2. C H A P T E R 14
Parallel
A.C. Circuits

3. Parallel A.C. Circuits C-14 Solving Parallel Circuits
Vector or Phasor Method
Admittance Method
Application of Admittance Method
Complex or Phasor Algebra
Series-Parallel Circuits
Series Equivalent of a Parallel Circuit
Parallel Equivalent of a Series Circuit
Resonance in Parallel Circuits
Graphic Representation of Parallel Resonance
Points to Remember
Bandwidth of a Parallel Resonant Circuit
Q-factor of a Parallel Circuit Parallel

4. Solving Parallel Circuits
When impedances are joined in parallel, there are three methods available to solve such circuits:
Vector or phasor Method
Admittance Method and
Vector Algebra

5. Vector or Phasor Method
Consider the circuits shown in Fig Here, two reactors A and B have been joined in parallel across an r.m.s. supply of V volts. The voltage across two parallel branches A and B is the same, but currents through them are different.

6. Admittance Method

7. Application of Admittance Method
Consider the 3-branched circuit of Fig Total conductance is found by merely adding the conductances of three branches. Similarly, total susceptance is found by algebraically adding the individual susceptances of different branches.

8. Complex or Phasor Algebra
Consider the parallel circuit shown in Fig The two impedances, Z1 and Z2, being in parallel, have the same p.d. across them.

9. Series – Parallel Circuits
By Admittance Method
In such circuits, the parallel circuit is first reduced to an equivalent series circuit and then, as usual, combined with the rest of the circuit.
By Symbolic Method
Consider the circuit of Fig First, equivalent impedance of parallel branches is calculated and it is then added to the series impedance to get the total circuit impedance.

10. Series Equivalent of a Parallel Circuit
Consider the parallel circuit of Fig (a). As discussed in Art. 14.5

11. Parallel Equivalent of a Series Circuit

12. Resonance in Parallel Circuits
We will consider the practical case of a coil in parallel with a capacitor, as shown in Fig
Such a circuit is said to be in electrical resonance when the reactive (or wattless) component of line current becomes zero.
The frequency at which this happens is known as resonant frequency. The vector diagram for this circuit is shown in Fig (b).

13. Graphic Representation of Parallel Resonance
We will now discuss the effect of variation of frequency on the susceptance of the two parallel branches. The variations are shown in Fig
Inductive susceptance ;
Capacitive susceptance ;
Net Susceptance B

14. Points to Remember
Following points about parallel resonance should be noted and compared with those about series resonance. At resonance.
Net susceptance is zero i.e. 1/XC = XL/Z2 or XL × XC = Z2 or L/C = Z2
The admittance equals conductance
Reactive or wattless component of line current is zero.
Dynamic impedance = L/CR ohm.
Line current at resonance is minimum and = but is in phase with the applied voltage.
Power factor of the circuit is unity.

15. Bandwidth of a Parallel Resonant Circuit
The bandwidth of a parallel circuit is defined in the same way as that for a series circuit. This circuit also has upper and lower half-power frequencies where power dissipated is half of that at resonant frequency.

16. Q – Factor of a Parallel Circuit