(a) Calculate the impedance Z of an infinite
chain of elements with impedances Z1 and Z2, as shown in
the top figure.
(b) Calculate Z1 and Z2 for the specific case shown in the bottom figure.
We treat the circuit as an infinite ladder network with characteristic impedance Z0.
Since the ladder is infinite, the impedance Z will not change if an additional section is added to the front of ladder.
Details of the calculation:
(a) An equivalent network with impedance Z is shown in the figure.
Z = Z1 + ZZ2/(Z + Z2), Z2 – Z1Z – Z1Z2 = 0, Z = Z½ + (Z12/4 + Z1 Z2)½.
(b) For the network in the bottom figure
Z1 = iωL – i/(ωC) = (i/(ωC))(ω2LC - 1), Z2 = (-iωL)/(ω2LC - 1).
Z12 = (-1/(ωC)2)(ω2LC - 1)2, Z1 Z2 = L/C.
Consider the "twin-T" band-pass filter as shown below. Find the highest and lowest frequency passed by this filter if it is properly terminated.
In order to act as a filter, the circuit must be terminated with the characteristic impedance Z0 of the ladder network. Z0 = (Z12/4 + Z1Z2)½ if the sections of the ladder look as shown below.
If Z02 > 0 the frequency is passed.
If Z02 < 0 the frequency is not passed. (The circuit absorbs no power.)