**I Introduction**

The LC circuit is a circuit composed of capacitors, inductors, resistors and other components and electronic devices that can generate oscillating current or have a filtering effect, and is also called a resonant circuit, tank circuit, or tuned circuit. The LC circuit formed by connecting the inductor L and the capacitor C is the simplest type of LC circuit.

LC circuits are widely used in radio technology and radio and television technology. The LC circuit is indispensable in various radio devices, equipment, measuring instruments, etc. This article will introduce what is the LC circuit, including its basic concepts, basic principles, working process and application circuit diagram.

**Catalog**

V Comparison of Two Types of LC circuits |

**II The Concept of LC Circuit and Resonance**

In an AC circuit with a resistor R, an inductor L, and a capacitor C, the phase of the voltage across the circuit and the current in it are generally different. If you adjust the parameters of the circuit components (L or C) or the power frequency, you can make them in the same phase, and the entire circuit appears purely resistive. When the circuit reaches this state, it is called resonance. In the resonant state, the total impedance of the circuit reaches or reaches the extreme value. According to different circuit connections, there are series LC circuit and parallel LC circuit.

The essence of resonance is that the electric field energy in the capacitor and the magnetic field energy in the inductance can be converted into each other. The sum of the electric field energy and the magnetic field energy remains constant at all times. The power supply does not need to convert energy back and forth with the capacitor or inductor, but only supplies the energy consumed by the resistance in the circuit.

Figure1. What is Resonance

The LC circuit is used to generate signals of a specific frequency, or to extract signals of a specific frequency only from more complex signals. It is suitable for important components such as oscillation circuits, filter circuits, tuners, and mixers. LC circuit is an ideal model, it ignores the energy dissipation caused by resistance.

Figure2. Energy Stored by a Capacitor

The LC circuit uses the energy storage characteristics of capacitors and inductors to alternately transform the two types of electromagnetic energy, that is to say, electrical energy and magnetic energy will have a maximum and minimum value, and there will be oscillation.

However, this is only an ideal situation. In fact, all electronic components will have losses. Energy will either be lost or leak out of the process of conversion between the capacitor and the inductor. The energy will continue to decrease, so the actual LC circuit needs An amplifying element that is either a triode or an integrated op amp and other electrical LC. Using this amplifying element, the continuously consumed oscillation signal is feedback amplified by various signal feedback methods, so as to finally output a signal with stable amplitude and frequency.

The frequency calculation formula is f = 1 / [2π√ (LC)],

Where f is the frequency and the unit is Hertz (Hz); L is the inductance and the unit is Henry (H); C is the capacitor and the unit is Farad (F).

Figure3. Energy Stored by an Inductor

**III Introduction of ****E****lectromagnetic ****P****rinciple of LC ****C****ircuit**

The concept of the electromagnetic field is highly generalized. This is a very rich concept. Although it includes the magnetic field of electrostatic field and electric current, the electromagnetic field is not a simple addition of electric field and magnetic field.

**(1) Several possible situations about the time-varying electric field generated by the magnetic field**

①A constant magnetic field does not generate an electric field: for example, the original coil of the transformer is always connected to the current power supply. Because the constant current generates a constant magnetic field, no induced current is generated in the secondary coil loop-no electric field that drives charge.

②The changing magnetic field generates an electric field: According to the knowledge of electromagnetic induction, when the magnetic field changes in the closed-loop, an induced current is generated in the loop. Maxwell has a deep insight that the conductor loop is only a tool to reflect the existence of an induced electric field. In essence, as long as there is a magnetic field that changes in space, an electric field will be generated-it is not an electric field generated by a charge.

③ A uniformly changing magnetic field produces a constant electric field: According to Faraday's law of electromagnetic induction, ε = Δф / Δt can be the same as above, and the conclusion can be drawn from Faraday's law of electromagnetic induction.

Figure4. Faraday’s Laws of Electromagnetic Induction

**(2) Regarding the generation of a magnetic field by an electric field, the following will be described in layers according to several possibilities of the time-varying electric field.**

①A constant electric field does not generate a magnetic field, for example, the space around a static charge has only an electrostatic field and no magnetic field-a constant electric field does not generate a magnetic field.

②The changing electric field generates a magnetic field. With his extraordinary genius, Maxwell believes that when the capacitor is charged and discharged, the conduction current is interrupted by the capacitor in another way-continuous, he pointed out that the change in the electric field in the capacitor is equivalent to the current-like the conduction current, it can Generate a magnetic field (but does not generate human Joule heat), that is, a changing electric field generates a magnetic field. Connect the parallel-plate capacitor used for large-scale demonstration to the induction coil, and place a free small magnetic needle between the capacitor plates. The deflection of the free small magnetic needle shows that the changing electric field generates a magnetic field.

- A uniformly changing electric field produces a constant magnetic field: if the charge on the capacitor changes uniformly, the conduction current I = ΔQ / Δt is a steady current, which generates a constant magnetic field in space. When the charge on the capacitor changes uniformly with time, it is necessary to cause a uniform change in the electric field between the plates. The uniformly changed electric field, like a steady conduction current, must generate a constant magnetic field in space.
- Unevenly changing electric field produces a changing magnetic field using a similar narrative method to draw conclusions.

Figure5. Magnetic Field Produced by Electric Current

(3) Electromagnetic field According to the reasoning of the above two aspects, the extension points out: In general, the magnetic field generated by an unevenly changing electric field (such as an oscillating current) also changes unevenly, and this magnetic field must also produce an unevenly changing electric field. It can be seen that the changing electric field and magnetic field are always related to each other, forming an inseparable unity, which is the electromagnetic field.

**Conditions for generating electromagnetic fields****:**

- Generated by static charge.
- Generated by a uniformly changing magnetic field.
- Produced by steady current.
- Generated by a uniformly changing electric field. Interdependent non-uniformly changing electric and magnetic fields.

Figure6. Electromagnetic Fields

**IV The Operation of LC Circuit**

**（1）****Charging completed (discharge start):** the electric field can reach the maximum, the magnetic field energy is zero, and the induced current i = 0 in the loop.

**（2）****Discharge completed (charging started):** the electric field energy is zero, the magnetic field can reach the maximum, and the induced current in the loop reaches the maximum.

**（3）****Charging process:** the electric field energy is increasing, the magnetic field energy is decreasing, the current in the loop is decreasing, and the electric capacity on the capacitor is increasing. From the perspective of energy: the magnetic field can be transformed into the electric field.

**（4）****Discharge process:** the electric field energy is decreasing, the magnetic field energy is increasing, the current in the loop is increasing, and the amount of electricity on the capacitor is decreasing. From the energy point of view: the electric field can be transformed into the magnetic field.

In the process of generating an oscillating current in an oscillating circuit, the charge on the plate of the capacitor, the current through the coil, and the magnetic field and electric field associated with the current and charge all periodically change. This phenomenon is called electromagnetic oscillation.

Figure7. Tuned Circuit

**V Comparison of ****T****wo ****Types of ****LC Circuits**

**5.1 Capacitive ****F****eedback ****O****scillation ****C****ircuit**

**5.1.1 Circuit Composition**

Figure8. Capacitive Feedback Oscillation Circuit

In order to obtain a better output voltage waveform, if the capacitor in the inductive feedback oscillation circuit is replaced with an inductor, the inductor is replaced with a capacitor, and after the conversion, the common terminal of the two capacitors is grounded, and the collector resistance Rc is increased, The capacitor feedback oscillation circuit is obtained, as shown on the right. Because the three terminals of the two capacitors are respectively connected to the three poles of the transistor, it is also called a capacitor three-point circuit.

**5.1.2 Working ****P****rinciple**

(1) According to the judgment method of the sine wave oscillation circuit, observe the circuit shown in the above figure, which includes four parts: the amplifier circuit, the frequency selection network, the feedback network and the nonlinear element (transistor);

(2) The amplifier circuit can work normally;

(3) Disconnect the feedback, add the input voltage with frequency f0, and given its polarity, determine the polarity of the feedback voltage obtained from C2 is the same as the input voltage. The polarity is as shown.

(4) As long as the circuit parameters are properly selected, the circuit can meet the amplitude condition and produce a sine wave oscillation.

**5.1.3 Oscillation ****F****requency and ****S****tarting ****C****onditions**

Oscillation frequency

Feedback coefficient

Vibration conditions

**5.1.4 Advantages and ****D****isadvantages**

The output voltage waveform of the capacitive feedback oscillation circuit is good, but if the oscillation frequency is adjusted by changing the capacitance method, it will affect the feedback coefficient and the starting condition of the circuit; and if the oscillation frequency is adjusted by changing the inductance method, it is more difficult; Commonly used in the occasion of fixed oscillation frequency. When the adjustable range of the oscillation frequency is not large, the circuit shown in the figure on the right can be used as the frequency selection network.

Figure9. Frequency Selective Network with Adjustable Frequency

**5.1.5 Measures to Stabilize the Oscillation Frequency**

To increase the frequency of the capacitive feedback oscillation circuit, the capacitance of C1 and C2 and the inductance of L must be reduced. In fact, when C1 and C2 are reduced to a certain degree, the interelectrode capacitance of the transistor and the stray capacitance in the circuit will be included in C1 and C2, thus affecting the oscillation frequency. These capacitors are equivalent to the input capacitance Ci and output capacitance Co of the amplifier circuit. The improved circuit and equivalent appliances are shown in the figure below. Because the inter-electrode capacitance is affected by temperature, the stray capacitance is difficult to determine. In order to stabilize the oscillation frequency, a small-capacity capacitor C3 is connected in series with the inductor branch, and C3 <

Oscillation frequency

Almost has nothing to do with C1 and C2, so does Ci and Co, so the frequency stability is high.

Figure10. Improvement of Capacitive Feedback Oscillation Circuit and Equivalent Circuit

**5.2 Inductive Feedback Oscillation Circuit**

**5.2.1 Circuit Composition**

In order to overcome the disadvantage that the primary coil and the secondary coil of the transformer are not tightly coupled in the feedback oscillation circuit of the transformer, N1 and N2 of the transformer feedback oscillation circuit can be combined into one coil. As shown in the figure, in order to strengthen the resonance effect, the capacitor C is connected across the entire coil to obtain an inductive feedback oscillation circuit.

Figure11. Inductive Feedback Oscillation Circuit

**5.2.2 ****Working Principle**

- Observe the circuit, it contains four parts of the amplifier circuit, frequency selection network, feedback network and nonlinear components (transistors), and the amplifier circuit can work normally.
- Use the instantaneous polarity method to judge whether the circuit meets the sine wave oscillation phase conditions:

① Disconnect the feedback, add the input voltage with frequency f0, and give its polarity

②It is judged that the polarity of the feedback voltage obtained from N2 is the same as the input voltage

③ Therefore, the circuit satisfies the phase condition of sine wave oscillation, and the instantaneous polarity of each point is as shown in the above figure.

- As long as the circuit parameters are properly selected, the circuit can satisfy the amplitude condition and produce a sine wave oscillation.

The following figure shows the AC path of the inductive feedback oscillation circuit. The three ends of the primary coil are connected to the three poles of the transistor, so the inductive feedback oscillation circuit is called an inductive three-point circuit.

Figure12. AC Path of Inductive Feedback Oscillation Circuit

**5.2.3 Oscillation Frequency and Starting Conditions**

Oscillation frequency

Feedback coefficient

Vibration conditions

**5.2.4 Advantages and Disadvantages**

In the inductive feedback oscillation circuit, the coupling between N2 and N1 is tight, the amplitude is large, and it is easy to oscillate; when C uses a variable capacitor, the oscillation frequency with a wide adjustment range can be obtained, and the highest oscillation frequency can reach tens of MHz. Because the feedback voltage is taken from the inductance, it has a large reactance to high-frequency signals. The feedback signal contains more harmonic components, and the output voltage waveform is not good.

**VI Series LC Circuit and Parallel LC Circuit**

**6.1 Series LC Circuit**

**6.1.1 Concept**

In the LC circuit, the corresponding frequency value when the inductive reactance and capacitive reactance are equal is called the resonance frequency, that is, XC = XL. As shown in the figure below, the voltage u and the current i in the circuit are in the same phase, and the circuit is resistive. This phenomenon is called series resonance. When the circuit has series resonance, the impedance of the circuit Z = √R ^ 2 + (XC-XL) ^ 2 = R, the total impedance in the circuit is the smallest, and the current will reach the maximum value.

Figure13. Series Resonance Frequency

**6.1.2 Characteristics of Series LC Circuit**

When the input signal passes through the series LC circuit, according to the characteristics of the inductor and the capacitor, the higher the signal frequency, the larger the impedance of the inductor, and the smaller the impedance of the capacitor. The larger the impedance, the greater the attenuation of the signal. The signal with a higher frequency will be greatly attenuated by the inductor, while the DC signal cannot pass through the capacitor. When the frequency of the input signal bow is equal to the frequency of the LC resonance, the impedance of the LC series circuit is minimum. Signals at this frequency easily output through capacitors and inductors. At this time, the LC series resonant circuit plays the role of frequency selection.

Figure14. The Frequency Characteristic for LC Series Resonant Circuits

**6.1.3 Formula**

When series resonance occurs:

Inductive reactance XL = capacitive reactance XC

Source voltage U = resistance voltage UR

Inductor voltage UL = capacitance voltage UC

Inductive reactive power QL = capacitive reactive power QC

The total impedance of the circuit ∣Z∣ = resistance R

Apparent power S = resistance power P

**6.2 Parallel LC Circuit**

**6.2.1 Concept**

The parallel LC resonance circuit is formed by connecting an inductor and a capacitor in parallel. In a parallel resonant circuit, if the current in the coil is equal to the current in the capacitor, the circuit reaches the state of parallel resonance. In this circuit, except for the LC parallel part, the impedance change of other parts has almost no effect on energy consumption. Therefore, the stability of this circuit is good, and it is used more than series resonance circuits.

Parallel resonance is a complete compensation. The power supply does not need to provide reactive power, only the active power required by the resistor. 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 may cause an accident that the circuit fuse blows or burns electrical equipment; but it is often used to select signals and eliminate interference in radio engineering.

Figure15. Parallel LC Circuit

**6.2.2 Characteristics of Parallel LC Circuit**

(1) The current and voltage phases are the same, and the circuit is resistive.

(2) The series impedance is the smallest and the current is the largest: Z = R, then I = U / R.

(3) The voltage at the inductor end and the voltage at the capacitor end are equal in magnitude, opposite in phase, and compensate each other. The voltage at the resistor end is equal to the power supply voltage.

(4) The ratio of the inductance (capacitance) terminal voltage to the power supply voltage at resonance is called the quality factor Q, which is also equal to the ratio of inductive reactance (or capacitive reactance) and resistance. When Q >> 1, the voltages on L and C are much larger than the power supply voltage (similar to resonance). This is called series resonance and is often used to amplify the signal voltage; however, series resonance should be avoided in the power supply circuit.

**VII Application of LC Circuit**

**7.1 Application Note of LC Circuit**

In amplifier circuits and other forms of signal processing circuits, parallel LC resonance circuits and series LC resonance circuits are used very frequently.

**(1) Frequency selection circuit or frequency selection amplifier**

The LC circuit can form a frequency selection circuit or a frequency selection amplifier circuit, which is used to select a signal of a desired frequency among a large number of signals for amplification. This circuit is widely used in radio, television and other circuits, as well as in sine wave oscillator circuits.

**(2) Absorption circuit**

The LC circuit can constitute an absorption circuit, which absorbs a signal of a certain frequency among signals of many frequencies, that is, performs attenuation, and removes signals of this frequency from signals of many frequencies.

**(3) Wave blocking circuit**

The LC circuit can form a wave blocking circuit, which prevents signals of a certain frequency from passing through amplifier circuits or other circuits from signals of many frequencies.

**(4) Phase shift circuit**

An LC parallel circuit is used to form a phase shift circuit, and the signal is phase shifted.

**7.2 LC ****A****pplication ****C****ircuit ****D****iagram**

LC parallel and series resonant circuits have many changes in application, which is a difficult point in circuit analysis.

**(1) LC free resonance circuit**

The figure below shows the LC free resonance circuit. L in the circuit is an inductor, C is a capacitor, and L and C form a parallel circuit.

Figure16. LC Free Resonance Circuit

**(2) LC parallel resonance phase shift circuit**

The following figure shows the phase shift circuit composed of LC parallel resonance circuit. VT1 in the circuit constitutes a primary amplifier; R1 is its base bias resistor; R3 is its emitter resistor; C4 is the emitter bypass capacitor; L1 and C3 constitute an LC parallel resonance circuit, and R2 is the damping resistor of this resonant circuit.

Figure17. LC Parallel Resonance Phase Shift Circuit

By adjusting the inductance of L1, the phase of the output signal voltage can be changed to achieve the purpose of phase shift.

**(3) LC series resonance absorption circuit**

The function of the absorption circuit is to remove the signal of a certain frequency in the input signal. The following figure shows the absorption circuit composed of LC series resonant circuit. VT1 in the circuit constitutes a primary amplifier. L1 and C1 form the LC series resonance absorption circuit, and the resonance frequency is connected between the input terminal of VT1 and the ground.

Figure18. LC Series Resonance Absorption Circuit

**(4) Series resonance high-frequency boost circuit**

The figure below shows a high-frequency boost circuit composed of LC series circuits. VT1 in the circuit constitutes a first-stage common-emitter amplifier, and L1 and C4 constitute an LC series resonance circuit, which is used to boost high-frequency signals. The resonant frequency of the series resonant circuit of L1 and C4 is higher than the highest frequency of the working signal of this amplifier.

Figure19. Series Resonance High Frequency Boost Circuit

Since the impedance of the L1 and C4 circuits at resonance is the smallest, and the negative feedback resistance is the smallest after paralleling with the emitter negative feedback resistance R4, the amplification factor at this time is the largest. In this way, the high-frequency signal close to the resonance frequency is improved.

For input signals with a frequency much lower than the resonant frequency, the L1 and C4 circuits have no boost effect on them, because the L1 and C4 circuits are in a detuned state and their impedance is very large, and the negative feedback resistance at this time is R4.

**(5) Input tuning circuit**

The radio selects the required radio stations from many radio stations by input tuning circuit. The input tuning circuit is also called antenna tuning circuit, because there is a cash register antenna in this tuning circuit.

The following figure shows a typical input tuning circuit. L1 in the circuit is the primary winding of the magnetic rod antenna, L2 is the secondary winding of the magnetic rod antenna; C1-1 is a connection of the double variable capacitor, which is the antenna connection, and C2 is the high-frequency compensation capacitor, which is the trimming capacitor. It is usually attached to a double variable capacitor.

Figure20. Input Tuning Circuit

**The working principle of input tuned circuit:**

The primary winding L1 of the magnetic rod antenna, variable capacitor c1-1, and trimmer capacitor C2 constitute LC series resonance circuit. When resonance occurs in the circuit, the energy in L1 is the largest, that is, the voltage amplitude of the signal of the resonant frequency at both ends of L1 is much larger than that of the signal of the non-resonant frequency. In this way, the amplitude of the resonant frequency signal output from the secondary winding L2 through magnetic coupling is the maximum.

The following figure shows the practical input tuning circuit.

Figure21. Pratical Input Tuning Circuit

**VIII One Quiz Related to LC Oscillator **

**8.1 Question**

The output of a LC oscillator is often fed into a common collector amplifier stage. The reason for this is:

- a) To provide extra voltage gain.
- b) To provide negative feedback.
- c) To reduce loading on the tank circuit.
- d) To convert the sine wave output to a square wave.

**8.2 Answer**

C

## Ⅸ FAQ

**1. What does an LC circuit do?**

LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal; this function is called a bandpass filter.

**2. How do you solve an LC circuit?**

Begin with Kirchhoff's circuit rule. Take the derivative of each term. The voltage of the battery is constant, so that derivative vanishes. The derivative of charge is current, so that gives us a second-order differential equation.

**3. What makes an ideal LC circuit?**

An LC circuit is an electronic circuit made up of an inductor and a capacitor. ... An ideal LC circuit does not have resistance. At the LC circuit energy saves in the capacitor's electric field. U is energy and q is electric charge.

**4. Why do LC circuits resonate?**

Resonance of a circuit involving capacitors and inductors occurs because the collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor, and then the discharging capacitor provides an electric current that builds the magnetic field in the inductor.

**5. What is the difference between RC and LC circuits?**

RC - a resistor and capacitor in series. Exhibits charging behavior with a characteristic time constant with DC voltage source. ... LC (and RLC) - an inductor and capacitor (and resistor) in series. If initially charged, has oscillatory behavior (damped if also has a resistor).

**6. What are the different properties of the LC circuit?**

An LC circuit is a closed loop with just two elements: a capacitor and an inductor. It has a resonance property like mechanical systems such as a pendulum or a mass on a spring: there is a special frequency that it likes to oscillate at, and therefore responds strongly to.

**7. Is an LC circuit first order?**

In electronics, the classic second-order system is the LC circuit. The LC circuit is one of the last two circuits we will solve with the full differential equation treatment.

**8. Where is energy stored in the LC circuit?**

The oscillations of an LC circuit can, thus, be understood as a cyclic interchange between electric energy stored in the capacitor, and magnetic energy stored in the inductor.

**9. What is the natural frequency of the LC circuit?**

The natural frequency of an LC - circuit is 1,25000 cycles per second.

**10. What is a parallel LC circuit?**

Parallel LC Circuit. The Voltage across each terminal of different elements in a parallel circuit is the same. Hence the voltage across the terminals is equal to the voltage across the inductor and the voltage across the capacitor.

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