Circuit Analysis

This document contains the contents of the books.

This document contains the preface of the chapter.

Resistances are the simplest of circuit elements. The equation determining the current through and the p.d. (potential difference) across a resistor is given by Ohm’s law (v = R i).

Alternating currents are more widely used in the field of power and electronics than direct current. By the term alternating, it is meant that the current is not unidirectional but assumes alternately positive and negative values. Unlike the flow of direct current, there is no electron drift in a conductor carrying alternating current, but they merely oscillate about their average positions inside the conductor. If we plot an alternating current changing with time as a graph, with time in the X-axis and current in the Y-axis, we get the waveform of the alternating current. Sinusoidal wave alternating current is what we use in A.C. power circuits. Other waveforms like a triangular wave, square wave are used only in electronic circuits.
Fig. 2.1(c) shows a random waveform. Such waveforms are not classified as alternating currents. Only those waves which are periodic i.e., repeat themselves in a definite pattern are amenable for study. Even among these periodic waves, the sinusoidal (sine and cosine) waveform is of fundamental importance. Fourier, a mathematician, showed that all periodic waveforms (sinusoidal or not) can be split up into a series of sinusoidal waves. So, a study of the behaviour (response) of circuit elements to steady sinusoidal voltages (or currents) will be made in this chapter. The response to any other a.c. wave of voltage (or current) will be the sum of the responses of the Fourier components of the waveform.

Electrical networks containing resistors and capacitors obey this principle which states:
1. The response to a number of independent sources (of voltage or current) may be found by adding up the responses due to each of them found separately.
2. If the sources are multiplied by a constant, the response is also multiplied by it.

A transient in a circuit is the short-time change in the current (or voltage) in part or whole of a circuit due to some switching action (or a similar disturbance) brought about in the circuit. The transient current has an initial value (before the switching) and a final value (when the transient has elapsed); between these, a current change that occurs is the transient current; because the current cannot change suddenly to the final value in a circuit which has either an inductance or capacitance or both.

Two coils kept side by side such that the magnetic flux due to a current in one links the other (either partly or wholly) have a mutual effect between them.

Any periodic wave which is non-sinusoidal can be resolved into a series of sinusoidal waves, by Fourier’s series expansion. 

For periodic signals, such as a square wave, we saw that it can be represented as a series of sine and cosine waves, from the fundamental and all the harmonics.

An exponentially decaying oscillation, as in a series R-L-C circuit after an excitation, can be represented as Am e–σt sin ωt

Suppose we have an R-L-C circuit to which we apply a step voltage and observe the transient current that passes. Let R be a variable resistance. Let us vary R from zero upto a high value. The current will first be oscillatory for low values of R, of the form e–at sin (bt + θ). Then at a critical R value, the response becomes non-oscillatory when a = b. Further increase of R makes the transient current exponential in nature.

We have so far been concerned with the Analysis problem of Electric Circuits—finding the output for given input and vice versa having the network in hand. In Synthesis, a relation between the input and output is given and the network is to be found.