Question 6: Explain the term impedance of an AC circuit. Find expression for the impedance of the RLC series circuit.

ANSWER

Impedance

An AC circuit may consist of a resistor R, a capacitor C and inductor L in combination. The combined opposition offered to the flow of AC current by these elements is called impedance. It is defined as,

“The ratio of the rms value of the applied voltage to the rms value of the resulting current.” Mathematically,

Z being the impedance of the AC circuit.

Unit of impedance

Unit of impedance is ohm, denoted by Ω. . Impedance of a circuit is said to be one ohm if a current of one rms Ampere flows in the circuit when the voltage across its terminals is one rms volts.

Impedance of the RLC series circuit

Consider resistance R, inductance L and capacitance C connected in series in an AC circuit. We note that;

1. Potential difference V_R across the resistance R is in phase with the current and given by

   

2. Voltage V_L across the inductance L leads the current by π/2 rads.

   

3. Voltage drop across C is lagging behind the current by π/2 rad.

   

Therefore, V_L and V_C are out of phase by π rads. This is shown in the vector diagram.

Impedance calculation

The applied voltage is equal to the vector sum of V_R, V_L and V_C. Let V_L>V_C, then their sum is V_L – V_C which is directed in the upward direction along V_L. The net applied voltage is, therefore, somewhere in the first quadrant between V_L and V_R as shown in the figure.

 The magnitude of the applied voltage is

Now we know that V_R = IR, V_L = IX_L and V_C = IX_C. Therefore, the above equation can be written as

Total impedance of the circuit is

For this particular arrangement (X_L>X_C), the voltage is leading the current by an angle φ, called the phase angle. However, when X_C>X_L, the voltage is lagging behind the current. Both the situations are depicted in the diagram to the right. These are called impedance triangles.