Coupled Inductances and Capacitances

Coupled inductances and capacitances defined, in vector form, by

$${\bf v} = {\bf L}\frac{d{\bf i}}{dt}\hspace{0.7in}{\bf i} = {\bf C}\frac{d{\bf v}}{dt}$$ (2.48)

for symmetric positive definite matrices ${\bf L}$ and ${\bf C}$ were first introduced in the WDF context by Nitsche [131]; they turn out to be essential to the construction of WDF-based numerical simulation algorithms for stiff distributed systems such as plates (see §5.4) and shells (see §5.5), as well for full three-dimensional elastic solid dynamics (see §5.6). Though these are best thought of as vector elements, they appear within larger scalar circuits, and it is convenient to have a representation for which the vectors of port quantities are separated out into scalar port-wise components.

Figure 2.15: (a) $q$ coupled inductances and (b) $q$ coupled capacitances.

We show an inductive coupling of $q$ loops in Figure 2.15(a); self-inductances are indicated by directed arrows, accompanied by an inductance $L_{jj}$, $j=1,\hdots, q$ (these are the diagonal elements of ${\bf L}$), and a mutual inductance between loops $j$ and $k$, $j\neq k$ is represented by an arrow and the associated inductance $L_{kj}$ (which is the $(k,j)$th or $(j,k)$th element of ${\bf L}$, and is not constrained to be positive). A coupled capacitance is shown in Figure 2.15(b).

A coupled inductance can be discretized through the use of the trapezoid rule applied directly to the vector equations of (2.48); in terms of wave variables defined by (2.43), we get

$${\bf b}(n) = -{\bf a}(n-1)\hspace{0.5in} {\bf R} = 2{\bf L}/T$$

which is a direct vector generalization of (2.23). Similarly, for a capacitor, we get

$${\bf b}(n) = {\bf a}(n-1)\hspace{0.8in} {\bf R} = T(2{\bf C})^{-1}$$

In practice, if a coupled inductance (or capacitance) appears in a circuit which is to be discretized using WDFs, we may treat it as a $q$-vector two-port made up of a series (or parallel) junction terminated on a vector wave digital inductor (or capacitor) of port resistance $2{\bf L}/T$ (or $T(2{\bf C})^{-1}$). See Figure 2.16 for the signal flow diagrams for these objects and the simplified representations that we will use. The port resistance at the opposing port will in general be diagonal, so that the vector wave variables entering and leaving the junction may be decomposed into scalar wave variables; this diagonal port resistance will be determined by the rest of the network to which the $q$-vector two-port is connected. See §4.2.6 for more information on this decomposition in the DWN context; we will return to vector/scalar connections in Chapter 5. We note that in the representations in Figure 2.16, we have not explicitly indicated the order in which the $q$ scalar incoming and outgoing vectors should be ``packed'' and ``unpacked'' from the vector wave variables at the lower ports of the vector junctions. In the applications in Chapter 5, for a given coupled inductance (say), self-inductances will all be identical, as will all mutual inductances; thus any ordering will do, as long as the $j$th elements of both ${\bf a}$ and ${\bf b}$ correspond to wave variables at the $j$th scalar port.

Figure 2.16: (a) Signal flow graph for a wave digital coupled inductance and a simplified representation. Here the $q\times q$ port resistance ${\bf R} = 2{\bf L}/T$, and ${\bf R}_{0}$ is a $q\times q$ diagonal matrix; the diagonal entries specify the port resistances at the $q$ scalar ports to which the element is connected. (b) The signal flow graph for a wave digital coupled capacitance (vector port resistance ${\bf R} = T(2{\bf C})^{-1}$), and its simplified representation. In either case, the wave variables at the lower port of the vector junction are simply defined by ${\bf a}_{0} = [a_{1}, \hdots,a_{q}]^{T}$ and ${\bf b}_{0} = [b_{1}, \hdots, b_{q}]^{T}$.