Circuit Analysis Using Impedance

For the circuit shown in Figure 6.3.3a, determine the transfer function between the input voltage $v_{s}$ and the output voltage $v_{o}$ .

Question

Circuit Analysis Using Impedance

For the circuit shown in Figure 6.3.3a, determine the transfer function between the input voltage [latex]v_{s}[/latex] and the output voltage [latex]v_{o} [/latex].

Accepted Answer

Note that R and C are in parallel. Therefore their equivalent impedance Z(s) is found from
[latex]\frac{1}{Z(s)} = \frac{1}{1/Cs} + \frac{1}{R} [/latex]
or
[latex]Z(s) = \frac{R}{RCs + 1} [/latex]
An impedance representation of the equivalent circuit is shown in Figure 6.3.3b. In this representation we may think of the impedance as a simple resistance, provided we express the relations in Laplace transform notation. Kirchhoff’s voltage law gives
[latex]V_{s}(s) − R_{1} I (s) − Z(s) I (s) = 0 [/latex]
The output voltage is related to the current by [latex]V_{o}(s) = Z(s) I (s) [/latex]. Eliminating I (s) from these two relations gives
[latex]V_{s} (s) − R_{1} \frac{V_{o}(s)}{Z(s)} − V_{o}(s) = 0 [/latex]

which yields the desired transfer function:
[latex]\frac{V_{o}(s)}{V_{s} (s)} = \frac{Z(s)}{R_{1} + Z(s)} = \frac{R}{RR_{1} Cs + R + R_{1}}[/latex]

This network is a first-order system whose time constant is
[latex]τ = \frac{R R_{1}}{R + R_{1}} [/latex]

Question 6.3.2: Circuit Analysis Using Impedance For the circuit shown in Fi...

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