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11.2 Transmission line transformers
RS
− v2 + i2
i2 RL
vs i1
i1 − v1 + Figure 11.10 A 4:1 Guannela transformer.
The voltage gain is given by Gv =
vL 2 2 = = jk = 2e− jk Z 0 vs e cos k + j sin k 2R S
(11.43)
where the last equality holds if we select Z 0 = 2R S . We see that the output voltage is twice the input voltage plus a delay. This relation is a broadband relationship for a low-loss circuit |G v | = 2
(11.44)
Using Eq. (11.36), we can also derive the input impedance 2vs e− jk + j Z 0 sin k i s = 2 cos k vs vs (2 cos k − 2e
− jk
) = j Z 0 sin k i s
(11.45) (11.46)
2Rs
Simplifying we have vs = Rs is
(11.47)
which shows that the circuit behaves like a 4 : 1 impedance matching circuit.
4:1 Unbalanced Guannella transformer Consider the Guannella transformer shown in Fig. 11.10. Intuitively, when the transmission line is electrically short, we can see that the source current is twice as large as the load current. We thus expect that an impedance match occurs for a load four times as large as the source. We can verify this at high frequency by applying KVL around the source and load loop vs = i s R S − v2 + i 2 R L = (i 1 + i 2 )R S − v2 + i 2 R L
(11.48)
We can also take a KVL loop around the source vs = i s R S + v1 = (i 1 + i 2 )R S + v1
(11.49)