The input and the output signals of four different...

The input and the output signals of four different amplifiers are given below. Obtain the voltage gain and the phase shift for each of the amplifiers. (a) $v_{\text {in }}(t)=3 \cos (\omega t) v_{\text {out }}(t)=27 \cos (\omega t)$ (b) $v_{\text {in }}(t)=5 \sin (10 t) \quad v_{\text {out }}(t)=15 \cos (10 t)$ (c) $v_{\text {it }}(t)=4 \sin (\omega t) \quad v_{\text {out }}(t)=20 \sin (\omega t+\pi / 2)$ (d) $v_{\text {in }}(t)=7 e^{20 j t} \quad v_{\text {out }}(t)=j 5 e^{20 j t}$

The input and the output signals of four different amplifiers are given below. Obtain the voltage gain and the phase shift for each of the amplifiers.
(a) $v_{\text {in }}(t)=3 \cos (\omega t) v_{\text {out }}(t)=27 \cos (\omega t)$
(b) $v_{\text {in }}(t)=5 \sin (10 t) \quad v_{\text {out }}(t)=15 \cos (10 t)$
(c) $v_{\text {it }}(t)=4 \sin (\omega t) \quad v_{\text {out }}(t)=20 \sin (\omega t+\pi / 2)$
(d) $v_{\text {in }}(t)=7 e^{20 j t} \quad v_{\text {out }}(t)=j 5 e^{20 j t}$

Key Concepts

Voltage Gain

Voltage gain is the ratio of the amplitude of the output signal to the amplitude of the input signal. It quantifies how much an amplifier increases the input voltage in magnitude, regardless of the signal's phase information. This concept is central to understanding and designing amplification systems in electrical engineering.

Phase Shift

Phase shift refers to the difference in phase between the input and output signals of a system. In AC circuits, a phase shift indicates a time delay or advancement between corresponding points of the sinusoidal input and output waveforms. It is important in analyzing the behavior of amplifiers, filters, and other signal processing components.

Sinusoidal Signal Representation

Sinusoidal signals can be represented mathematically using sine or cosine functions, which describe periodic oscillations. These representations are fundamental in analyzing and processing AC signals, vibrations, and waves in various engineering applications. Recognizing the inherent properties of these functions is key to understanding amplitude scaling and phase shifting in systems.

Phasor and Complex Exponential Representation

Phasors and complex exponential representations simplify the analysis of sinusoidal functions by converting differential equations into algebraic ones. The method involves expressing sinusoidal signals as complex numbers, which encapsulate both magnitude (amplitude) and phase information. This approach is widely used in circuit analysis, signal processing, and control systems.

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