## Introduction

RLC circuit is a circuit structure composed of resistance (R), inductance (L), and capacitance (C). The LC circuit is a simple example. RLC circuits are also called second-order circuits. The voltage or current in the circuit is the solution of a second-order differential equation, and its coefficients are determined by the circuit structure.

If the circuit components are regarded as linear components, an RLC circuit can be regarded as an electronic harmonic oscillator.

The natural frequency of this circuit is generally expressed as: (unit: Hz)

RLC circuits are often used as band-pass filters or band-stop filters, and the Q factor can be obtained by the following formula:

There are generally two types of RLC circuit composition: series and parallel.

The animation above demonstrates the operation of the LC circuit (RLC circuit without resistors). The charge is transferred back and forth between the capacitor plate and the inductor. The energy oscillates back and forth between the electric field (E) of the capacitor and the magnetic field (B) of the inductor. The RLC circuit works similarly. The difference is that the oscillating current decays to zero over time due to the resistance in the circuit.

## Catalog

1.1 What is Series RLC Circuit? 1.2 What is Transient Response of RLC Circuit? |

III The Difference Between Series Resonant Circuit and Parallel Resonant Circuit |

IV Application of RLC Circuit Resonance |

## I RLC Series Circuit

### 1.1 What is Series RLC Circuit?

Figure1. RLC Series Circuit

V-supply voltage

I-circuit current

R-resistance

L-Inductance

C-capacitance

In this circuit, all three elements are connected in series with the voltage. The main differential equations can be obtained by substituting the constitutive equations of the three elements into Kirchhoff's voltage law (KVL). From Kirchhoff's voltage law:

are the voltages across R, L, and C respectively, and V(t) is the voltage of the power supply that changes with time. Substituting the constitutive equation to get:

In the case of a constant supply voltage, take the derivative of the above formula and divide by L to obtain the following second-order differential equation:

This equation can be written in a more common form:

α is called "attenuation", which is used to measure the attenuation rate of the transient response of this circuit when the external input is removed. ω0 is the angular resonance frequency. These two coefficients are given by:

The damping coefficient ζ is another commonly used parameter, defined as the ratio of α to ω0:

### 1.2 What is Transient Response of RLC Circuit?

Figure2. Transient Response

The figure shows the underdamped and overdamped responses of the series RLC circuit. The critical damping is drawn with a thick red curve. These drawings are unified when L = 1, C = 1 and ω_{0}=1. According to the value of different damping coefficient ζ, the solution of the differential equation has three different situations, namely: under-damping (ζ<1), over-damping (ζ>1), and critical damping (ζ=1).

The characteristic equation of the differential equation is:

The roots of this equation are:

The general solution of this differential equation is the linear superposition of two exponential functions:

The coefficients A1 and A2 are given by the boundary conditions of the specific problem.

The following video introduces how to analyze RLC circuits by way of second order differential equations. Both parallel and series RLC configurations are discussed in it, looking primarily at Natural Response, but also touching on Step Response.

RLC Circuit Response Explanation

#### 1.2.1 Over-damped response

The over-damped response (ζ>1) is:

Overdamping response is a transient current without oscillation attenuation.

#### 1.2.2 Underdamped response

The underdamped response (ζ<1) is:

Through the trigonometric identities, these two trigonometric functions can be expressed by a phased sine function:

The underdamped response is an attenuated oscillation with a frequency of ωd. The rate of oscillation decay is α. The α in the index describes the envelope function of the oscillation. B1 and B2 (or B3 and phase difference φ in the second form) are arbitrary constants and are determined by boundary conditions. The frequency ω_{d }is given by:

This is the so-called damped resonance frequency or damped natural frequency. It is the frequency at which the circuit naturally vibrates when driven by no external source. The resonant frequency ω0 is the resonant frequency of the circuit when it is driven by an external source, and is often called the undamped resonant frequency in order to facilitate the distinction.

#### 1.2.3 Critical damping response

The critical damping response (ζ=1) is:

### 1.3 Laplacian Domain

The Laplace transform can be used to analyze the AC transient and steady-state behavior of the RLC series circuit. If the waveform generated by the above voltage source is V(s) after Laplace transform (where s is the complex frequency s=σ+iω), then Kirchhoff’s voltage law is applied in the Laplace domain:

Among them, I(s) is the current after Laplace transform. Solve for I(s):

After rearranging, the following formula can be obtained:

#### 1.3.1 Laplace admittance

Solve for Laplace admittance Y(s):

The above formula can be simplified by using the parameters α and ωo defined in the above content, and we can get:

#### 1.3.2 Pole and zero

The zero point of Y(s) is s such that Y(s)=0: s=0 and |s|⟶ ∞; the pole of Y(s) is s such that Y(s)⟶ ∞. Solve the quadratic equation. Get:

The poles of Y(s) are the roots s1 and s2 of the characteristic equation of the differential equation mentioned above.

#### 1.3.3 Sine steady state

The sine steady state can be represented by letting s=jω, where j is the imaginary unit. Substitute this into the amplitude of the above equation:

The function of the current with ω as the variable isThere is a peak.

In this special case, ω in this peak is equal to the undamped natural resonance frequency:

### 1.4 RLC Series Resonance Formula

The so-called series resonance formula refers to the study of the energy value of the voltage and current of the series circuit to reach the same phase, and the inductance of the inductance in the circuit and the capacitive reactance in the capacitor are equal in value. Therefore, in the study of the resistance characteristics of the circuit, In the case of a given terminal voltage, the maximum current is released, and the active power consumed will also be the maximum.

Figure3. RLC series resonance formula

**Resonance definition:** The energy of the L and C_ elements in the circuit are equal. When a reactance element in the circuit releases energy, the other reactance element must absorb the same energy, that is, energy pulsation occurs between the two reactance elements.

When series resonance occurs:

Inductive reactance XL = capacitive reactance XC

Source voltage U = resistance voltage UR

Inductor voltage UL = Capacitor voltage UC

Inductor's reactive power QL = Capacitor's reactive power QC

Total circuit impedance Z=resistance value R

Apparent power S = resistance power P

Explanation: When the circuit resonates, it must have two components: inductor L and capacitor C, and the frequency corresponding to resonance is called "resonant frequency" or resonant frequency, generally we use fr to indicate.

### 1.5 Phasor Diagram of RLC Series Circuit

(1) Phasor diagram of voltage and current

U^{&}=U^{&}_{R}+U^{&}_{L}+U^{&}_{C}

_{}

_{Figure4. Phasor diagram of voltage and current}

_{}

_{Figure5. Phasor diagram of voltage and current}

(2) Voltage triangle

The relationship between the voltage triangle and the impedance triangle: divide the effective value of the voltage triangle by I to get the impedance triangle.

Figure6. Voltage triangle

● The relationship between the total voltage and the effective value of each part of the voltage:

● The effective value relationship between total voltage and total current: U=I|Z|

● The phase difference relationship between total voltage and total current:

## II RLC Parallel Circuit

Figure7. RLC Parallel Circuit

V-supply voltage

I-circuit current

R-resistance

L-Inductance

C-capacitance

The characteristics of the RLC parallel circuit can be handled by the duality (electrical circuits) of the circuit. The RLC parallel circuit is treated as the dual impedance of the RLC series circuit, so it can be analyzed in a similar way to the RLC series circuit.

The attenuation α of the RLC parallel circuit can be obtained by the following formula:

If the factor of 1/2 is not considered, the damping coefficient of the RLC parallel circuit is exactly the reciprocal of the damping coefficient of the RLC series circuit.

**Frequency domain**

Add the admittance of each element in parallel to obtain the admittance of this circuit:

After capacitors, resistors, and inductors are connected in parallel, the impedance at the resonance frequency is the maximum, which is the opposite of the case where capacitors, resistors, and inductors are connected in series. The RLC parallel circuit is an antiresonator.

In the figure below, it can be seen that if a constant voltage is used for driving, the frequency response of the current has a minimum value at the resonance frequency ω0=1/√LC. If it is driven by a constant current, the frequency response of the voltage has a maximum value at the resonance frequency, which is similar to the frequency response graph of the current in an RLC series circuit.

Figure8. Sinusoidal steady state analysis

Normalize with R = 1 ohm, C = 1 Farad, L = 1 Henry, and V = 1.0 Volt

## III The Difference Between Series Resonant Circuit and Parallel Resonant Circuit

In an AC circuit containing resistance, inductance and capacitance, the voltage at both ends of the circuit and its current are generally out of phase. If the circuit parameters or the power supply frequency are adjusted to make the current and the power supply voltage in phase, the circuit is resistive, which is called resonance for the working state of the circuit at this time.

Resonance is a specific phenomenon of sinusoidal AC circuits. It is widely used in electronics and communication engineering. However, in power systems, resonance may damage the normal operation of the system.

Resonance is generally divided into series resonance and parallel resonance. As the name implies, series resonance is the resonance that occurs in a series circuit. Parallel resonance is the resonance that occurs in a parallel circuit.

### 3.1 Series Resonance

#### 3.1.1 Introduction

In a series circuit composed of resistance, inductance and capacitance, when the capacitive reactance XC and the inductive reactance XL are equal, that is, XC=XL, the voltage U and the current I in the circuit have the same phase, and the circuit presents pure resistivity. This phenomenon is called series resonance. When the circuit is in series resonance, the total impedance in the circuit is the smallest, and the current will reach the maximum.

#### 3.1.2 Conditions for the occurrence of series resonance

In order to resonate in a series circuit, certain conditions must be met.

When UL=UC, that is, XL=XC,. Voltage and current are in phase, and series resonance occurs in the circuit. From ωL=1/ωC, ω0=1/√LC can be obtained, and the resonance frequency is f=f0=1/2π√LC.

#### 3.1.3 Characteristics of series resonance circuit

● Minimum total impedance

● When the power supply voltage is constant, the current is the largest

● The circuit is resistive, and the voltage on the capacitor or inductor may be higher than the power supply voltage

#### 3.1.4 Energy changes in the circuit at resonance

The circuit absorbs Q=0 from the power supply, and the circuit energy exchanges between the electric field and the magnetic field inside the circuit during resonance. The power supply only provides energy to R.

High voltage may damage the device. Series resonance should be avoided in the power system. And series resonance is widely used in radio engineering.

### 3.2 Parallel Resonance

#### 3.2.1 Introduction

In a circuit where an inductance and a capacitor are connected in parallel, when the size of the capacitor just makes the voltage and current in the circuit have the same phase, that is, when the power supply is consumed by resistance and becomes a resistance circuit, it is called parallel resonance.

Parallel resonance is a complete compensation. The power supply does not need to provide reactive power, only the active power required by the resistance. At resonance, the total current of the circuit is the smallest, and the current of the branch is often greater than the total current of the circuit. Therefore, parallel resonance is also called current resonance.

When parallel resonance occurs, a large current flows in the inductance and capacitance components, which will cause the fuse of the circuit to blow or burn the electrical equipment; however, it is often used in radio engineering to select signals and eliminate interference.

#### 3.2.2 Parallel resonance conditions

In the following two types of circuits

Figure9. Two types of circuits

The resonant frequency formula of (a) has been discussed above, and (b) is determined by

,

We can get.

Under normal circumstances, the coil resistance R is much smaller than XL, therefore, ignoring R we can getthat is f=f0=1/2π√LC.

#### 3.2.3 Features of parallel resonant circuit

● When the voltage is constant, the current is the smallest at resonance

● Maximum total impedance

● The circuit is resistive, and the branch current may be greater than the total current

## IV Application of RLC Circuit Resonance

### 4.1 Application of Series Resonance Circuit

The use of series resonance to generate power frequency high voltage, which is used in high voltage technology to do withstand voltage test for power equipment such as transformers, can effectively find dangerous concentrated defects in the equipment, and is the most effective and direct way to test the insulation strength of electrical equipment Methods. Used in radio engineering, series resonance is often used to obtain a higher voltage.

In the radio, the series resonance circuit is often used to select the radio signal. This process is called tuning. The following figure shows a typical circuit.

Figure10. A typical circuit for tuning

When the electric waves of various signals of different frequencies generate electric signals of different frequencies on the antenna, they are induced to the coil 2L through the coil 1L. If the oscillation circuit resonates to a certain signal frequency, the current of the signal in the loop is the largest, and a voltage CU higher than the signal voltage Q times is generated across the capacitor. For other signals of various frequencies, because no resonance occurs, the current in the loop is very small, which is suppressed by the circuit. Therefore, the capacitor C can be changed to change the resonant frequency of the loop to select the desired radio signal.

### 4.2 Application of Parallel Resonance Circuit

The application of LC parallel resonant circuit in communication electronic circuit is determined by its characteristics. Specifically, it mainly includes three categories. One is working in resonance, as a frequency-selective network application. At this time, it appears as a large resistance and outputs a larger voltage under the excitation of current; the second is working in detuning The state, present as inductive or capacitive at this time, together with other inductances and capacitors in the circuit, satisfies the oscillation conditions of the three-point oscillation circuit to form a sine wave oscillator; the third is to work in a detuned state, that is, to work on the amplitude-frequency characteristic curve Or one side of the phase-frequency characteristic curve to realize amplitude-frequency conversion, frequency-amplitude conversion, frequency-phase conversion, and phase-frequency conversion to form an angle modulation and demodulation circuit.

**(1) LC parallel resonant circuit used as frequency selective matching network**

Frequency selection is to select useful frequency components from the input signal and suppress useless frequency components or noise. In communication electronic circuits, the LC parallel resonant circuit is the most commonly used as a frequency selection network. It is widely used in high-frequency small-signal amplifiers, Class C high-frequency power amplifiers, mixers and other circuits. The common feature of these circuits is that the LC resonant circuit is not only a frequency-selective network. Through the connection of the transformer, it also plays the role of impedance transformation, reducing the impact of the amplifier tube or the load on the resonant circuit, and obtaining better selectivity. .

**(2) The LC parallel resonant circuit of the overtone crystal oscillator as a capacitor**

Under the action of the applied alternating voltage, in the mechanical vibration generated by the quartz crystal, in addition to the fundamental frequency mechanical vibration, there are many odd frequency overtones. When a crystal oscillator with a very high operating frequency is required, overtone crystal oscillators are often used. The figure below shows the overtone crystal oscillator.

Figure11. Circuit composition and reactance curve of L1C1 circuit

In the above figure, the quartz crystal and the CL branch are inductive. The quartz crystal, C2, and L1C1 loop together form a three-point oscillator. According to the composition principle of the three-point oscillator (shooting the same), the L1C1 resonant circuit should be capacitive. Assuming that the quartz crystal in the figure is working at the 5th overtone frequency, the nominal frequency is 5 MHz. In order to suppress the parasitic oscillation of the fundamental frequency and 3rd overtone, the L1C1 loop should be tuned between the 3rd and 5th overtone frequency, that is, 3~ Between 5 MHz. From the reactance characteristic curve of the L1C1 resonant circuit shown in Figure (b), it can be seen that for the 5th overtone frequency of 5 MHz, the L1C1 circuit is capacitive, and the circuit meets the three-point oscillation condition and can oscillate. For the fundamental and third harmonics that are less than the resonance frequency of the L1C1 loop, the loop has an inductive characteristic, which does not conform to the principle of different components and cannot produce oscillation. For overtones of 7 times and above, although the L1C1 circuit is also capacitive, the equivalent capacitance at this time is too large, the amplitude starting conditions cannot be met, and the oscillation cannot be generated.

**(3) LC parallel resonant circuit that realizes the functions of amplitude-frequency conversion and frequency-phase conversion**

The phase-frequency characteristic of the impedance of the LC parallel resonant circuit is a monotonous curve with a negative slope. The linear part of the curve can be used to perform a linear conversion between frequency and phase. This is mainly used in the phase frequency discrimination circuit; the same, the LC parallel resonant circuit The linear part of the impedance's amplitude-frequency characteristic curve can also perform the linear conversion between frequency and amplitude, so it has also been applied in the slope frequency discrimination circuit.

## V Frequently Asked Questions about RLC Circuit

1. Is LCR and RLC circuit same?

Yes. A RLC circuit (also known as a resonant circuit, tuned circuit, or LCR circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. This configuration forms a harmonic oscillator.

2. What is resonant frequency of RLC circuit?

What is Resonance in RLC circuit? Resonance is the phenomenon in the electrical circuit, where the output of the circuit is maximum at one particular frequency. And that frequency is known as the resonant frequency. At the resonant frequency, The capacitive reactance and inductive reactance are equal.

3. Is RLC circuit linear?

In RLC circuit, the most fundamental elements of a resistor, inductor and capacitor are connected across a voltage supply. All of these elements are linear and passive in nature.

4. What is bandwidth of RLC circuit?

The bandwidth of any system is the range of frequencies for which the current or output voltage is equal to 70.7% of its value at the resonant frequency, and it is denoted by BW.

5. What is second order circuit?

A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements.

6. What is first order circuit?

A first-order circuit can only contain one. energy storage element (a capacitor or an. inductor). The circuit will also contain.

7. What is half power frequency?

The frequencies for which current in a series RLC (or a series tuned) circuit is equal to 1/√2 (i.e. 70.71%) of the maximum current (current at resonance)are known as Half Power Frequencies.

8. What is natural response of RC circuit?

The natural response tells us what the circuit does as its internal stored energy (the initial voltage on the capacitor) is allowed to dissipate. It does this by ignoring the forcing input (the voltage step caused by the switch closing). The "destination" of the natural response is always zero voltage and zero current.

9. What is the difference between first order and second order filters?

The main difference between a 1st and 2nd order low pass filter is that the stop band roll-off will be twice the 1st order filters at 40dB/decade (12dB/octave) as the operating frequency increases above the cut-off frequency ƒc, point as shown.

10. What is the use of resonant circuit?

One use for resonance is to establish a condition of stable frequency in circuits designed to produce AC signals. Usually, a parallel (tank) circuit is used for this purpose, with the capacitor and inductor directly connected together, exchanging energy between each other.

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