Resonant Circuits: Resonant Frequency and Q Factor

A resonant circuit refers to an electrical circuit using circuit elements such as an inductor (\(L\)) and a capacitor (\(C\)) to cause resonance at a specific frequency. There are two types of resonant circuits: series resonant circuits and parallel resonant circuits. In a series resonant circuit, the impedance of the circuit reaches its minimum value at resonance, whereas in a parallel resonant circuit, the impedance reaches its maximum value. These characteristics are widely applied as the foundation of various electronic circuits such as in wireless communication, filter design, and signal processing.

This page provides a detailed explanation of how to calculate the resonant frequency and the Q factor (Quality Factor). Accurate calculation of the resonant frequency is essential for the design and optimization of resonant circuits, and the Q factor is a crucial indicator for evaluating the selectivity and energy loss of the circuit. By understanding these concepts, it is possible to fully utilize the performance of resonant circuits and design more efficient and precise electronic circuits.

What is Resonant Frequency?

Resonant frequency refers to the natural frequency at which an object or system tends to vibrate. When energy is supplied to a system at this frequency, electrical resonance occurs, amplifying the system’s vibrations. Understanding this phenomenon requires a grasp of the concept of natural vibration. Natural vibration means that an object vibrates at its own frequency when subjected to an external shock. This frequency is determined by the object’s shape, mass, and elasticity, and the object vibrates solely at this frequency, regardless of the shock’s intensity. For example, different percussion instruments produce unique sounds because each vibrates at its distinct frequency.

When resonance occurs at the resonant frequency, the system absorbs and stores energy from the external source very efficiently. A classic example is the act of swinging on a swing, where minimal effort can significantly increase the swing’s amplitude due to resonance. This happens because energy is supplied to the swing in sync with its natural frequency, thus amplifying the vibration.

In electrical circuits, the combination of resistors, inductors (\(L\)), and capacitors (\(C\)) establishes a resonant frequency. When the circuit operates at this frequency, electromagnetic energy transfers efficiently between the inductor and capacitor, maximizing the energy stored in the circuit. This principle is utilized in various fields, including wireless communication, filtering, and sensor technology. By accurately understanding and controlling the resonant frequency, the performance of these systems can be maximized, achieving high efficiency and precision.

Resonant Frequency and Resonance

To understand resonant circuits, it is essential to first grasp the concepts of “resonant frequency” and “resonance.”

“Resonant frequency” refers to the phenomenon where an object, when subjected to an impact, vibrates at a frequency that is intrinsic to the object. To illustrate, consider a xylophone (a musical instrument); when you strike each key or tap the rim of a glass filled with water, a sound is produced. Regardless of who performs the action, the same pitch is always heard.

The reason for this consistent pitch is that each key of the xylophone has its own specific material and size, and the glass has its own material, thickness, and the water level. These factors determine the vibration frequency when impacted (the resonant frequency).

Furthermore, if a resonant frequency is continuously applied to an object, the vibration synchronizes with the applied frequency and amplifies significantly. This phenomenon is known as “resonance.” For instance, when you continuously push a swinging swing, the swing’s motion gradually increases due to resonance.

Resonant Frequency in LC Circuit

The resonant frequency in a resonant circuit refers to the specific frequency at which the circuit impedance is at its minimum or maximum. Resonant circuits can be designed in either series or parallel configurations, and the formula used to calculate the resonant frequency is the same for both configurations.

Resonant Frequency and Angular Resonant Frequency

The resonant frequency is expressed by the following equation:

\(f_0 = \displaystyle \frac{1}{2\pi \sqrt{LC}} \, \text{[Hz]}\)

Additionally, the angular resonant frequency (\(ω_0\)) is given by:

\(\omega_0 = \displaystyle \frac{1}{\sqrt{LC}}\)

Proving Resonant Frequency from Reactance

Understanding resonant circuits requires a basic comprehension of both the resonant frequency and the angular resonant frequency. The resonant frequency (\(f_0\)) is the frequency at which a circuit most efficiently transfers energy, resulting in maximum voltage or current. The angular resonant frequency (\(ω_0\)) is the resonant frequency expressed in radians per second (rad/s), which is used in calculations and analyses.

By grasping these fundamental concepts, one can gain deeper insight into how resonant circuits operate and their characteristics. This knowledge is crucial for the design and application of resonant circuits, and thus, will be explained here.

Reactance is related to whether resonance occurs. Capacitive reactance (\(X_C\)) and inductive reactance (\(X_L\)) are defined as follows:

\(X_C = \displaystyle \frac{1}{2\pi f C}\)

\(X_L = 2\pi f L\)

These equations show that the reactance varies with frequency.

The condition for resonance occurs when the inductive reactance and the capacitive reactance are equal.

\(X_L = X_C\)

Substituting the formulas for inductance and capacitance into this equation, we get:

\(2\pi f L = \displaystyle \frac{1}{2\pi f C}\)

Solving this equation for frequency (\(f\)), we find the resonant frequency (\(f_0\)).

\(f_0^2 = \displaystyle \frac{1}{4\pi^2 LC}\)

\(f_0 = \displaystyle \frac{1}{2\pi \sqrt{LC}}\)

Furthermore, the resonant angular frequency (\(ω_0\)) is given by:

\(\omega_0 = \displaystyle \frac{1}{\sqrt{LC}}\)

Resonant circuits are used in various fields such as communication, broadcasting, and analog electronic devices, where their characteristics enable the amplification of signals at a specific frequency.

Series Resonant Circuit

A series resonant circuit is a circuit that connects an inductor and a capacitor in series, minimizing impedance at the resonance frequency. This circuit functions as a filter, selectively allowing only specific frequencies to pass. It is widely used in devices such as radio receivers and audio equipment to emphasize particular signals.

The resonance frequency of a series resonant circuit is determined by the values of inductance and capacitance, allowing precise frequency control. Designing a series resonant circuit enables the operation of high-performance electronic devices.

Impedance of the Series RLC Resonant Circuit

Resonance occurs in a circuit when a resistor (\(R\)), inductor (\(L\)), and capacitor (\(C\)) are connected in series. Resonance happens at the frequency where the reactance of the capacitor and inductor are equal.

This is known as a series RLC resonant circuit. In a series RLC circuit, the impedance (\(Z\)) is given by:

\(|Z| = \sqrt{R^2 + X^2} = \sqrt{R^2 + (X_L – X_C)^2}\)

The condition for resonance is when the reactance of the inductance and capacitance are equal. Thus:

In other words, in a resonant RLC series circuit, the impedance (\(Z\)) is as follows.

\(|Z| = \sqrt{R^2 + 0^2} = R\)

From the complex impedance perspective, it can be understood that the impedance consists only of the resistive component when the reactance of the capacitor and inductor are equal.

\(Z = R + j(X_L – X_C)\)

\(Z = R + j(\omega_0 L – \displaystyle \frac{1}{\omega_0 C}) = R + j\left(\displaystyle \frac{L}{\sqrt{LC}} – \displaystyle \frac{\sqrt{LC}}{C}\right) = R + j\left(\displaystyle \frac{\sqrt{L}}{\sqrt{C}} – \displaystyle \frac{\sqrt{L}}{\sqrt{C}}\right) = R\)

In other words, in a resonant state of the series RLC circuit, the impedance (\(Z\)) is:

\(Z=R\)

Inductive reactance and capacitive reactance cancel each other out at the resonant frequency, resulting in a condition where only resistance (\(R\)) appears to be present.

As a result, the impedance is minimized, and the current within the circuit reaches its maximum.

\(I = \displaystyle \frac{V}{\sqrt{R^2 + (X_L – X_C)^2}} = \displaystyle \frac{V}{\sqrt{R^2 + 0}} = \displaystyle \frac{V}{R}\)

Applications of Series Resonant Circuits

Series resonant circuits play a crucial role in electrical and electronic circuits. They are incorporated into various electrical products, such as AC power filters, noise filters, and the tuning circuits of radios and televisions, to create selective tuning circuits for receiving specific frequency channels. This allows for the accurate selection and reception of signals at various frequencies.

What is a Parallel Resonant Circuit?

A parallel resonant circuit is a circuit in which inductance and capacitance are connected in parallel, exhibiting a phenomenon called resonance at a particular frequency. This type of circuit is commonly used in various electronic equipment such as wireless communication devices and filter circuits, where selecting or emphasizing a specific frequency is necessary. At the resonance frequency, the reactance of the inductor and capacitor cancel each other out, resulting in the circuit’s overall impedance reaching its maximum value, thereby amplifying a specific signal.

Impedance of an RLC Parallel Resonant Circuit

An RLC parallel resonant circuit consists of a resistor (\(R\)), inductor (\(L\)), and capacitor (\(C\)) connected in parallel. The impedance of this circuit is given by the following formula:

\(\displaystyle \frac{1}{Z} = \displaystyle \frac{1}{R} + \displaystyle \frac{1}{j \omega L} + j \omega C = \displaystyle \frac{1}{R} + \displaystyle \frac{-j}{\omega L} + j \omega C = \displaystyle \frac{1}{R} + j(\omega C – \displaystyle \frac{1}{\omega L})\)

In a parallel RLC resonant circuit, similar to a series RLC resonance circuit, at resonance, the reactance of the inductor (\(L\)) and the capacitor (\(C\)) cancel each other out.

\(\displaystyle \frac{1}{Z} = +j\left(\displaystyle \frac{\sqrt{C}}{\sqrt{L}} – \displaystyle \frac{\sqrt{C}}{\sqrt{L}}\right) = \displaystyle \frac{1}{R}\)

It can be seen that at resonance in a parallel RLC circuit, the impedance (\(Z\)) becomes purely resistive.

\(Z = R \)

In a parallel RLC circuit at resonance, unlike a series RLC circuit, the impedance is at its maximum (infinity), and the current is at its minimum (no current flows, making it similar to an open circuit).

Differences Between Series Resonance and Parallel Resonance

The relationship between impedance and current differs in series resonance circuits and parallel resonance circuits. In a series RLC resonance circuit, the current flowing through the circuit is the result of dividing the supply voltage by the impedance. At resonance, the impedance reaches its minimum value (equal to R), and the circuit current is at its maximum.

On the other hand, in a parallel RLC resonance circuit, at resonance, the imaginary part of the admittance becomes zero, and the impedance of the circuit reaches its maximum (equal to ∞). When the impedance is at its maximum, the current flowing through the circuit is limited, effectively causing the circuit to behave as if it is open.

Although the method for finding the resonance frequency and the fact that \(Z ＝ R\) at resonance are common to both series RLC and parallel RLC resonance circuits, it is important to note that in a series RLC resonance circuit, the impedance is minimized, and the current is maximized at resonance. In contrast, in a parallel RLC resonance circuit, the impedance is maximized (infinite) and the current is minimized (no current flows, as if the circuit is open) at resonance.

Quality Factor of Resonant Circuits

The ” Quality Factor (Q factor)” is a crucial metric that quantifies the sharpness and selectivity of a signal at the resonance frequency. A high Q factor indicates a sharp signal characteristic, while a low Q factor suggests a broader characteristic. This value reflects the circuit’s ability to efficiently store energy and minimize losses, significantly impacting the precision of electronic filters, oscillators, and antenna designs.

High-Q circuits can precisely capture narrowband signals, making them essential for advanced communication technologies and precise sensor designs. The appropriate selection and adjustment of the Q factor function as critical elements in the design of high-performance electronic devices.

Additionally, the bandwidth, which is inversely related to the Q factor, indicates the range of frequencies that the resonant circuit can effectively pass.

\(Q = \displaystyle \frac{\text{Signal strength at resonance frequency}}{\text{Bandwidth (BW)}}\)

\(Q = \displaystyle \frac{\omega_0}{\omega_2 – \omega_1} = \displaystyle \frac{f_0}{f_2 – f_1}\)

Q-Factor in a Series RLC Resonant Circuit

The method for calculating the Q-factor in a series RLC resonant circuit focuses on understanding the behavior of the current and the interaction between the inductor (\(L\)) and capacitor (\(C\)) when the circuit is in resonance. At resonance, the current I is constant, and the inductor and capacitor exchange energy, resulting in the reactance of these elements canceling each other out.

In this state, the Q-factor is defined as the ratio of the inductive reactance to the resistance R of the resistor, representing the selectivity and sharpness of the resonance.

\(Q = \displaystyle \frac{\omega_0}{\Delta \omega} = \displaystyle \frac{\omega_0}{\omega_2 – \omega_1} = \displaystyle \frac{1}{R} \sqrt{\displaystyle \frac{L}{C}} = \displaystyle \frac{\omega_0 L}{R}\)

For example, the figure below illustrates the frequency response of the current I in a series RLC resonant circuit. At the resonance frequency (angular frequency) where the inductor (\(L\)) and capacitor (\(C\)) resonate, the impedance is minimized.

Moreover, the Q-factor functions as a coefficient indicating the voltage magnification in the resonant circuit, showing how much the voltage across the inductor is amplified by the resistance component, or the voltage multiplier relative to the supply voltage.

You can observe that when plotted against angular frequency, the current reaches its maximum value.

In a series RLC resonant circuit, the bandwidth varies with the Q-factor, whereas in a parallel resonant circuit, the bandwidth remains fixed.

Q-Factor in a Parallel Resonant Circuit

The method for determining the Q factor in a parallel resonant circuit is deeply rooted in the resonance phenomenon and the relationship between the current and voltage at that time. When resonance occurs, the supply voltage \(V\) across the entire circuit remains constant, and energy circulates between the inductor (\(L\)) and the capacitor (\(C\)), with the reactance of these components canceling each other out. In this state, the circulating current through the inductor and the capacitor is equal and opposite, resulting in zero power consumption between these components.

The key factor in determining the Q factor is the ratio of the circulating current flowing through the inductor to the effective current flowing through the resistor (\(R\)). This ratio quantifies the circuit’s selectivity and low energy loss, indicating that a higher Q factor means the circuit has a higher selectivity for a narrow bandwidth signal and lower energy loss. The characteristic of maintaining a constant voltage during resonance shows that a parallel resonant circuit can efficiently store energy and process signals near the resonance frequency with high precision.

\(Q = \displaystyle \frac{I_C}{I_R} = \displaystyle \frac{I_L}{I_R}\)

\(Q = \displaystyle \frac{V}{\omega L} \div \displaystyle \frac{V}{R} = \displaystyle \frac{R}{\omega L} = \displaystyle \frac{R}{2\pi f L}\)

\(Q = \omega C ÷ \displaystyle \frac{V}{R} = \omega CR = 2\pi f CR\)

One characteristic of parallel RLC circuits is that the impedance becomes very large at resonance, which consequently limits the current flowing through the circuit. This occurs because the inductor and capacitor exchange energy at resonance, each generating currents in opposite directions, making the circuit exhibit high impedance from an external perspective. This phenomenon allows the circuit to efficiently pass signals at the resonance frequency while blocking signals at other frequencies, acting as a filter.

Moreover, when compared to series RLC circuits, the role of resistance in parallel LC circuits is to provide a damping effect on the circuit’s bandwidth. This damping effect controls the sharpness of the resonance, preventing excessively sharp resonance and resulting in more stable circuit operation.

Importantly, the Q factor of a parallel resonant circuit is related differently from that of a series resonant circuit. Specifically, the Q factor of a parallel resonant circuit is defined as the reciprocal of the Q factor formula for a series circuit. This difference arises because the Q factor in a series circuit is expressed as the ratio of inductance to resistance, while in a parallel circuit, it is characterized by the reciprocal of this ratio. This distinction stems from the fundamental differences in how both types of circuits handle energy.

Examples of Resonant Circuit Applications

How are the series and parallel resonant circuits, which we have explained so far, actually applied in our daily lives?

For example, applications of series resonant circuits include television and radio receivers.

By utilizing the characteristic of series resonance where the current becomes maximum at a particular frequency, a circuit is pre-designed to have a resonance frequency equal to the desired frequency to be received. This allows large currents to flow at the target frequency, while almost no current flows at undesired frequencies.

Series resonant circuits are also applied in AC power filters and noise filters.

Examples of parallel resonant circuit applications include trap circuits used in multiband antennas and wideband amplification circuits.

For more information on noise countermeasures and EMC countermeasures using capacitors and coils, please refer to the following.

https://techweb.rohm.com/know-how/nowisee/

Electronic Components to Suppress Resonance

As electronic components continue to miniaturize and increase in density, noise control becomes crucial when utilizing RLC circuits. Therefore, a thorough understanding of proper noise control using inductors and frequency management is essential.

In electronic circuits, unintended resonance circuits can result in very high currents and voltages when the resonance frequency is reached. This makes noise interference more likely to occur. Hence, it is important to eliminate unintended resonance from circuits as much as possible. To suppress unintended resonance, damping resistors are used.

Additionally, there is a method that uses ferrite beads to bypass noise and convert it into heat.

For more detailed information about ferrite beads, the basic characteristics of inductors and their relationship to noise control, and their operation as low-pass filters, please refer to this article.

https://techweb.rohm.com/know-how/nowisee/8138/

Utilize Resonant Circuits in Circuit Design

A resonant circuit is one of the essential electronic circuits that exhibits resonance phenomena at a specific frequency. It is employed to enhance the quality of many electrical products’ designs.

As the miniaturization and high integration of electronic components progress, the importance of noise countermeasures is increasing. In this regard, understanding noise countermeasures using inductors is becoming crucial. A deep understanding of RLC resonant circuits is a significant point in gaining knowledge about inductor self-resonance and noise countermeasures.

Furthermore, the application of resonant circuits in product design becomes a central circuit when considering solutions to noise issues. By referring to the explanations and related information provided in this article, let’s appropriately utilize resonant circuits to improve the quality of electronic circuits.