Analysis of ac circuit

ANALYSIS OF AC CIRCUIT

It is assumed that the sinusoidally varying voltage is connected to

(a) purely resistive

(b) purely inductive and

(c) purely capacitive circuits.

In each case it is required to find the following quantities

1. The expression for the instantaneous current.

2. The polar form of voltage and current

3. The phasor diagram.

4. Ratio of voltage to current

5. Average power

6. Phase angle between voltage and current and hence power factor.

1. Purely Resistive Circuit Excited by a Sinusoidally Varying Voltage

Let, v = Vm sin ot be the applied voltage

R = Resistance of the circuit

i = instantaneous current flowing in the circuit

By observing equations (i) and (ii) we can conclude that the voltage applied and the current flowing, in a purely resistive circuit are in phase (it is because both V and I are sinusoidally varying quantities and also the angles are same).

To represent V and I in polar form

Let V = RMS value of the voltage = Vm/√2

I = RMS value of the current = Im/√2

Now, writing equations (i) and (ii) in polar form, we get

V = | V | ∠o ... (iii)

I = |V| ∠ o. .. (iv)

With the help of the above equations, phasor diagram can be drawn as shown in the fig 2.28 (b).

Impedance In the case of A.C circuits, ratio of V to I is called impedance (Z)

From equations (iii) and (iv)

If the angle of Z is zero as shown above,,

Z = R

i.e; Z = R ∠ 0

This Z is independent of frequency. The instantaneous power = w.

Maximum instantaneous value at (θ = π/2) = V_m^I_m

Average power P = W

(i.e) Average power = W = VI watts

The angle between voltage and current is called phase angle and is denoted by ϕ. cos ϕ is called power factor.

In this circuit, ϕ = 0.

Power factor (p.f) = cos ϕ = cos 0 = 1

(i.e)., the circuit is said to have unity power factor (u.p.f).

2. Circuit Consisting of Pure Inductance Excited by a Sinusoidally Varying Voltage

Let v = V_m sin ot be the applied voltage ... (i)

i = instantaneous current

L = self-inductance of the coil.

We know that e = - L di/dt … (ii)

But, according to Lenz's law, e opposes the applied voltage

i.e.,

From equations (ii) and (iii),

Both equations (i) and (iv) are sinusoidally varying quantities. Hence, we can compare these equations for knowing the phase relation. As the angle of i is less than that of v bу π/2, we say that current lags behind voltage by π/2.

As already mentioned minus angle (for alternating quantities) indicates lagging.

To represent V and I in polar form

We take V as reference. With the help of equations (i) and (iv) we write,

V = |V| ∠ 0 … (v)

and I = | I | ∠ - π/2 … (vi)

Phasor diagram

The phasor diagram can be drawn as shown in fig 1.154 (b) with the aid of equations (v) and (vi).

The phasor diagram drawn in fig 1.154(b) can also be shown as in fig 1.154(c). It is obtained by rotating the phasor diagram in 1.154(b) by π/2 radians in anti-clockwise direction. Still, we say that current lags behind the voltage by л/2.

(Note: The reader should know as how to draw the phasor diagram by taking both current and voltage as reference. This knowledge is necessary to draw the phasor diagrams of series and parallel circuits. For single elements we can take either V or I as reference. For series combinations I is taken as reference. For parallel circuit V is taken as reference).

The instantaneous power w varies at twice the supply frequency.

Average power =

Thus, the average power dissipated by the pure inductor = 0.

X_L is a function of frequency. X_L varies linearly with frequency. For D.C. circuits, frequency is zero and hence X_L

The variation of XL with frequency is as shown in fig.

Purely Capacitive Circuit Excited by Sinusoidally Varying Voltage

Let v=  V_m sin ωt

C = Capacitance of the capacitor

i = instantaneous current

By definition, charge q = Cv

By observing equations (i) and (ii), we can say that I leads V by angle π/2. (This comparison is possible because both equations are varying in the same nature). This condition is the prerequisite for the comparison. The angle of i is positive.

To represent V and I in the polar form:

Taking V as reference,

V = | V | ∠ 0, ... (iii)

and I = | I | 2π/2 ... (iv)

Phasor diagram:

Equations (iii) and (iv) can be represented as phasors shown in the fig. 1.156(a).

For a series circuit, current is same through all the elements. For drawing phasor diagram for series circuits, as already mentioned, I is taken as reference. Thus, the phasor diagram in Fig.1.156(a) can be re-drawn as in Fig.1.156(b). Still, the current leads the voltage bу л/2.

The frequency of instantaneous power is twice that of voltage or current. Maximum instantaneous power

Thus the average power dissipated by purely capacitive circuit = 0.

We know that Xc = 1 / ωC

X_c is a function of frequency.

As frequency increases, X_c decreases.

The relation between the two will be a rectangular hyperbola.