phasor representation of alternating quantities – resistive, inductive, capacitive

The Phasor Representation of all sinusoidal electrical quantities

Purely resistive circuits

A purely resistive or a non-inductive circuit is a circuit which has so small inductance, at normal frequency its reactance – is negligible as compared to its resistances. Ordinary filament lamps, water resistances etc. are the examples of non-inductive resistances. If the circuit is purely non- inductive, no reactance emf (i.e. self-induced or back emf) is set up and whole of the applied voltage is utilised in overcoming the ohmic resistance of the circuit. Consider an ac circuit containing a non-inductive resistance of R ohms connected across a sinusoidal voltage represented by v = Vm sin wt, as shown in figure (a).

When the current flowing through a pure resistance changes, no back emf is set up, therefore, applied voltage has to overcome the ohmic drop of iR only i.e. iR=v

From the expressions, of instantaneous applied voltage and instantaneous current, it is evident that in a pure resistive circuit/ the applied voltage and current are in phase with each other, as shown by wave and phasor diagrams in figure (b) and (c) respectively.

Purely inductive circuits

An inductive circuit is a coil with or without an iron core having negligible resistance. Practically pure inductance can never be had as the inductive coil has always small resistance. However, a coil of thick copper wire wound on a laminated iron core has negligible resistance and is known as a chock coil.

When an alternating voltage is applied to a purely inductive coil, an emf, known as self induced emf, is induced in the coil which opposes the applied voltage. Since coil has no resistance, therefore, at every instant applied voltage has to overcome this self-induced emf only.

Let the applied voltage v = Vm sin wt

and self inductance of coil = L henry

self induced emf in the coil,

From the expressions of instantaneous applied voltage and instantaneous current flowing through a purely inductive coil it is observed that the current lags behind the applied voltage by pie/2.

as shown in figure (b)  by wave diagram and in figure (c) by phasor diagram.

Purely capacitive circuits

When a b is impressed across the plates of a perfect condenser, it will become charged to full voltage almost instantaneously. charging current will flow only during the period of “build up” and will cease to flow as soon as the capacitor has attained the steady voltage of the source. This implies that for a direct current, a capacitor is a break in the circuit or an infinitely high resistance. When a capacitor is connected to an alternating current circuit the capacitor is charged and discharged during alternate quarter cycles. Let an alternating voltage represented by v = Vm sin wt be applied across a capacitor of capacitance C farads. The expression for instantaneous charge is given as q = CVmax sin wt .

Since the capacitor current is equal to the rate of change of charge, therefore, the capacitor current may be obtained by differentiating the above equation .

From the expressions of instantaneous applied voltage and instantaneous current flowing through capacitance, it is observed that the current leads the applied voltage by pie/2, as shown in figure (b) and (c) by wave and phasor diagrams respectively.


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