(a) Calculate the impedance Z of an infinite
chain of elements with impedances Z_{1} and Z_{2}, as shown in
the top figure.

(b) Calculate Z_{1} and Z_{2} for the specific case shown in
the bottom figure.

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Solution:

- Concepts:

Ac circuits, ladder networks -
Reasoning:

We treat the circuit as an infinite ladder network with characteristic impedance Z_{0}.

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 = Z_{1}+ ZZ_{2}/(Z + Z_{2}), Z^{2}– Z_{1}Z – Z_{1}Z_{2}= 0, Z = Z_{½}+ (Z_{1}^{2}/4 + Z_{1}Z_{2})^{½}.

(b) For the network in the bottom figure

Z_{1}= iωL – i/(ωC) = (i/(ωC))(ω^{2}LC - 1), Z_{2}= (-iωL)/(ω^{2}LC - 1).

Z_{1}^{2}= (-1/(ωC)^{2})(ω^{2}LC - 1)^{2}, Z_{1}Z_{2}= 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.

Solution:

- Concepts:

Ac circuits, ladder networks -
Reasoning:

In order to act as a filter, the circuit must be terminated with the characteristic impedance Z_{0}of the ladder network. Z_{0}= (Z_{1}^{2}/4 + Z_{1}Z_{2})^{½}if the sections of the ladder look as shown below.If Z

_{0}^{2}> 0 the frequency is passed.

If Z_{0}^{2}< 0 the frequency is not passed. (The circuit absorbs no power.) - Details of the calculation:

We treat the circuit as an infinite ladder network with characteristic impedance Z_{0}.

Z_{0}= (Z_{1}^{2}/4 + Z_{1}Z_{2})^{½}.

Z_{1}= iωL + 1/(iωC), Z_{2}= 1/(iωC').

Z_{1}^{2}= -ω^{2}L^{2}- 1/(ω^{2}C^{2}) + 2L/C.

Z_{1}Z_{2}= L/C' - 1/(ω^{2}C'C).

Z_{0}^{2}= L/2C + L/C' - ω^{2}L^{2}/4 - 1/(4ω^{2}C^{2}) - 1/(ω^{2}C'C)

= (LC'+2LC)/(2CC') - ω^{2}L^{2}/4 - (¼ + C/C')/(ω^{2}C^{2}).

Set Z_{0}^{2}= 0, then ω^{2}(LC'+2LC)/(2CC') - ω^{4}L^{2}/4 - (¼ + C/C')/C^{2}= 0.

Quadratic equation in ω^{2}: ω^{2}= 1/(LC) +2/(LC') ± 2/(LC').

ω_{min}^{2}= 1/(LC), ω_{max}^{2}= 1/LC + 4/(LC')

If ω ≈ 0, Z_{0}^{2}< 0 the frequency is not passed.

If ω --> ∞,Z_{0}^{2}< 0 the frequency is not passed.

Frequencies between ω_{min}^{2}= 1/(LC) and ω_{max}^{2}= 1/LC + 4/(LC') are not filtered out.