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- Network Theory - Series Resonance
- Parallel Resonance
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# Network Theory - Parallel Resonance

In the previous chapter, we discussed the importance of series resonance. Now, let us discuss parallel resonance in RLC circuits.

## Parallel Resonance Circuit Diagram

If the resonance occurs in parallel RLC circuit, then it is called as **Parallel Resonance**. Consider the following **parallel RLC circuit**, which is represented in phasor domain.

Here, the passive elements such as resistor, inductor and capacitor are connected in parallel. This entire combination is in **parallel** with the input sinusoidal current source.

Write **nodal equation** at node P.

$$- I + I_R + I_L + I_C = 0$$

$$\Rightarrow - I + \frac{V}{R} + \frac{V}{j X_L} + \frac{V}{-j X_C} = 0$$

$$\Rightarrow I = \frac{V}{R} - \frac{jV}{X_L} + \frac{jV}{X_C}$$

$\Rightarrow I = V[\frac{1}{R} + j \lgroup \frac{1}{X_C} - \frac{1}{X_L} \rgroup]$**Equation 1**

The above equation is in the form of ** I = VY**.

Therefore, the **admittance Y** of parallel RLC circuit will be

$$Y = \frac{1}{R} + j \lgroup \frac{1}{X_C} - \frac{1}{X_L} \rgroup$$

## Parameters & Electrical Quantities at Resonance

Now, let us derive the values of parameters and electrical quantities at resonance of parallel RLC circuit one by one.

### Resonant Frequency

We know that the **resonant frequency, f_{r}** is the frequency at which, resonance occurs. In parallel RLC circuit resonance occurs, when the imaginary term of admittance, Y is zero. i.e., the value of $\frac{1}{X_C} - \frac{1}{X_L}$ should be equal to zero

$$\Rightarrow \frac{1}{X_C} = \frac{1}{X_L}$$

$$\Rightarrow X_L = X_C$$

The above resonance condition is same as that of series RLC circuit. So, the **resonant frequency, f_{r}** will be same in both series RLC circuit and parallel RLC circuit.

Therefore, the **resonant frequency, f_{r}** of parallel RLC circuit is

$$f_r = \frac{1}{2 \pi \sqrt{LC}}$$

Where,

- L is the inductance of an inductor.
- C is the capacitance of a capacitor.

The **resonant frequency, f_{r}** of parallel RLC circuit depends only on the inductance

**and capacitance**

*L***. But, it is independent of resistance**

*C***.**

*R*### Admittance

We got the **admittance Y** of parallel RLC circuit as

$$Y = \frac{1}{R} + j \lgroup \frac{1}{X_C} - \frac{1}{X_L} \rgroup$$

Substitute, $X_L = X_C$ in the above equation.

$$Y = \frac{1}{R} + j \lgroup \frac{1}{X_C} - \frac{1}{X_C} \rgroup$$

$$\Rightarrow Y = \frac{1}{R} + j(0)$$

$$\Rightarrow Y = \frac{1}{R}$$

At resonance, the **admittance**, Y of parallel RLC circuit is equal to the reciprocal of the resistance, R. i.e., $\mathbf{\mathit{Y = \frac{1}{R}}}$

### Voltage across each Element

Substitute, $\frac{1}{X_C} - \frac{1}{X_L} = 0$ in Equation 1

$$I = V [\frac{1}{R} + j(0)]$$

$$\Rightarrow I = \frac{V}{R}$$

$$\Rightarrow V = IR$$

Therefore, the **voltage** across all the elements of parallel RLC circuit at resonance is ** V = IR**.

At resonance, the admittance of parallel RLC circuit reaches to minimum value. Hence, **maximum voltage** is present across each element of this circuit at resonance.

### Current flowing through Resistor

The current flowing through resistor is

$$I_R = \frac{V}{R}$$

Substitute the value of ** V** in the above equation.

$$I_R = \frac{IR}{R}$$

$$\Rightarrow I_R = I$$

Therefore, the **current flowing through resistor** at resonance is $\mathbf{\mathit{I_R = I}}$.

### Current flowing through Inductor

The current flowing through inductor is

$$I_L = \frac{V}{j X_L}$$

Substitute the value of ** V** in the above equation.

$$I_L = \frac{IR}{j X_L}$$

$$\Rightarrow I_L = -j \lgroup \frac{R}{X_L} \rgroup I$$

$$\Rightarrow I_L = -jQI$$

Therefore, the **current flowing through inductor** at resonance is $I_L = -jQI$.

So, the **magnitude** of current flowing through inductor at resonance will be

$$|I_L| = QI$$

Where, Q is the **Quality factor** and its value is equal to $\frac{R}{X_L}$

### Current flowing through Capacitor

The current flowing through capacitor is

$$I_C = \frac{V}{-j X_C}$$

Substitute the value of ** V** in the above equation.

$$I_C = \frac{IR}{-j X_C}$$

$$\Rightarrow I_C = j \lgroup \frac{R}{X_C} \rgroup I$$

$$\Rightarrow I_C = jQI$$

Therefore, the **current flowing through capacitor** at resonance is $I_C = jQI$

So, the **magnitude** of current flowing through capacitor at resonance will be

$$|I_C| = QI$$

Where, Q is the **Quality factor** and its value is equal to $\frac{R}{X_C}$

**Note** − Parallel resonance RLC circuit is called as **current magnification** circuit. Because, the magnitude of current flowing through inductor and capacitor is equal to *Q* times the input sinusoidal current *I*.