Transfer functions via impedance

Now the true power of impedance-based modeling is revealed: we can skip a time-domain model (e.g. state-space or io differential equation) and derive a transfer-function model, directly! Before we do, however, let’s be sure to recall that a transfer-function model concerns itself with the forced response of a system, ignoring the free response. If we care to consider the free response, we can convert the transfer function model to an io differential equation and solve it.

There are two primary ways impedance-based modeling is used to derive transfer functions. The first and most general is described, here. The second is a shortcut most useful for relatively simple systems; it is described in .

In what follows, it is important to recognize that, in the Laplace-domain, every elemental equation is just¹ \[\begin{aligned} \mathcal{V} = \mathcal{F} Z, \end{aligned}\] where the across-variable, through-variable, and impedance are all element-specific.

This algorithm is very similar to that for state-space models from linear graph models, presented in . In the following, we consider a connected graph with \(B\) branches, of which \(S\) are sources (split between through-variable sources \(S_T\) and across \(S_A\)). There are \(2 B - S\) unknown across- and through-variables, so that’s how many equations we need. We have \(B-S\) elemental equations and for the rest we will write continuity and compatibility equations. \(N\) is the number of nodes.

1. Derive \(2 B-S\) independent Laplace-domain, algebraic equations from Laplace-domain elemental, continuity, and compatibility equations.

   1. Draw a normal tree.

   2. Write a Laplace-domain elemental equation for each passive element.²

   3. Write a continuity equation for each passive branch by drawing a contour intersecting that and no other branch.³

   4. Write a compatibility equation for each passive link by temporarily “including” it in the tree and finding the compatibility equation for the resulting loop.⁴

2. Solve the algebraic system of \(2B\) equations and \(2B\) unknowns for outputs in terms of inputs, only. Sometimes, solving for all unknowns via the usual methods is easier than trying to cherry-pick the desired outputs.

3. The solution for each output \(Y_i\) depends on zero or more inputs \(U_j\). To solve for the transfer function \(Y_i/U_j\), set \(U_k=0\) for all \(k\ne j\), then divide both sides of the equation by \(U_j\).