval);
    gval=polyval(g,lmbda);
    greal=real(gval);
    gimag=imag(gval);
    S21=fval/gval;
    MS21(i)=abs(S21);
    phase_f=atan(fimag/freal);
    phase_g=atan(gimag/greal);
    phase_S21(i)=phase_f-phase_g;
    ph(i)=teta;
    fr(i)=fr_opt(i);
end
figure(2)
plot(fr,MS21, fr_opt,mag)
xlabel('Frequency')
ylabel('MS21, modula')
title('Arbitraray amplitude approximation via SRFT')
%----------------------------------------
% Comprehensively obtain the required microstrip circuit
[Z_imp]=UE_sentez(h,g)

The running results, the picture shows the comparison between ideal and actual:

5. ADS verification

Circuit diagram construction:

The simulation results are as follows, which show that they are basically consistent with the design goals: