Projecting the impedance matrix

The equations for projection of the impedance matrix were derived in section 2.6. Remember that the labels (0), (1) and (2) refer to planes 0, 1 and 2 in figure 4.1.

Figure 4.1: Detail of a waveguide consisting of straight sections of length $d$ joined discontinuously

By way of summary, the equation for projection across a discontinuity is

$$Z^{(1)} = F Z^{(2)} F^{\mathrm{T}}, \mbox{\hspace{1cm}} S_1 < S_2$$ (4.3)

where $S_1 = \pi R_1^2$, $S_2 = \pi R_2^2$ are the cross-sectional areas and

$$Z^{(1)} = V^{-1} Z^{(2)} (V^{\mathrm{T}})^{-1}, \mbox{\hspace{1cm}} S_1 > S_2.$$ (4.4)

The projection matrices are given by

$$F_{nm}(\beta) = \frac{2 \beta \gamma_m J_1(\beta \gamma_m)} {(\beta^2 \gamma_m^2 - \gamma_n^2)J_0(\gamma_m)}$$ (4.5)

where $\beta = R_1/R_2$ with $F(0,0)=1$ and $V_{nm}(\beta) = F_{nm}(1/\beta)$.

The equation for projection through a distance $d$ is (N.B. This is a correction to the original version of my thesis and D₂ is given in equation (2.41))

$$Wrong: Z^{(0)} = (Z^{(1)} + i D_3 Z_c)(i D_3 Z_c^{-1} Z^{(1)} + I)^{-1}. Right: Z^{(0)} = (i D_3)^{-1} Z_c - D_2^{-1} Z_c (Z^{(1)} + (i D_3)^{-1} Z_c)^{-1} D_2^{-1} Z_c$$ (4.6)

where

$$D_3(n,m) = \left\{ \begin{array} {r@{\quad:\quad}l} \tan(k_n d) & n = m, \\ 0 & n \neq m. \end{array} \right.$$ (4.7)

Here

$$k_n = \left\{ \begin{array} {r@{\quad:\quad}l} -\sqrt{k^2 ... ...{R}\right)^2} & k > \frac{\gamma_n}{R}. \end{array} \right.$$ (4.8)

is the wavenumber of the $n$th mode along the tube neglecting the effect of losses; the corresponding expression for lossy propagation is given in section 2.4.1. $\gamma_n$ is the $n$th zero of the Bessel function $J_1$ as tabulated in appendix A.