Series resonance

One of the most important phenomena in ac circuits, especially in radio-frequency engineering, is the property of resonance. You’ve already learned that resonance is a condition that occurs when capacitive and inductive reactance cancel each other out. Resonant circuits and devices have a great many different applications in electricity and electronics.
Recall that capacitive reactance, X_C, and inductive reactance, X_L, can sometimes be equal in magnitude. They are always opposite in effect. In any circuit containing an inductance and capacitance, there will be a frequency at which X_L = −X_C. This is resonance.
Sometimes X_L= −X_C at just one frequency; in some special devices it can occur at many frequencies. Generally, if a circuit contains one coil and one capacitor, there will be one resonant frequency.
Refer to the schematic diagram of Fig. 1. You might recognize this as a series RLC circuit. At some particular frequency, X_L = −X_C. This is inevitable, if L and C are finite and nonzero. This is the resonant frequency of the circuit. It is abbreviated f_o.

Fig. 1: A series RLC circuit.

At f_o, the effects of capacitive reactance and inductive reactance cancel out. The result is that the circuit appears as a pure resistance, with a value very close to R. If R=0, that is, the resistor is a short circuit, then the circuit is called a series LC circuit, and the impedance at resonance will be extremely low. The circuit will offer practically no opposition to the flow of alternating current at the frequency f_o. This condition is series resonance.