The study of circuits involves three basic types of units (R, L, C i.e., resistance, reactance and capacitance respectively) and four possible series combination of them. The latter, in turn, may be arranged in many kinds of parallel, series-parallel, parallel-series or other complex circuits.

**4.1. A.C. Through Pure Ohmic Resistance Alone**

The circuit containing a pure resistance R is shown in Fig. 23 (a). Let the applied voltage be given by the equation,

U = V max sin θ = V max sin wt

Then the instantaneous value of current flowing through the resistance R will be,

i = u / R = V _{max} sin wt / R

The value of current will be maximum

when sin wt = 1 or (wt = 90°)

I _{max} = V _{max} / R

Substituting this value in eqn. (ii), we get

i = [max sin rot

Comparing (i) and (iii), we find that alternating voltage and current are in phase with each other as shown in Fig. 23 (b), also shown vectorially in Fig. 23 (c).

**Power.** Refer Fig. 23 (c)

Instantaneous power,

p = Vi = V _{max} sin wt x _{max} sin wt = V _{max} [_{max} sin^{2} wt

= V _{max} I _{max} / 2 x 2 sin 2 wt = V _{max} I _{max} / 2 ( 1 – cos 2 wt)

For a complete cycle the average of V _{max} / √2. I_{ max} / √2 cos 2 wt is zero.

Hence, power for the whole cycle

P = V _{max} / √2 . . I_{ max} / √2

P=Watt

where V = R.M.S. value of applied voltage, and

I = R.M.S. value of the current.

It may be observed from the Fig. 23 (c) that no part of the power cycle at any time becomes negative. In other words the power in a purely resistive circuit never becomes zero.

Hence in pure resistive circuit we have :

1. Current is in phase with the voltage.

2. Current I= V / R where I and V are r.m.s. values of current and voltage.

3. Power in the circuit, P = W =FR.