Question 10.8: Determining Complex Gain The input voltage to a certain ampl...

Recall that the phasor for the input voltage is a complex number whose magnitude is the peak value of the sinusoidal signal and whose angle is the phase angle of the sinusoidal signal. Thus,

\pmb{\text{V}}_i=0.1 \underline{/-30}^\circ

Similarly,

\pmb{\text{V}}_o=10 \underline{/15}^\circ

Now, we can find the complex voltage gain as

|A_v|=\frac{\pmb{\text{V}}_o}{\pmb{\text{V}}_i} =\frac{10 \underline{/15}^\circ}{0.1 \underline{/-30}^\circ} \\ \quad =100 \underline{/45}^\circ

The meaning of this complex voltage gain is that the output signal is 100 times larger in amplitude than the input signal. Furthermore, the output signal is phase shifted by 45° relative to the input signal.
To express gain in decibels, we first find the magnitude of the gain by dropping the angle and then compute decibel gain:

|A_v|_{dB}=20 \log|A_v|=20 \log(100)=40 \text{ dB}