Circle Diagram for a Series R- L Circuit

Circle Diagram for a Series R- L Circuit

Consider a series R-L circuit with a variable R as shown in the Fig. 6.2.1. It is excited by an alternating source of V volts. The frequency of the source is f Hz.

Let I = current flowing through the circuit

Z = Impendance of the circuit

Z = R + j X_L where X_L = 2π fL

Now R is variable while X_L is fixed.

The phasor diagram is shown in the Fig. 6.2.2 (a). The current I lags voltage V by angle ϕ as the circuit is inductive. The impedance triangle is shown in the Fig. 6.2.2 (b). 

From the impedance triangle we can write,

Sin ϕ = X_L / Z

Substituting in the expression for I,

I = [V/ X_L ] sin ϕ … (6.2.1)

This is the equation of a circle in polar co-ordinates with a diameter equal to (V/X_L).

When the resistance R = 0, then ϕ = 90° hence sin ϕ = 1.

I = I_m = V / X_L

This is the maximum value of current.

As R increases, the phase angle ϕ decreases thus decreasing sin ϕ . Effectively current I also decreases. When R → ∞ the ϕ → 0° and current becomes zero.

The locus obtained of extremities of a current phasor plotted for various values of R is a semicircle. The semicircle is shown in the Fig. 6.2.3.

The voltage axis is taken as vertical axis as a reference, with respect to which the various current phasors are plotted.

The power factors at various conditions are cos ϕ₁, cos ϕ₂ etc. As varies only from 0° to 90°, the diagram is semicircle, infact it is a half part of a circle hence it is known as circle diagram.

This theory of series R-L circuit can be easily extended to a three phase induction motor.