When the inductive reactance and capacitive reactance are equal, the circuit resonates, that is to say, when 2 π fL=1/2 π fC
Where L is the inductance in Henry units and C is the capacitance in Farads units.
For a given L and a given C, this only occurs at one frequency: f=1/2 π√ (LC)
This frequency is called the resonant frequency, and the resonance in a circuit is the frequency at which the capacitive reactance equals the inductive reactance.
Let's take the problems in the problem pool as an example and calculate some resonance frequencies:
If R is 22 ohms, L is 50 microhenries, and C is 40 picofarads. The series resonant frequency of the series RLC circuit is 3.56 MHz.
f = 1 /2π√(LC)= 1 /(6.28 x√(50×10 -6 x 40×10 -12))= 1 /(2.8 x 10 -7)= 3.56 MHz
Please note that the resistance value is not important, and the resonant frequency is R=220 ohms or 2.2 megaohms.
If R is 33 ohms, L is 50 microhenries, and C is 10 picofarads. The parallel resonant frequency of the parallel RLC circuit is 7.12 MHz.
f = 1 /2π√(LC)= 1 /(6.28x√(50×10 -6 x 10×10 -12))= 1 /(1.4×10 -7)= 7.12 MHz
When an inductor and a capacitor are connected in series, the impedance of the series circuit at the resonant frequency is zero because the reactance is equal and opposite at that frequency. If there is a resistor in the circuit, it will affect the impedance alone. Therefore, the impedance of the series RLC circuit at resonance is approximately equal to the circuit resistance.
As the frequency goes through resonance, the current amplitude at the input of the series RLC circuit is maximized, because neither the capacitor nor the inductor increases the overall circuit impedance at the resonance frequency.
When an inductor and a capacitor are connected in parallel, the impedance becomes equal and opposite to each other again at the resonant frequency. However, since they are connected in parallel, the circuit is actually open circuit. Therefore, at resonance, the impedance of the circuit with parallel resistors and inductors and capacitors is approximately equal to the circuit resistance.
Because parallel LC circuits effectively open at resonance, the magnitude of the current at the input of the resonant parallel RLC circuit is minimized. The maximum circulating current in the components of a parallel LC circuit during resonance can cause the voltage across the series reactance to be greater than the voltage applied to them.
Another result of the mutual cancellation of inductance and capacitance reactance is that there is no phase shift at the resonant frequency, and the phase relationship between the current and voltage flowing through the series resonant circuit during resonance is that the voltage and current are in phase.
In an ideal situation, the impedance of a series LC circuit at the resonant frequency is zero, while the impedance of a parallel LC circuit at the resonant frequency is infinite. However, in the real world, resonant circuits do not function in this way. To describe the similarity between the behavior of a circuit and an ideal resonant circuit, we use the quality factor or Q. Since the inductance reactance is equal to the capacitance reactance at the resonant frequency, the Q of an RLC parallel circuit is the resistance divided by the inductance reactance. Inductance or capacitance:
Q=R/X L or R/X C
The Q of RLC series resonance (also known as frequency conversion resonance) circuit is the reactance of the inductor or capacitor divided by the resistance:
Q=X L/R or X C/R
Basically, the higher the Q, the more the resonant circuit behaves like an ideal resonant circuit, and the higher the Q, the lower the resistance loss in the circuit. Lower losses will increase the Q of inductors and capacitors, and the effect of increasing Q in the resonant circuit is to increase the internal voltage and circulating current.
Q is an important parameter in designing impedance matching circuits. The result of increasing Q in impedance matching circuits is a decrease in matching bandwidth, and circuits with lower Q will produce wider bandwidth, but at the cost of increased losses.
The parameter of the resonant circuit related to Q is the half power bandwidth, which refers to the bandwidth where the series resonant circuit will pass half of the input signal power, and the parallel resonant circuit will reject half of the input signal power.
We can use the Q of the circuit to calculate the half power bandwidth: bandwidth=f/Q