3.3: Inverter Network Scaling

Figure \(\PageIndex{1}\): Inverter network: (a) network with two identical admittance inverters with an inserted shunt element of admittance, \(y\); (b) equivalent network using equivalence shown in Figure 2.8.5; (c) scaled original network; and (d) scaled equivalent network. Element values are impedances except for \(J\), \(y\) and \(y_{1}\), which are admittances. The use of the equivalent networks in filter synthesis is illustrated in the case study in Section 3.4.

A filter that uses transmission lines is nearly always synthesized from a filter prototype that contains inverters. The synthesis of a filter begins with a normalized prototype that is transformed to the desired frequency, impedance, and type. It is the purpose of this section to show how to use such scaling when there are inverters in the prototype. In particular, it is shown that scaling the admittance of the network of Figure \(\PageIndex{1}\)(a), a common inverter subnetwork, by a factor \(x\) results in the network in Figure \(\PageIndex{1}\)(c).

The nodal admittance matrix of the network in Figure \(\PageIndex{1}\)(a) is

\[\label{eq:1}Y=\left[\begin{array}{ccc}{0}&{-\jmath J}&{0}\\{-\jmath J}&{y}&{-\jmath J}\\{0}&{-\jmath J}&{0}\end{array}\right] \]

Assigning nodal voltages to Terminals \(1\), \(2\), and \(3\), the nodal admittance matrix equation is

\[\label{eq:2}\left[\begin{array}{ccc}{0}&{-\jmath J}&{0}\\{-\jmath J}&{y}&{-\jmath J} \\ {0}&{-\jmath J}&{0}\end{array}\right]\left[\begin{array}{c}{v_{1}}\\{v_{2}}\\{v_{3}}\end{array}\right]=\left[\begin{array}{c}{J_{1}}\\{0}\\{J_{3}}\end{array}\right] \]

and by eliminating Node \(2\) using network condensation (see Section 1.A.16) of [31], this reduces to

\[\label{eq:3}\left[\begin{array}{cc}{J^{2}/y}&{J^{2}/y}\\{J^{2}/y}&{J^{2}/y}\end{array}\right]\left[\begin{array}{c}{v_{1}}\\{v_{3}}\end{array}\right]=\left[\begin{array}{c}{J_{1}}\\{J_{3}}\end{array}\right] \]

and this describes the external characteristics of the subnetwork in Figure \(\PageIndex{1}\)(a).

The nodal admittance matrix of the scaled network in Figure \(\PageIndex{1}\)(c) is

\[\label{eq:4}Y'=\left[\begin{array}{ccc}{0}&{-\jmath J\sqrt{x}}&{0}\\{-\jmath J\sqrt{x}}&{yx}&{-\jmath J\sqrt{x}}\\{0}&{-\jmath J\sqrt{x}}&{0}\end{array}\right] \]

That is,

\[\label{eq:5}\left[\begin{array}{ccc}{0}&{-\jmath J\sqrt{x}}&{0}\\{-\jmath J\sqrt{x}}&{yx}&{-\jmath J\sqrt{x}}\\{0}&{-\jmath J\sqrt{x}}&{0}\end{array}\right]\left[\begin{array}{c}{v_{1}}\\{v_{2}}\\{v_{3}}\end{array}\right]=\left[\begin{array}{c}{J_{1}}\\{0}\\{J_{3}}\end{array}\right] \]

and by eliminating Node \(2\) this reduces to

\[\label{eq:6}\left[\begin{array}{cc}{(J^{2}x)/(yx)} &{(J^{2}x)/(yx)} \\ {(J^{2}x)/(yx)}&{(J^{2}x)/(yx)}\end{array}\right]\left[\begin{array}{c}{v_{1}}\\{v_{3}}\end{array}\right]=\left[\begin{array}{cc}{J^{2}/y}&{J^{2}/y}\\{J^{2}/y}&{J^{2}/y}\end{array}\right]\left[\begin{array}{c}{v_{1}}\\{v_{3}}\end{array}\right]=\left[\begin{array}{c}{J_{1}}\\{J_{3}}\end{array}\right] \]

Thus the original network shown in Figure \(\PageIndex{1}\)(a) has the same external electrical characteristics as the scaled network of Figure \(\PageIndex{1}\)(c), with the characteristic admittance of the inverters scaled by \(\sqrt{x}\) and the shunt admittance scaled by \(x\).

A generalization of this result (which is useful when there are additional connections between Nodes \(1\) and \(3\)) is that multiplying a row and a column of the nodal admittance matrix by the same factor results in identical external characteristics. Note that the element sharing a row and column is multiplied twice.