Extract S-Parameters from Circuit

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 $\mathit{V}\mathit{–}\mathit{I}\times {\mathit{Z}}_0=\mathit{S}(\mathit{V}+\mathit{I}\times {\mathit{Z}}_0)$.

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 ${\mathit{Z}}_0$. Then, the RF Toolbox independently drives each port j, with $\frac{1}{{\mathit{Z}}_0}$ and solves for the port voltages ${\mathit{V}}_{\mathit{ij}}$. Driving with current sources is the Norton equivalent of driving with a 1 V source and a series resistance of ${\mathit{Z}}_0$.

Measure the port voltage ${\mathit{V}}_{\mathit{ij}}$ at node i when node j is driven.

• If i $\ne$ j, the S-parameter entry ${\mathit{S}}_{\mathit{ij}}$ is simply twice the port voltage ${\mathit{V}}_{\mathit{ij}}$, and this is given using the equation ${\mathit{S}}_{\mathit{ij}}=2\times {\mathit{V}}_{\mathit{ij}}$.

• The diagonal entries of S-parameters when $\mathit{i}=\mathit{j}$ are given using the equation ${\mathit{S}}_{\mathit{ij}}=2\times {\mathit{V}}_{\mathit{ij}}-1$.

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 ${\mathit{I}}_{\mathit{S}}$, ${\mathit{I}}_1$, ${\mathit{I}}_2$, and ${\mathit{I}}_{\mathit{L}}$ and two node voltages ${\mathit{V}}_{11}$ and ${\mathit{V}}_{21}$. 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 = 
$$\left(\begin{array}{c}{V}_1-\mathrm{IS}\hspace{0.17em}{Z}_0\\ V_1-I_1\hspace{0.17em}{Z}_0-S_{11}\hspace{0.17em}\left(V_1+I_1\hspace{0.17em}{Z}_0\right)-S_{12}\hspace{0.17em}\left(V_2+I_2\hspace{0.17em}{Z}_0\right)\\ V_2-I_2\hspace{0.17em}{Z}_0-S_{21}\hspace{0.17em}\left(V_1+I_1\hspace{0.17em}{Z}_0\right)-S_{22}\hspace{0.17em}\left(V_2+I_2\hspace{0.17em}{Z}_0\right)\\ V_2-\mathrm{IL}\hspace{0.17em}{Z}_0\\ I_1+\mathrm{IS}\\ I_2+\mathrm{IL}\end{array}\right)$$

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 = 
$$\left(\begin{array}{cccccc}-Z_0& 0& 0& 0& 1& 0\\ 0& -Z_0-S_{11}\hspace{0.17em}{Z}_0& -S_{12}\hspace{0.17em}{Z}_0& 0& 1-S_{11}& -S_{12}\\ 0& -S_{21}\hspace{0.17em}{Z}_0& -Z_0-S_{22}\hspace{0.17em}{Z}_0& 0& -S_{21}& 1-S_{22}\\ 0& 0& 0& -Z_0& 0& 1\\ 1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\end{array}\right)$$

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 = 
$$\left(\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ \frac{1}{Z_0}& 0\\ 0& \frac{1}{Z_0}\end{array}\right)$$

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 = 
$$\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right)$$

Related Topics

• Extract S-Parameters from Mutual Inductor
• De-Embedding S-Parameters
• Bisect S-Parameters of Cascaded Probes