RLC parallel circuit

When an AC circuit contains elements of resistance, inductance, and capacitance connected in parallel, the voltage dropped across each element is the same. The currents flowing through each branch will be out of phase with each other. The current flowing through a pure resistive element will be in phase with the applied voltage. The current flowing through a pure inductive element lags the applied voltage by 90 electrical degrees, and the current flowing through a pure capacitive element will lead the voltage by 90 electrical degrees. The phase angle difference between the applied voltage and the total current is determined by the ratio of resistance, inductance, and capacitance connected in parallel. As with an RLC series circuit, if the inductive VARs is greater than the capacitive VARs, the current will lag the voltage and the power factor will be lagging. If the capacitive VARs is greater, the current will lead the voltage and the power factor will be leading.

Impedance

The impedance of the circuit is the reciprocal of the sum of the reciprocals of the legs. Since these values are out of phase with each other, vector addition must be used and we have:

Z = 1/√((1/R)² + ((1/X_L)-(1/X_C))²)

The amount of total current flow in the circuit can be computed by vector addition of the current flowing through each leg of the circuit. The inductive current is 180° out of phase with the capacitive current. These two currents tend to cancel each other, resulting in the elimination of the smaller and reduction of the larger. The total circuit current is the hypotenuse of the resulting right triangle. I_T = √(I_R² + (I_L-I_C)²)
I_T = E/Z

The apparent power or VAs, can be found using the formula:
VA = E x I_T
VA = √(P² + (VARs_L-VARs_C)²)

The power factor is PF = W/VA x 100 [%] and for angle θ we know that cos ∠θ = PF.

When values of inductive reactance and capacitive reactance become equal, they are said to be resonant. In a parallel circuit, inductive current and capacitive current cancel each other because they are 180° out of phase with each other. This produces minimum line current at the point of resonance. An LC parallel circuit is often referred to as tank circuit.

Computing the values for a parallel resonant circuit is a bit more involved than computing the values for a series resonant circuit. In theory, when a parallel circuit reaches resonance, the total circuit current should reach zero and total circuit impedance should become infinite because the capacitive current and inductive currents cancel each other. In practice, the quality (Q) of the circuit components determines total circuit current and, therefore, total circuit impedance,  Q = I_TANK/I_LINE .
The amount of current circulating inside the tank is equal to the product of the line current and the Q of the circuit.

Graphs illustrating the decrease of current and increase of impedance in a parallel resonant circuit are shown in figures below.

Bandwidth

The bandwidth for a parallel resonant circuit is determined in a manner similar to that used for a series resonant circuit. The bandwidth of a parallel circuit is determined by computing the frequency on either side of resonance at which the impedance is 0.707 of maximum.