Consider an amplifier in which the input voltage is...

Consider an amplifier in which the input voltage is sinusoidal with an amplitude of $100 \mathrm{mV}$ and $\omega=500$. The complex gain is given by $20 \angle 30^{\circ}$. Write an expression for the output voltage as a function of time.

Key Concepts

Sinusoidal Signals

A sinusoidal signal is a continuous waveform described by sine or cosine functions. It is characterized by its amplitude, frequency (or angular frequency), and phase. These signals are fundamental in analyzing and designing systems in electrical engineering because they are periodic and behave predictably under linear operations.

Angular Frequency

Angular frequency is a measure of how fast the sinusoidal signal oscillates, measured in radians per second. It sets the time scale of the oscillation and is critical in converting between time-domain equations and frequency-domain representations.

Complex Gain

Complex gain represents the amplification and phase shift applied by a system to an input signal. Expressed in polar form, it consists of a magnitude, which scales the input amplitude, and a phase angle, which shifts the input signal in time. This provides a concise way to describe the behavior of a system, particularly in frequency-domain analysis.

Phase Shift

A phase shift refers to the horizontal displacement of a sinusoidal waveform in time. When an input signal is multiplied by a complex gain, the phase angle component causes the output waveform to be shifted in time relative to the input, affecting the waveform’s start point within its cycle.

Time-Domain Conversion

Time-domain conversion involves translating the effects of complex gain into a time-based expression of the output signal. This process typically involves applying the gain’s magnitude to the amplitude of the input signal and adjusting the phase of the sinusoid by adding the phase angle, resulting in an output function that clearly shows how the original signal has been altered over time.

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