The output impedance of a non-ideal operational amplifier is denoted by Z_out . The variation in the magnitude of Z_out with increasing frequency, f , in the circuit shown below, is best represented by

The correct representation of the closed-loop output impedance variation with frequency is given by the plot showing an initially constant low impedance value, followed by an increasing slope, and eventually saturating at higher frequencies:

1. Understanding the Circuit Configuration:
The given circuit diagram shows a non-ideal operational amplifier connected in a unity-gain voltage follower configuration with negative feedback:

For a voltage follower, the feedback factor is $\beta =1$.

2. Closed-Loop Output Impedance Derivation:
Due to negative feedback (voltage-series feedback), the closed-loop output impedance $Z_{out}$ of an op-amp is reduced by the loop gain factor:

$Z_{out}\left(s\right)=\frac{R_o}{1+A\left(s\right)\beta }$

where $R_o$ is the open-loop output resistance of the non-ideal op-amp and $A\left(s\right)$ is the open-loop frequency-dependent gain of the op-amp.

3. Open-Loop Gain Frequency Response:
A typical non-ideal op-amp exhibits a dominant pole frequency response:

$A\left(f\right)=\frac{{\mathrm{Amo}}_{}}{1+j(f/f_p)}$

- At very low frequencies ($f«f_p$), the open-loop gain is large and constant ($A\approx {\mathrm{Amo}}_{}$). Thus, the closed-loop output impedance is constant and extremely low:

$\left|Z_{out}\right|\approx \frac{R_o}{1+{\mathrm{Amo}}_{}}$

- At intermediate frequencies ($f_p«f«f_T$), the open-loop gain drops by 20 dB/decade ($A\propto 1/f$). As the denominator $1+A\left(f\right)$ decreases with frequency, $\left|Z_{out}\right|$ increases proportionally with frequency (a positive slope of +20 dB/decade on a logarithmic plot).

- At very high frequencies ($f»f_T$), the open-loop gain approaches zero ($A\left(f\right)\to 0$). The feedback loop loses control, and the output impedance approaches the internal open-loop output resistance:

$\left|Z_{out}\right|\to R_o$

4. Conclusion:
On a log-log scale of $\log \left(\right|Z_{out}\left|\right)$ versus $\log \left(f\right)$, the curve starts flat at a low magnitude, increases linearly with frequency, and then flattens out again at higher frequencies. Therefore, the fourth graph correctly illustrates the frequency dependence of $Z_{out}$.