Q. Sinusoidal steady-state phasor analysis?

The kind of response of a physical system in an applied excitation depends in general on the type of excitation, the elements in the system and their interconnection, and also on the past history of the system. The total response generally consists of a forced response determined by the particular excitation and its effects on the system elements, and a natural response dictated by the system elements and their interaction. The natural response caused by the energy storage elements in circuits with nonzero resistance is always transient; but the forced response caused by the sources can have a transient and a steady-state component. The boundary conditions (usually initial conditions), representing the effect of past history in the total response, decide the amplitude of the natural response and reflect the degree of mismatch between the original state and the steady-state response. However, when excitations are periodic or when they are applied for lengthy durations, as in the case of many applications, the solution for the forced response is all that is needed, whereas that for the natural response becomes unnecessary. When a linear circuit is driven by a sinusoidal voltage or current source, all steady-state voltages and currents in the circuit are sinusoids with the same frequency as that of the source. This condition is known as the sinusoidal steady state. Sinusoidal excitation refers to excitation whose waveform is sinusoidal (or cosinusoidal). Circuits excited by constant currents or voltages are called dc circuits, whereas those excited by sinusoidal currents or voltages are known as ac circuits.Sinusoids can be expressed in terms of exponential functions with the use of Euler's identity, where j represents the imaginary number √-1. The reader is expected to be conversant with complex numbers.

If we are able to find the response to exponential excitations, e^jθ or e^-jθ , we can use the principle of superposition in order to evaluate the sinusoidal steady-state response. With this in mind let us now study the response to exponential excitations.