[Click here for a PDF of this post with nicer formatting]

To get a feel for how to generate the MLN equations for a circuit that has both RLC and non-linear components, consider the circuit of fig. 1.

The KCL equations for this circuit are

- \( 0 = i_s – i_d \)
- \( i_L + \frac{v_2 – v_3}{R} = i_d \)
- \( \frac{v_3 – v_2}{R} + C \frac{dv_3}{dt} = 0 \)
- \( -v_2 + L \frac{d i_L}{dt} = 0 \)
- \( i_d = I_0 \lr{ e^{(v_1 – v_2)/v_T} – 1} \)

FIXME: for the diode, is that the right voltage sign with respect to the current direction?

With \( Z = 1/R \), these can be put into the standard MLN matrix form as

\begin{equation}\label{eqn:diodeRLCSample:20}

\begin{bmatrix}

0 & 0 & 0 & 0 \\

0 & Z & -Z & 1 \\

0 & -Z & Z & 0 \\

0 & -1 & 0 & 0 \\

\end{bmatrix}

\begin{bmatrix}

v_1 \\

v_2 \\

v_3 \\

i_L \\

\end{bmatrix}

+

\begin{bmatrix}

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 \\

0 & 0 & C & 0 \\

0 & 0 & 0 & L \\

\end{bmatrix}

{\begin{bmatrix}

v_1 \\

v_2 \\

v_3 \\

i_L \\

\end{bmatrix}}’

=

\begin{bmatrix}

I_0 & 1 \\

-I_0 & 0 \\

0 & 0 \\

0 & 0 \\

\end{bmatrix}

\begin{bmatrix}

1 \\

i_s(t) \\

\end{bmatrix}

+

\begin{bmatrix}

-I_0 \\

I_0 \\

0 \\

0 \\

\end{bmatrix}

\begin{bmatrix}

e^{(v_2 – v_3)/v_T}

\end{bmatrix}

\end{equation}

Let’s write this as

\begin{equation}\label{eqn:diodeRLCSample:40}

\BG \BX(t) + \BC \dot{\BX}(t) = \BB \Bu(t) + \BD \Bw(t).

\end{equation}

Here \( \Bu(t) \) collects up all the unique signature sources (for example sources with each different frequency in the system), and \( \Bw(t) \) is a vector of all the unique non-linear (time dependent) terms.

Assuming a bandwidth limited periodic source we know how to express any of the time dependent variables \( v_1, … \) in terms of their (discrete) Fourier transforms. Suppose that

the Fourier coefficients for \( v_a(t), u_b(t), w_c(t) \) are given by

\begin{equation}\label{eqn:diodeRLCSample:60}

\begin{aligned}

v_a(t) &= \sum_{n = -N}^N V_n^{(a)} e^{j \omega_0 n t} \\

u_b(t) &= \sum_{n = -N}^N U_n^{(b)} e^{j \omega_0 n t} \\

w_c(t) &= \sum_{n = -N}^N W_n^{(c)} e^{j \omega_0 n t}.

\end{aligned}

\end{equation}

For example, in this circuit, if the source was zero phase signal at the fundamental frequency of our Fourier basis (\( i_s(t) = e^{j \omega_0 t} \)), the only non-zero Fourier components \( U_n^{(a)} \) would be \( U_0^{(1)} = 1, U_1^{(2)} = 1 \).

Equation \ref{eqn:diodeRLCSample:40} then becomes

\begin{equation}\label{eqn:diodeRLCSample:80}

0 = \sum_{n=-N}^N

e^{j n \omega_0 t}

\lr{

\lr{

\BG + j \omega_0 n \BC

}

\begin{bmatrix}

V_n^{(1)} \\

V_n^{(2)} \\

\vdots

\end{bmatrix}

– \BB

\begin{bmatrix}

U_n^{(1)} \\

U_n^{(2)} \\

\end{bmatrix}

– \BD

\begin{bmatrix}

W_n^{(1)} \\

\end{bmatrix}

}

\end{equation}

The time dependence in the linear terms is nicely taken of by this transformation to the frequency domain. However, we have a fairly messy structure with sums of Fourier components instead of the nice Fourier component vectors that we see in \S A.4 of [1]. That reference does consider multivariable problems like this one, so it looks like fully digesting that methodology is the next step.

# References

Giannini and Giorgio Leuzzi. *Nonlinear Microwave Circuit Design*. Wiley Online Library, 2004.