Resonance In Series LCR Circuit

Resonance in a series LCR circuit occurs when the inductive reactance (XL) and capacitive reactance (XC) balance each other out, causing the circuit to resonate at a specific frequency known as the resonant frequency. At this point, the impedance of the circuit is minimized, and the circuit allows maximum current to flow. This phenomenon is crucial in alternating current (AC) circuits, as it underpins the operation of many electronic devices. In real life, resonance is utilized in tuning radios and televisions to select desired frequencies, in improving the efficiency of electrical transformers, and in designing audio equipment for optimal sound quality.

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This Story also Contains

1. Resonance in Series LCR Circuit
2. Solved Examples Based on Resonance in Series LCR Circuit
3. Example 1: The self-inductance of the motor of an electric fan is 10 H. In order to impart maximum power at 50 Hz, it should be connected to a capacitance of
4. Summary

Resonance in Series LCR Circuit

At resonance, the voltage across the inductor and capacitor is maximized, but they are equal in magnitude and opposite in phase, leading to a net voltage of zero across these reactive components. This condition is critical in many practical applications, including tuning circuits for radios and televisions, where precise frequency selection is required. Additionally, resonance in LCR circuits plays a significant role in improving the efficiency of electrical circuits and filters used in audio and signal processing.

As we have discussed when

$\omega L=\frac{1}{\omega C}$ ,

then tanφ is zero i.e. phase angle (φ) is zero and voltage and current are in phase. We have called it electric resonance. So, if $X_L=X_C$, then the equation of impedance becomes

$Z=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}=R$

So, we get a minimum value of Z.

In this case, impedance is purely resistive and minimum and currents have its maximum value. Now as

$\omega L=\frac{1}{\omega C}$

So,

$\omega=\frac{1}{\sqrt{L C}}$

As, $\omega=2 \pi f_o$. Where $f_o$ is the frequency of the applied voltage

So,

$f_o=\frac{1}{2 \pi \sqrt{L C}}$

This frequency is called the resonant frequency of the circuit.

Peak current in this case is given by

$i_o=\frac{V_o}{R}$

We will now discuss the resonance curve and its nature. We will show the variation in circuit current (peak current i₀) with a change in frequency of the applied voltage

This figure/graph shows the variation of current with the frequency.

Conclusions From the Graph

1. If R has a small value, the resonance is sharp which means that if the applied frequency is lesser than resonant frequency f₀, the current is high otherwise

2. If R is large, the curve is broadsided which means that those is limited change in current for resonance and non-resonance conditions

Note

The natural or resonant frequency is Independent of the resistance of the circuit.

$\begin{aligned} X_L=X_c & =\omega_0 L=\frac{1}{\omega_0 c} \\ \nu_0 & =\frac{1}{2 \pi \sqrt{L c}}(H z)\end{aligned}$

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Solved Examples Based on Resonance in Series LCR Circuit

Example 1: The self-inductance of the motor of an electric fan is 10 H. In order to impart maximum power at 50 Hz, it should be connected to a capacitance of

1) $1 \mu F$
2) $2 \mu \mathrm{F}$
3) $4 \mu F$
4) $8 \mu F$

Solution

If $X_L=X_c$

This is the resonance condition

For maximum power, $L \omega=\frac{1}{C \omega}$

$\begin{aligned} & \therefore C=\frac{1}{L \omega^2}=\frac{1}{10 \times(2 \pi \times 50)^2} \\ & =\frac{1}{10 \times 10^4 \times(\pi)^2}=10^{-6} \mathrm{~F} \\ & \text { or } C=1 \mu \mathrm{F}\end{aligned}$

Hence, the answer is the option (1).

Example 2: At resonance the impedance of the given circuit

1) $\sqrt{R^2+\left(X_L-X_C\right)^2}$
2) $\sqrt{R^2+X_L^2}$
3) $\sqrt{R^2+X_C^2}$
4) $R$

Solution:

At resonance (series resonant and circuit) If X_{L}=X_{c}

$Z_{\min }=R$

wherein

The circuit behaves as a resistive circuit.

$\begin{aligned} & \text { At resonance, } X_L=X_C \\ & \Rightarrow Z=\sqrt{R^2+\left(X_L-X_C\right)^2} \\ & \Rightarrow Z=\sqrt{R^2} \\ & \mathrm{z}=\mathrm{R}\end{aligned}$

Hence, the answer is the option (4).

Example 3: In a L-C-R circuit , if V_L = V_C = V , then applied voltage would be

1) V
2) $\sqrt{V_R^2+V^2}$
3) $V_R$
4) None

Solution:

$\begin{aligned} & \text { If } V_L=V_c=V \\ & V=V_R\end{aligned}$

wherein

The whole applied voltage appears across the resistance.

In the LCR circuit, the applied voltage $V_a=\sqrt{V_R^2+\left(V_L-V_C\right)^2}$ If $\mathrm{V}_{\mathrm{L}}=\mathrm{V}_{\mathrm{C}}=\mathrm{V}$
$
\begin{aligned}
& V_a=\sqrt{V_R^2} \\
& \mathrm{v}_{\mathrm{a}}=\mathrm{v}_{\mathrm{R}}
\end{aligned}
$

Hence, the answer is the option (3).

Example 4: The natural frequency of the circuit depends on which of the following ( consider resonance case )

1) Resistance & capacitance

2) Inductance & Resistance

3) Inductance & Capacitance

4) Resistance only

Solution:

Resonant frequency (natural frequency)

$\begin{aligned} & X_L=X_c=\omega_0 L=\frac{1}{\omega_0 c} \\ & \nu_0=\frac{1}{2 \pi \sqrt{L c}}(H z)\end{aligned}$

wherein

Independent from the resistance of the circuit.

$\begin{aligned} & \text { At resonance, } \mathrm{x}_{\mathrm{L}}=\mathrm{x}_{\mathrm{C}} \\ & \omega_o L=\frac{1}{\omega_o C} \\ & \omega_o^2=\frac{1}{L C} \\ & \left(2 \pi f_o\right)^2=\frac{1}{L C} \\ & f_o=\frac{1}{2 \pi \sqrt{L C}}\end{aligned}$

Hence, the answer is the option (3).

Example 5: In a LCR circuit capacitance is changed from C to 2C. For the resonant frequency to remain unchanged, the inductance should be changed from L to

1) 4 L

2) 2L

3) L/2

4) L/4

Solution:

RLC circuit Voltage
$
V_R=i R \quad V_L=i X_L \quad V_C=i X_C
$

wherein

At resonance, $\omega=\frac{1}{\sqrt{L C}}$ when $\omega$ is constant,
$
\begin{aligned}
& \therefore \frac{1}{L_1 C_1}=\frac{1}{L_2 C_2} \Rightarrow \frac{1}{L C}=\frac{1}{L_2(2 C)}=\frac{1}{2 L_2 C} \\
& \therefore L_2=L / 2
\end{aligned}
$

Summary

Resonance in a series LCR circuit occurs when the inductive reactance and capacitive reactance cancel each other out, resulting in minimum impedance and maximum current at a specific resonant frequency. This phenomenon is essential for various applications, such as tuning radios and televisions for precise frequency selection and improving the efficiency of electrical transformers and audio equipment. The resonant frequency is determined by the circuit's inductance and capacitance and is independent of resistance.