What is resonance

Resonance Definition

Resonance is a simple harmonic vibration in physics. An object vibrates under the action of a restoring force that is proportional to the displacement from the equilibrium position and always points to the equilibrium position. Its dynamic equation is F=-kx.
The phenomenon of resonance is that the current increases and the voltage decreases. The closer to the center of resonance, the faster the rotation and change of the ammeter, voltmeter, and power meter. However, the difference from a short circuit is that there will be no zero-sequence quantity.
In physics, there is a concept called resonance: when the frequency of the driving force is equal to the natural frequency of the system, the amplitude of the forced vibration of the system is the largest. This phenomenon is called resonance. The resonance in the circuit actually means the same thing: when the frequency of the circuit excitation is equal to the natural frequency of the circuit, the amplitude of the electromagnetic oscillation of the circuit will also reach a peak value. In fact, resonance and resonance express the same phenomenon. This phenomenon with the same essence is just called differently in different fields.
The radio uses the resonance phenomenon. When you turn the knob of the radio, you are changing the natural frequency of the circuit inside. Suddenly, at a certain point, the frequency of the circuit is equal to the frequency of the electromagnetic wave in the air that was originally invisible, so they resonate. The distant sound came from the radio. This sound is the product of resonance.
·Resonant circuit
Circuits composed of inductance L and capacitance C that can resonate at one or several frequencies are collectively called resonant circuits. In electronic and radio engineering, we often need to select the electrical signals we need from many electrical signals, while suppressing or filtering out the electrical signals we don't need. For this purpose, we need a selection circuit, namely a resonant circuit. On the other hand, in power engineering, some hazards may occur due to resonance in the circuit, such as overvoltage or overcurrent. Therefore, the study of resonant circuits is of great significance both in terms of utilization and in terms of limiting its hazards.

§9.1 Series resonant circuit
1. Resonance and resonant conditions
2. Intrinsic resonant frequency of the circuit
3. Resonant impedance, characteristic impedance and quality factor

1. Resonance and resonant conditions
The resonant circuit composed of inductance L and capacitance C in series is called a series resonant circuit, as shown in Figure 9-1-1. Where R is the total resistance of the circuit, that is, R=RL+RC, RL and RC are the resistances of the inductor and capacitor respectively; is the voltage of the voltage source, and ω is the angular frequency of the power supply. The input impedance of the circuit is
Where X=ωL-1/ωC. Therefore, the modulus and phase angle of Z are From
formula (9-1-2), it can be seen that when X=ωL-1/ωC=0, φ=0, which is the same as. At this time, we say that the circuit resonates. The condition for the circuit to reach resonance is
X=ωL-1/ωC=0 (9-1-3)
Figure 9-1-1 Series resonant circuit

2. The inherent resonant frequency of the circuit
From formula (9-1-3),
ω0 is called the inherent resonant angular frequency of the circuit, or resonant angular frequency for short, because it is only determined by the parameters L and C of the circuit itself. The resonant frequency of the circuit is
3. Resonant impedance, characteristic impedance and quality factor
The input impedance of the circuit at resonance is called the resonant impedance, represented by Z0. Since the reactance X=0 at resonance, the resonant impedance is
Z0=R from formula (9-1-1)
. It can be seen that Z0 is a pure resistor and its value is the smallest.
The inductive reactance XL0 and the capacitive reactance XC0 at resonance are called the characteristic impedance of the circuit, represented by ρ. That is,
It can be seen that ρ is only related to the circuit parameters L and C, but not to ω, and XL0=XC0.
The quality factor is represented by Q, which is defined as the ratio of the characteristic impedance ρ to the total resistance R of the circuit, that is,
Q=ρ/R=XL0/R=XC0/R
. In electronic engineering, the Q value is generally between 10-500. From the above formula,
ρ=XL0=XC0=QR.
Therefore, another expression of the resonant impedance is
Z0=R=ρ/Q.
In circuit analysis, the quality factor of circuit components is generally used. The quality factors of inductance components and capacitance components are defined as
That is, the quality factor Q of the circuit, which can actually be considered as the quality factor QL of the inductance component. If the quality factor Q is mentioned in the future, it refers to QL.

4. Characteristics of the circuit at resonance
V. Frequency characteristics of the circuit

IV. Characteristics of the circuit at resonance
The characteristics of the resonant circuit at resonance are:
1. The resonant impedance Z0 is a pure resistor, and its value is the minimum, that is, Z0=R.
2. The current is in phase with the power supply voltage, that is, φ=ψu-ψi=0.
3. The modulus of the current reaches the maximum value, that is, I=I0=US/R0, and I0 is called the resonant current.
4. High voltage may appear at both ends of L and C, that is,
UL0=I0XL0=US/R XL0=QUS
UC0=I0XC0=US/R XC0=QUS
It can be seen that when Q?1, UL0=UCO?US, so series resonance is also called voltage resonance. This phenomenon of high voltage is extremely useful in radio and electronic engineering, but it is harmful in power engineering and should be prevented.
From the above two formulas, we can get another expression and physical meaning of Q, that is,
Q=UL0/US=UC0/US
5. The vector diagram of the circuit at resonance is shown in Figure 9-1-2. As can be seen from the figure, the voltages at both ends of L and C are equal in magnitude and opposite in phase, canceling each other out. Therefore, .

5. Frequency characteristics of circuits
The functional relationship between the physical quantities of a circuit and the power supply frequency ω is called the frequency characteristics of the circuit. The purpose of studying the frequency characteristics of a circuit is to further study the selectivity and passband of the resonant circuit.
1. The frequency characteristics of impedance are as follows: The inductive reactance XL, the capacitive reactance XC, the reactance X, and the impedance
modulus XL, XC, and XC are respectively Their frequency characteristics are shown in Figure 9-1-3 (a), collectively referred to as the frequency characteristics of impedance. As can be seen from the figure, when ω=0, , when 0<ω<ω0, X<0, the circuit is capacitive; when ω=ω0, X=0, the circuit is purely resistive, ; when ω0<ω<∞, X>0, the circuit is inductive; when ω→∞, .
The phase-frequency characteristics of impedance are the relationship between the impedance angle φ and ω, that is,
when ω=0, φ=-π/2; when ω=ω0, φ=0; when ω=∞, φ=π/2. Its curve is shown in Figure 9-1-3 (b), which is called the phase frequency characteristic.
2. Current frequency characteristic
When ω=0, I=0; when ω=ω0, I=I0=US/R; when ω=∞, I=0. Its curve is shown in Figure 9-1-3 (c), which is called the current frequency characteristic
3. Voltage frequency characteristic The effective values ​​of capacitor and inductor voltage are
UC=I/ωC
UL=IωL
. Since Q?1, ω0 is very high in electronic engineering, and ω changes around ω0, 1/ωC≈1/ω0C, ωL≈ω0L. Therefore, the above two equations can be written as
UC=UL≈I/ω0C=Iω0L
, that is, UC and UL are approximately proportional to the current I. The frequency characteristics of UC and UL are similar to the frequency characteristics of current I, as shown in Figure 9-1-3 (d). In the figure, UL0=UCO=I0X=I0XC0.

VI. Selectivity and passband
4. Relative frequency characteristics
From formula (9-1-5), we can see that the current I is not only related to R, L, C, but also to US, which makes it difficult for us to accurately compare the influence of circuit parameters on the circuit frequency characteristic curve. For this reason, we will study the relative current frequency characteristics.
The functional relationship between the relative current value I/I0 and ω/ω0 (or f/f0) described in the above formula is the relative current frequency characteristic. It can be seen that the right side of the above formula has nothing to do with US, and its frequency characteristics are shown in Figure 9-1-4.
Figure 9-1-4 Relative frequency characteristics
5. Relationship between Q value and frequency characteristics
According to formula (9-1-6), the relative current frequency characteristic curve at different Q values ​​can be drawn, as shown in Figure 9-1-5. As can be seen from the figure, the higher the Q value, the sharper the curve; the lower the Q value, the flatter the curve. That is, the sharpness of the curve; is proportional to the Q value.
Figure 9-1-5 Relationship between Q value and frequency characteristics VI. Selectivity and passband 1. Selectivity The selectivity of a resonant circuit is the ability to select useful electrical signals. As shown in Figure 9-1-6, when many voltage signals of different frequencies are connected to the series circuit of R, L, and C, if the inherent resonant frequency ω0 of the circuit is adjusted (here, the capacitor C is adjusted), the frequency signal we need (such as ω2) can resonate with the circuit, even if ω0=ω2, so that the current in the circuit reaches the maximum value (resonant current). When the Q value of the circuit is very high, the voltage UC (or UL) output from both ends of C (or both ends of L) is also the maximum; while the current generated by the electrical signal we do not need (such as the voltage of ω1 and ω3) in the circuit is very small, and its output voltage is of course also small. This achieves the purpose of selecting the useful electrical signal ω2. Obviously, the higher the Q value of the circuit, the sharper the frequency characteristic, and thus the better the selectivity. Figure 9-1-6 Selectivity of series resonant circuit 2. Passband (1). Definition: When the ω (or f) of the power supply changes, the frequency range of the current (or the passband) is called the passband of the circuit, as shown in Figure 9-1-7. The passband is represented by Δω or Δf, that is, ω=ω2-ω1 or f=f2-f1 (2). Calculation formula It can be seen that Δω (or Δf) is inversely proportional to the Q value, that is, it contradicts the selectivity. The relative passband is defined as Δω/ω0=Δf/f0=1/Q Figure 9-1-7 Definition of circuit passband (3). Half-power point frequency We call f1 (or ω1) the lower boundary frequency and f2 (or ω2) the upper boundary frequency. Since the power consumed in the circuit at resonance is P0=I02R, and at f1 and f2, the power consumed in the circuit. It can be seen that at the upper and lower boundary frequencies f1 and f2, the power consumed in the circuit is equal to half of P0, so the upper and lower boundary frequencies are also called half-power point frequencies. Under sinusoidal excitation, for a passive circuit containing both L and C, if its terminal voltage and input current are in phase, such a specific circuit working state is called resonance. Usually, a sinusoidal AC circuit with voltage leading current is called an inductive circuit. At this time, the reactive power absorbed by the circuit reflects the rate of magnetic field energy exchange between the external power supply and the circuit. On the contrary, if the voltage lags behind the current, the reactive power reflects the rate of electric field energy exchange between the external power supply and the circuit, and the circuit is capacitive. In the resonant state, the voltage and current are in phase, and the reactive power is zero, indicating that there is no exchange of electric field energy or magnetic field energy between the circuit and the external power supply. Of course, this does not mean that the circuit does not contain electric field energy or magnetic field energy, but only indicates that when vibrating, the magnetic field energy in the circuit L and the electric field energy in C just form a system and exchange inside the circuit. [Edit this paragraph] Resonance analysis Resonant circuits all have a characteristic, the capacitive reactance is equal to the inductive reactance, and the circuit is resistive: then there is ωL=1/ωC because LC is a known condition, so the resonant frequency point can be calculated. Quality factor Q=ωL/R, the so-called quality factor is 28, then the parallel resonant circuit is a 28-fold reduction in current; if it is a series resonant circuit, then the voltage is increased by 28 times. Now the voltage at the series resonance point is the applied voltage multiplied by the quality factor. If the known conditions tell you that the applied voltage is the peak value, then just multiply it directly; if the known conditions tell you that the applied voltage is the effective value, then you also need to multiply the calculated voltage by 1.414 to get the peak value.

Additional answer:
Think about it, because there is a prerequisite ωL=1/ωC
quality factor Q=ωL/R, I considered the inductor, so shouldn't the capacitor also be considered?
First of all, you need to know what equipment will be used in the actual application of series resonance:
to resonate, of course, ωL=1/ωC must be satisfied. Among them, we can change three parameters to achieve resonance, capacitor C, inductor L and frequency ω. Then the test object in the actual application is a capacitor, and the size of the capacitor is fixed. We can change the size of the capacitor by connecting capacitors in series and parallel, but it is very troublesome; then we can change the inductor L. Adjustable inductors have been used before, but practical applications are very inconvenient and the volume is relatively large, so the most used later is to change the frequency, that is, frequency modulation power supply.
In the resonant circuit, first connect the power supply to the adjustable power supply, and the input voltage from the adjustable power supply to the secondary end of the excitation transformer, and the excitation transformer is transformed to a primary high voltage and then connected in series with the inductor, and the other end of the inductor is connected to the test object. Here, the multiple of the voltage increased by the quality factor Q refers to the actual voltage applied to the test object, that is, the voltage at the other end of the inductor divided by the high-voltage side voltage of the excitation transformer.
Of course, the resonant transformer will also saturate. The excitation transformer is a transformer. As long as it is a transformer, it will have the problem of core saturation. In actual applications, we need to calculate the rated current of this transformer to see if it exceeds the actual capacity. If it exceeds the rated current of the inductor or the excitation transformer, it is not only a saturation problem, but also a problem of damaging the test equipment.
For example, if the capacitance of the test product is 0.24μF, the inductance is 500H, the primary rated current of the excitation transformer is 2A, and the rated current of the inductor is also 2A, then we calculate, ωL=1/ωC, then the resonant frequency is 91.28HZ, calculate, if I add 17.4KV voltage to the test product, then the primary current is equal to
I=ωCU=2πf CU=2*3.14*91.28*0.24*0.000001*17400=2.39A,
at this time the current exceeds the rated current of the test equipment, at this time we can calculate, connect the same inductor in series, the inductance becomes 1000H, the resonant frequency becomes 64.55HZ, and the primary current becomes 1.69A.
In our actual application, if the current is definitely greater than 2A, then we can generally do this, and then connect a reactor in parallel. At this time, the reactor can withstand 4A, and of course the inductance is also reduced by half, and then the primary current of the excitation transformer is changed to 4A. (The primary current of the excitation transformer can be changed by connecting windings in series and parallel) At this time, if the resonant frequency cannot meet your requirements, you can achieve it by connecting capacitors in parallel and so on. §12-3 The resonant circuit is a single-port network containing inductor, capacitor and resistor elements. At certain operating frequencies, when the port voltage and current waveforms are in phase, the circuit is said to resonate. A circuit that can resonate is called a resonant circuit. Resonant circuits are widely used in electronics and communication engineering. This section discusses the characteristics of the most basic RLC series and parallel resonant circuits when they resonate. 1. RLC series resonant circuit
Figure 12-15 (a) shows the RLC series resonant circuit, and Figure 12-15 (b) is its phasor model. From this, the driving point impedance is calculated as
· Circuit resonance

The phenomenon that a passive (meaning no independent power supply) linear time-invariant circuit containing an inductor and a capacitor shows pure resistance properties to the outside under the action of an external power supply of a certain frequency. This specific frequency is the resonant frequency of the circuit. A circuit with resonance as the main working state is called a resonant circuit. Radio equipment uses resonant circuits to complete functions such as tuning and filtering. The power system needs to prevent resonance to avoid overcurrent and overvoltage.
The resonance in the circuit includes linear resonance, nonlinear resonance and parametric resonance. The former is the resonance that occurs in a linear time-invariant passive circuit, with the resonance in a series resonant circuit as a typical example. Nonlinear resonance occurs in a circuit containing nonlinear elements. A circuit composed of an iron core coil and a linear capacitor in series (or parallel) (commonly known as a ferromagnetic resonance circuit) can cause nonlinear resonance. Under the action of sinusoidal excitation, fundamental resonance, high-order harmonic resonance, subharmonic resonance, and current (or voltage) amplitude and phase jump phenomena will occur in the circuit. These phenomena are collectively called ferromagnetic resonance. Parametric resonance is the resonance that occurs in a circuit containing time-varying elements. Parametric resonance may occur in a circuit with a capacitive load of a salient pole synchronous generator.
According to circuit theory, resonance is a sinusoidal voltage applied to an ideal (parasitic resistance-free) inductor or capacitor series circuit. When the sinusoidal frequency is a certain value, the capacitive reactance is equal to the inductive reactance, the impedance of the circuit is zero, and the circuit current reaches infinity; if a sinusoidal voltage is applied to a parallel circuit of an inductor and a capacitor, when the sinusoidal voltage frequency is a certain value, the total admittance of the circuit (admittance is the inverse of impedance) is zero, and the voltage on the inductor and capacitor components is infinite. The former is called series resonance, and the latter is called parallel resonance. It
can be expressed as:
Z=R+j(XL-XC) where Z is impedance, R is resistance, and XL-XC=X is inductive reactance + capacitive reactance = reactance. It can be clearly seen from the middle of the formula: when the inductive reactance XL is equal to the capacitive reactance XC, Z only contains the real component R, that is, pure resistance. This is resonance.
·Resonator

When a vibrating object is regarded as a particle (or particle, point charge) without considering the volume, the vibrating object is called a harmonic oscillator.
The so-called resonance is a simple harmonic vibration in kinematics. This vibration is the movement of an object back and forth near a position away from the vibration center position (called the equilibrium position). In this vibration form, the magnitude of the force on the object is always proportional to the distance it deviates from the equilibrium position, and the direction of the force always points to the equilibrium position.
Electrical resonance refers to the intensity of electromagnetic physical quantities fluctuating around a median value, which is also similar to kinematic resonance.
Vibration is another form of particle motion. The vibration of a harmonic oscillator is also the simplest ideal vibration model. Here, the steady-state Schrödinger equation will be applied to one-dimensional harmonic oscillator and three-dimensional harmonic oscillator systems to solve their wave functions and energy.

·Resonator
Quartz crystal resonator
1. Concept:
Quartz crystal resonator is also called quartz crystal, commonly known as crystal oscillator. It is a resonant element made using the piezoelectric effect of quartz crystal. When used with semiconductor devices and resistor-capacitor components, a quartz crystal oscillator can be formed.
Piezoelectric effect:
Applying mechanical force to certain dielectrics causes the relative displacement of the positive and negative charge centers inside them, generating polarization, which leads to the appearance of bound charges of opposite signs on the surfaces of both ends of the dielectric. Within a certain stress range, the mechanical force and charge are linearly reversible. This phenomenon is called the piezoelectric effect.
Function: Provide system oscillation pulses, stabilize frequency, and select frequency.
2. Main parameters:
a. Nominal frequency: The resonant center frequency of the crystal oscillator under specified conditions.
b. Adjustment frequency difference: The maximum deviation value of the operating frequency at the reference temperature relative to the nominal frequency under specified conditions. (ppm).
c. Temperature frequency difference: Under specified conditions, within the entire operating temperature range, the allowable deviation value of the operating frequency relative to the reference temperature.
d. Load resonant resistance: The resistance value of the crystal oscillator in series with the specified external capacitor at the load resonant frequency.
e. Load capacitance: It refers to the effective external capacitance that determines the load resonant frequency together with the crystal oscillator. Common standard values ​​are: 12pF, 16pF, 20pF, 30pF.
The resonator plays a resonant role. It can act purely resistively with capacitive or inductive loads.