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Average Power Associated with LCR Circuit in Series

In an electrical circuit, energy is supplied by the source of EMF, stored by the capacitive and inductive elements and dissipated in resistive elements.

Conservation of energy requires that, at any particular time, the rate at which energy is supplied by the source of EMF must equal the rate at which it is stored in the capacitive and inductive elements; plus the rate at which it is dissipated in the resistive element (we assume ideal capacitive and inductive elements have no internal resistances).

Let E = E_osin wt. Since current lags in inductive dominated circuit by an angle q.

If dw be the work done by a source of EMF e on a charge dq, then

The power

Using equations (1) and (2),

I = I_o sin (wt -q)

If this instantaneous power remains constant for small time dt. then

Total work done or energy spent in maintaining current over one full cycle.

The second term is zero over complete cycle i.e.,

Therefore,

The quantity cosq is called the power factor of the AC circuit and we know

Recall the diagram to determine effective voltage in LCR circuit.

This can also be represented in terms of the resistance and reactances as follows (Since I is same through all resistive element).

Such a right-angled triangle whose sides represent the resistance and reactance an AC circuit is called the impedance triangle.

From this triangle:

Sub Topics

Special cases
Case 1:
Case 2:
Case 3:
Power Factor
Resonance Curve
Q Factor of a Series Resonant Circuit
Wattless Current
Choke Coil

Special cases

Case 1:

For a pure resistance in AC circuit

q = O therefore in

r = Erms I_r_m_s

The shaded portion in the graph indicates that power delivered to the a.c circuit by the source of e.m.f is maximum and the circuit is purely resistive.

Case 2:

For a pure inductor in AC circuit, cosq = 90^o P = O. i.e., Average power over a complete cycle of AC through an ideal inductor is zero. i.e., whatever energy is needed in building up current in L is returned back during the decay of the current.

The shaded portion of the graph indicates that on adding portion I, II, III and IV, vector p is equal to zero.

Case 3:

When a pure capacitance is contacted in AC circuit f= 90^o or r = 0. i.e., Average power supplied to an ideal capacitor by the source over a complete cycle of A.C. is also zero.

Power Factor

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P = E_r_m_s I_r_m_s cosq. The product E_r_m_s I_r_m_s is called apparent power, whereas the P is called the true power. The term cosf is called the power factor.

The current through pure L or pure C, which consumes no power for its maintenance in the circuit is called wattless or idle current.

Resonance Curve

The graph shows the variation in the peak circuit I_o with the applied frequency of the a.c source in different circuits (L and C for both are same).

Q Factor of a Series Resonant Circuit

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Q stands for quality. The voltage magnification produced by series resonance is termed as Q-factor of the series resonance circuit.

Hence 'Q' factor can be increased by

(i) Increasing the inductance of the coil ohmic

(ii) By decreasing the resistance of the coil

Hence the Q factor depends entirely upon the design of the coil.

Wattless Current

Choke Coil

At most places, the household electric power is supplied at 220V, 50H. If such a source is directly connected to the tube, the tube will be damaged. To reduce the current, a choke coil is connected in series with the tube - representing the tube by a resistance and the choke coil by an ideal inductor. The equivalent circuit is as follows:

Choke coil is an electrical appliance used to control current in an AC circuit. If resistance alone was used, a lot of energy would be wasted. A choke coil consists of a number of turns of thick copper wire wound closely over a laminated soft iron core. A choke coil is basically a coil of high inductance L and a small resistance r connected in series.

The Average power consumed over a complete cycle in a practical choke coil is

P = E_r_m_s I_r_m_s cosf

This comes out to be smaller than power loss (I²R) if a resistance R alone is used for reducing AC.

The advantages of using a choke coil is not only to reduce the voltage, but also to prevent loss of electrical energy in the form of heat.