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Resonance In Series LCR Circuit

Resonance In Series LCR Circuit

Edited By Vishal kumar | Updated on Jul 02, 2025 05:57 PM IST

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.

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
Resonance In Series LCR Circuit

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 VL = VC = 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.

Frequently Asked Questions (FAQs)

1. Why is power factor correction important in AC circuits, and how does resonance relate to it?
Power factor correction is important to maximize power transfer and efficiency in AC circuits. Resonance can be used for power factor correction by adding a capacitor or inductor to bring the circuit closer to unity power factor, where voltage and current are in phase.
2. What is the role of resonance in impedance matching networks for RF circuits?
Resonance plays a crucial role in impedance matching networks for RF circuits. By using resonant circuits, the impedance of a source can be matched to that of a load, ensuring maximum power transfer. This is particularly important in antenna systems and RF power amplifiers to optimize efficiency and performance.
3. What is the importance of understanding resonance in power distribution systems?
Understanding resonance in power distribution systems is crucial for preventing harmful voltage and current amplifications. Resonance can occur due to interactions between inductance and capacitance in the system, potentially leading to equipment damage, power quality issues, and system instability if not properly managed.
4. What is the role of resonance in wireless power transfer systems?
Resonance plays a crucial role in wireless power transfer systems by allowing efficient energy transfer between two coupled resonant circuits. By matching the resonant frequencies of the transmitter and receiver, power can be transferred over greater distances with higher efficiency.
5. How does the concept of resonance apply to the operation of wireless charging systems for electronic devices?
Resonance is fundamental to wireless charging systems. These
6. What is resonance in a series LCR circuit?
Resonance in a series LCR circuit occurs when the inductive reactance equals the capacitive reactance, resulting in maximum current flow and minimum impedance. At resonance, the circuit behaves purely resistively, and the voltage and current are in phase.
7. How does the resonant frequency of an LCR circuit relate to its components?
The resonant frequency (f_r) of an LCR circuit is given by the formula: f_r = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. This shows that resonant frequency depends inversely on both inductance and capacitance.
8. Why does the current reach its maximum value at resonance?
At resonance, the inductive and capacitive reactances cancel each other out, leaving only the resistance to impede current flow. This minimizes the total impedance, allowing the current to reach its maximum value for a given voltage.
9. How does the phase angle between voltage and current change at resonance?
At resonance, the phase angle between voltage and current becomes zero. This means the voltage and current are in phase, as the inductive and capacitive effects cancel out, leaving only the resistive component.
10. What happens to the impedance of the circuit at resonance?
At resonance, the impedance of the circuit reaches its minimum value, becoming equal to the resistance R. This is because the inductive and capacitive reactances cancel each other out, leaving only the resistive component.
11. How does the resonant frequency change if you double the inductance in the circuit?
If you double the inductance (L) in the circuit, the resonant frequency (f_r) will decrease by a factor of √2 (approximately 0.707 times the original frequency). This is because f_r is inversely proportional to the square root of L, as given by f_r = 1 / (2π√(LC)).
12. How does the concept of resonance apply to antennas in radio communication?
In radio communication, antennas are often designed to be resonant at their operating frequency. A resonant antenna has a purely resistive impedance, maximizing power transfer and radiation efficiency. The length of the antenna is typically related to the wavelength of the resonant frequency.
13. How does the concept of impedance matching relate to resonance in LCR circuits?
Impedance matching in LCR circuits often involves using resonance to transform impedances. At resonance, the circuit's impedance is purely resistive and at its minimum, which can be used to match the impedance of a source to a load for maximum power transfer.
14. How does the power dissipation in the resistor change as the circuit approaches resonance?
As the circuit approaches resonance, the power dissipation in the resistor increases. At resonance, the power dissipation reaches its maximum because the current through the circuit (and thus through the resistor) is at its peak value.
15. What is the significance of the "resonance curve" in analyzing LCR circuits?
The resonance curve graphically represents the relationship between frequency and either current amplitude or impedance in an LCR circuit. It helps visualize the circuit's behavior around resonance, showing the peak response, bandwidth, and how sharply the circuit is tuned.
16. What is meant by "anti-resonance" in the context of LCR circuits?
Anti-resonance occurs in parallel LCR circuits when the circuit presents maximum impedance to the source. This happens when the inductive and capacitive currents are equal but opposite, resulting in minimum line current and maximum voltage across the parallel combination.
17. How does the presence of mutual inductance affect the resonance in coupled LCR circuits?
Mutual inductance in coupled LCR circuits can shift the resonant frequency and create multiple resonance peaks. It introduces energy transfer between the circuits, which can either enhance or suppress resonance depending on the coupling strength and the relative tuning of the circuits.
18. How does changing the resistance affect the resonant frequency?
Changing the resistance does not affect the resonant frequency of an LCR circuit. The resonant frequency depends only on the inductance (L) and capacitance (C), as given by the formula f_r = 1 / (2π√(LC)).
19. What is the quality factor (Q) of a resonant circuit, and what does it indicate?
The quality factor (Q) is a dimensionless parameter that describes how under-damped a resonator is. It is defined as Q = (1/R) * √(L/C). A higher Q indicates a narrower bandwidth and a more selective circuit, with lower energy losses per cycle.
20. How does the bandwidth of a resonant circuit relate to its Q factor?
The bandwidth of a resonant circuit is inversely proportional to its Q factor. A higher Q factor results in a narrower bandwidth, while a lower Q factor leads to a wider bandwidth. The relationship is given by: Bandwidth = Resonant Frequency / Q.
21. What is the significance of the half-power frequencies in a resonant circuit?
Half-power frequencies are the frequencies on either side of the resonant frequency where the power drops to half its maximum value. They define the bandwidth of the circuit and are used to calculate the Q factor. The difference between these frequencies is the bandwidth.
22. How does the current magnitude change as frequency varies around the resonant frequency?
As the frequency varies around the resonant frequency, the current magnitude forms a bell-shaped curve. It reaches its maximum at the resonant frequency and decreases symmetrically on either side. The width of this curve is related to the circuit's Q factor.
23. What is the phase response of an LCR circuit around resonance?
The phase response of an LCR circuit changes rapidly around resonance. Below resonance, the circuit is capacitive, and the current leads the voltage. Above resonance, the circuit is inductive, and the current lags the voltage. At resonance, the phase difference is zero.
24. What happens to the impedance of the circuit at frequencies much lower than the resonant frequency?
At frequencies much lower than the resonant frequency, the impedance is dominated by the capacitive reactance. The circuit behaves mostly like a capacitor, with current leading voltage, and the impedance decreases as frequency increases.
25. How does the impedance behave at frequencies much higher than the resonant frequency?
At frequencies much higher than the resonant frequency, the impedance is dominated by the inductive reactance. The circuit behaves mostly like an inductor, with current lagging voltage, and the impedance increases as frequency increases.
26. How does the voltage across the inductor compare to the voltage across the capacitor at resonance?
At resonance, the voltage across the inductor (V_L) is equal in magnitude but opposite in phase to the voltage across the capacitor (V_C). This is why they cancel each other out, leaving only the voltage across the resistor.
27. What is voltage magnification in a series LCR circuit, and when does it occur?
Voltage magnification occurs when the voltage across either the inductor or capacitor is greater than the applied voltage. This happens at or near resonance and is more pronounced in circuits with high Q factors. The magnification factor is approximately equal to the Q factor.
28. How does the sharpness of the resonance curve relate to the circuit's Q factor?
The sharpness of the resonance curve is directly related to the circuit's Q factor. A higher Q factor results in a sharper, more peaked resonance curve, indicating a more selective circuit. Lower Q factors produce broader, less pronounced resonance curves.
29. What is the significance of the time constant in an LCR circuit?
The time constant (τ) in an LCR circuit determines how quickly the circuit responds to changes. It is given by τ = L/R for the inductive part and τ = RC for the capacitive part. At resonance, these time constants are equal, affecting the circuit's transient response.
30. How does damping affect the behavior of an LCR circuit at resonance?
Damping, primarily caused by resistance in the circuit, affects the sharpness of the resonance peak and the circuit's response time. Higher damping (more resistance) leads to a broader resonance peak and faster settling time, while lower damping results in a sharper peak and longer oscillations.
31. What is the relationship between the resonant frequency and the natural frequency of an LCR circuit?
In an LCR circuit, the resonant frequency is equal to the natural frequency of the system. Both are given by the same formula: f_n = f_r = 1 / (2π√(LC)). This frequency represents the system's preferred oscillation frequency when disturbed.
32. How does the energy distribution between the inductor and capacitor change during a cycle at resonance?
At resonance, energy oscillates between the inductor and capacitor. When the energy stored in the magnetic field of the inductor is at its maximum, the energy in the electric field of the capacitor is zero, and vice versa. This energy exchange occurs at the resonant frequency.
33. What is the significance of the resonant frequency in radio tuning circuits?
The resonant frequency is crucial in radio tuning circuits as it allows the selection of a specific radio station frequency. By adjusting the L or C values, the circuit can be tuned to resonate at the desired frequency, effectively filtering out other frequencies.
34. How does the phase difference between voltage and current change as you sweep through frequencies in an LCR circuit?
As frequency increases from below resonance to above, the phase difference changes from positive (current leading voltage) through zero at resonance (in phase) to negative (current lagging voltage). This phase shift occurs over a range centered on the resonant frequency.
35. What is meant by the "selectivity" of a resonant circuit, and how is it related to the Q factor?
Selectivity refers to a resonant circuit's ability to discriminate between signals of different frequencies. A higher Q factor indicates greater selectivity, meaning the circuit can better isolate a specific frequency while attenuating nearby frequencies.
36. How does the resonant frequency of an LCR circuit change if you increase both L and C by the same factor?
If both inductance (L) and capacitance (C) are increased by the same factor, the resonant frequency will decrease. Specifically, if L and C are both multiplied by a factor k, the new resonant frequency will be 1/√k times the original frequency, as f_r = 1 / (2π√(LC)).
37. What is the significance of the "half-power bandwidth" in resonant circuits?
The half-power bandwidth is the frequency range between the two points where the power drops to half its maximum value (or where the voltage drops to 1/√2 of its peak). It's a measure of the circuit's selectivity and is inversely proportional to the Q factor.
38. How does the presence of resistance in a series LCR circuit affect its resonant behavior?
Resistance in a series LCR circuit affects the sharpness of the resonance peak and the maximum current at resonance. Higher resistance leads to a broader resonance curve, lower Q factor, and lower maximum current, while also increasing the circuit's bandwidth.
39. What is meant by "forced oscillations" in the context of an LCR circuit?
Forced oscillations in an LCR circuit occur when an external alternating voltage is applied. The circuit responds by oscillating at the frequency of the applied voltage, but the amplitude of these oscillations depends on how close this frequency is to the circuit's natural resonant frequency.
40. What is the difference between series and parallel resonance in LCR circuits?
In series resonance, current is maximum and impedance is minimum at the resonant frequency. In parallel resonance, current is minimum and impedance is maximum. Series resonance is used for current selection, while parallel resonance is used for voltage selection.
41. How does the concept of resonance apply to mechanical systems, and what are the analogies to electrical LCR circuits?
Mechanical resonance is analogous to electrical resonance. Mass corresponds to inductance, spring constant to the inverse of capacitance, and damping to resistance. Just as an LCR circuit has a natural frequency, mechanical systems have natural frequencies of vibration determined by their physical properties.
42. How does the quality factor (Q) of an LCR circuit affect its transient response?
The quality factor (Q) affects the transient response of an LCR circuit. A higher Q results in a longer settling time and more oscillations before reaching steady state. Lower Q circuits have faster settling times but less pronounced resonance effects.
43. What is the relationship between the bandwidth and the rise time of a resonant circuit?
The bandwidth and rise time of a resonant circuit are inversely related. A wider bandwidth corresponds to a faster rise time, while a narrower bandwidth results in a slower rise time. This relationship is important in designing circuits for specific speed and frequency response requirements.
44. How does temperature affect the resonance characteristics of an LCR circuit?
Temperature changes can affect the resonance characteristics by altering the values of L, C, and R. Typically, resistance increases with temperature, potentially lowering the Q factor. Capacitance and inductance may also change, shifting the resonant frequency. These effects are important in designing temperature-stable circuits.
45. What is the significance of the "loaded Q factor" versus the "unloaded Q factor" in resonant circuits?
The unloaded Q factor represents the quality factor of a resonant circuit without any external load, while the loaded Q factor includes the effect of the load. The loaded Q is always lower than the unloaded Q and is more representative of the circuit's behavior in practical applications.
46. How does resonance in LCR circuits relate to the concept of impedance transformation?
Resonance in LCR circuits can be used for impedance transformation. At resonance, the circuit's impedance is purely resistive. By choosing appropriate L and C values, the effective impedance seen by the source can be transformed, which is useful in matching circuits and filter design.
47. What is the importance of understanding resonance in the context of electromagnetic interference (EMI) and compatibility (EMC)?
Understanding resonance is crucial in EMI/EMC because resonant circuits can amplify unwanted signals at specific frequencies. This knowledge helps in designing filters to suppress interference and in avoiding unintentional antennas or resonant structures that could emit or be susceptible to electromagnetic interference.
48. How does the concept of resonance apply to the design of band-pass and band-stop filters?
Resonance is fundamental to the design of band-pass and band-stop filters. Band-pass filters use resonance to allow a specific frequency range to pass while attenuating others. Band-stop filters use anti-resonance to reject a specific frequency range. The center frequency and bandwidth of these filters are determined by the resonant properties of the LCR components.
49. How does the presence of parasitic capacitance and inductance affect the resonant behavior of real-world circuits?
Parasitic capacitance and inductance can shift the actual resonant frequency from the calculated value and create multiple resonance points. They can also affect the Q factor and bandwidth of the circuit. Understanding and accounting for these parasitic elements is crucial in high-frequency circuit design and layout.
50. What is the significance of resonance in the context of power factor correction in AC power systems?
Resonance is significant in power factor correction because it can be used to cancel out the reactive power in a system. By adding a resonant circuit (usually a capacitor bank) tuned to the system frequency, the inductive reactance of motors and transformers can be offset, improving the power factor and system efficiency.
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