In an AC circuit, when the active element r, capacitive C, and inductive L are connected in series, phenomena such as series resonance occur. This phenomenon can be exploited (for example, in radio engineering), but it can also cause serious harm (in high-voltage electrical equipment, series resonance can lead to serious consequences).
The schematic diagram and vector diagram of series resonance are shown below:
Including all three elements of this circuit in sequence will satisfy the following conditions:
It must also be remembered that resonance may only occur at φ=0, where in series connection, it is equivalent to the relationship X=ω L -1/(ω C)=0, that is, the condition ω L=1/(ω C) or ω 2 LC=1. Series resonance can be achieved in three ways:
Pick up coil inductance;
Choose the capacitance of the capacitor;
Select angular frequency ω 0;
In addition, the following formula can be used to determine all of these frequencies, capacitance, and inductance values:
The frequency 0 of ω is called the resonant frequency. If the voltage or active resistance r in the circuit remains unchanged, the current in the circuit will be maximum and equal to U/r during series resonance. This means that the current will be completely independent of the reactance of the circuit. When the reactance X C=X L exceeds the value of the resistance r, the voltage across the coil and capacitor terminals will begin to appear, greatly exceeding the voltage across the circuit terminals. The condition for the voltage on the circuit terminal to be lower than the voltage of the reactive element is:
value
For the convenience of calculation and with the size of resistance, we specify that ρ is referred to as the wave impedance of the circuit.
The multiplicity of excess voltage on the terminals of capacitive and inductive components relative to the network can be determined by the following expression:
The value of Q determines the resonance characteristics of a circuit, which is called the quality factor of the circuit. Similarly, resonance characteristics can be characterized by a value of 1/Q-circuit attenuation.
The instantaneous power of inductance and capacitance will be equal to p=U=Isin2 ω t and p C ^=- U Ç Isin2 ω t. Under series resonance, when U L=U C, these powers will always be equal and have opposite signs. This means that in this circuit, there will be energy exchange between the magnetic field of the coil and the electric field of the capacitor, while there will be no energy exchange between the energy of the magnetic field and the energy of the electrical energy (power source). This is because p L+p C=dW m/dt+dW e/dt and W m+W e=const, which means the total energy of the field in the circuit is constant. During the operation of this system, when the current on the coil increases and the voltage across the capacitor decreases, energy from the capacitor will enter the coil within a quarter of the cycle. In the next quarter of this cycle, the situation is exactly the opposite - the coil current will decrease and the capacitor voltage will increase, meaning that energy from the inductor will flow to the capacitor. In this case, the electrical energy supplied to the circuit will only cover the energy consumption related to the presence of active resistors r in the circuit.