Extract S-Parameters from Circuit

This example uses:

This example uses the Symbolic Math Toolbox™ to explain how the RF Toolbox™ extracts two-port S-parameters from an RF Toolbox circuit object.

Consider a two-port network as shown in the figure 1that you want to characterize with S-parameters. S-parameters are defined as $V - I \times Z_{0} = S (V + I \times Z_{0})$ .

Figure 1: Two-Port Network

To extract the S-parameters from a circuit into an sparameters object, the RF Toolbox terminates each port with the reference impedance $Z_{0}$ . Then, the RF Toolbox independently drives each port j, with $\frac{1}{Z_{0}}$ and solves for the port voltages $V_{ij}$ . Driving with current sources is the Norton equivalent of driving with a 1 V source and a series resistance of $Z_{0}$ .

Measure the port voltage $V_{ij}$ at node i when node j is driven.

If i $\neq$ j, the S-parameter entry $S_{ij}$ is simply twice the port voltage $V_{ij}$ , and this is given using the equation $S_{ij} = 2 \times V_{ij}$ .
The diagonal entries of S-parameters when $i = j$ are given using the equation $S_{ij} = 2 \times V_{ij} - 1$ .

Figure 2: Circuit Driven at Port 1 with Current Source

Write Constitutive and Conservative Equations of Circuit

Circuits are represented in node-branch form in the RF Toolbox. There are four branches in the circuit represented in figure 2, one for the input port, two for the two-port nport object, and one for the output port. This means that the circuit has four branch current unknowns $I_{S}$ , $I_{1}$ , $I_{2}$ , and $I_{L}$ and two node voltages $V_{11}$ and $V_{21}$ . To represent the circuit described in figure 2 in node-branch form, you need four constitutive equations to represent the branch currents and two conservative equations to represent the node voltages.

syms F IS I1 I2 IL V1 V2 Z0
syms S11 S12 S21 S22

nI = 4; % number of branch currents
nV = 2; % number of node voltages

% F = [Fconstitutive; Fconservative]
F = [
    V1 - Z0*IS
    V1 - Z0*I1 - S11*(V1+Z0*I1) - S12*(V2+Z0*I2)
    V2 - Z0*I2 - S21*(V1+Z0*I1) - S22*(V2+Z0*I2)
    V2 - Z0*IL
    IS+I1
    I2+IL
    ]

F = 
 $(\begin{array}{c} V_{1} - IS Z_{0} \\ V_{1} - I_{1} Z_{0} - S_{11} (V_{1} + I_{1} Z_{0}) - S_{12} (V_{2} + I_{2} Z_{0}) \\ V_{2} - I_{2} Z_{0} - S_{21} (V_{1} + I_{1} Z_{0}) - S_{22} (V_{2} + I_{2} Z_{0}) \\ V_{2} - IL Z_{0} \\ I_{1} + IS \\ I_{2} + IL \end{array})$

Jacobian Evaluation of Circuit

Use the jacobian function from the Symbolic Math Toolbox to compute the matrix of derivatives of the function F with respect to the six unknowns (four branch currents and two node voltages)

J = jacobian(F,[IS; I1; I2; IL; V1; V2])

J = 
 $(\begin{array}{c} - Z_{0} & 0 & 0 & 0 & 1 & 0 \\ 0 & - Z_{0} - S_{11} Z_{0} & - S_{12} Z_{0} & 0 & 1 - S_{11} & - S_{12} \\ 0 & - S_{21} Z_{0} & - Z_{0} - S_{22} Z_{0} & 0 & - S_{21} & 1 - S_{22} \\ 0 & 0 & 0 & - Z_{0} & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \end{array})$

Solve S-Parameters of Circuit

Create a two-column right-hand side vector, rhs, to represent the driving of each port.

syms rhs [nI+nV 2]
syms x v S

% Compute S-parameters of cascade
rhs(:,:) = 0;
rhs(nI+1,1) = 1/Z0;  % rhs for driving input port
rhs(nI+nV,2) = 1/Z0  % rhs for driving output port

rhs = 
 $(\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \frac{1}{Z_{0}} & 0 \\ 0 & \frac{1}{Z_{0}} \end{array})$

To solve for the voltages, back solve the rhs with the Jacobian. The S-parameter matrix that MATLAB outputs represents the two-port circuit shown in figure 1.

x = J \ rhs;
v = x(nI+[1 nV],:);
S = 2*v - eye(2)

S = 
 $(\begin{array}{c} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array})$

Extract S-Parameters from Circuit

Write Constitutive and Conservative Equations of Circuit

Jacobian Evaluation of Circuit

Solve S-Parameters of Circuit

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