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General Inverters, Resonators, and End Couplings 291
Inverter
,--------l 2
0__1 : ,-'''-1-+--1-'-/~o
Yl1 1 Y21 Y21 I Yn
0-0_1_-;..1_1
I 1
L- ~
1---;---1_.L-I~o
Figure 8.13. A reciprocal short·circuit~parameter equivalent circuit when Y12 =Y21. The dashed
lines enclose the inverter portion of the network.
number. A sufficient condition for Y21 being an imaginary number is that the
network be lossless. In that case, the maximum possible efficiency is unity,
and (7.64) shows that stability factor K = 1. Then the definition of K in (7.57)
reveals that K = I when gil = 0 = gZ2 and YZI =jB, thus proving the sufficient
condition.
Note that susceptances YII and Y" in Figure 8.13 become part of the
synchronous reactance in their respective resonators, as seen from the particu.
lar case in Figure 8.12b. The frequency behavior of the resulting resonators
will be equated to the ideal LC resonator in Section 8.3.4.
Example 8. 2 Iden tify the inverter in a 3D-degree length of a lO0-0hm lossless
transmission line, and resonate each end to make a direct-coupled filter
between the 50-ohm terminations. The ABCD parameters of a transmission
line in (4.13)-(4.15) can be converted to Y parameters according to (3.86);
YIl= -jYocotO, and Y21= +jYo/sinO. Therefore, inverter YoI2 =Yo/sinO and
half of each resonator must be composed of admittance -jYocotO. Figure
8.14 shows the equivalent circuit in impedance terms. The equivalent of a
shorted-stub transmission line at each end of the inverter must be resonated to
make a direct-coupled filter. In this example, ZOIZ =50 ohms, and the shunt
reactances associated with Yll and Y" are +j57.74 ohms. Capacitors on either
end having the negative of the latter value are required, as shown in Figure
8.15. Programs A6-1 and A6-2 may be used to confirm the impedances noted
in Figure 8.15. It can be shown that the voltages across the capacitors have a
:_}t",
,--------------,
I Z012 "Zosine I
i 0 ", )\-----L...i-;1'--,t,,~<
Figure 8.14.
I at W o I
L
J
An equivalent circuit for a lossless transmission line having Zo ohms and (J length.