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Equations & Formulas For RLC Circuits (Series & Parallel)

RLC Circuits – Series & Parallel Equations & Formulas

RLC Circuit:

When a resistor, inductor and capacitor are connected together in parallel or series combination, it operates as an oscillator circuit (known as RLC Circuits) whose equations are given below in different scenarios as follow:

Equations & Formulas For RLC Circuits (Series & Parallel)

Parallel RLC Circuit

When they are connected in parallel combination

Impedance:

Total impedance of the circuit is;

Total impedance of parallel RLC circuit

Where

  • XL = Inductive reactance
  • XC = Capacitive reactance

Power Factor:

The power factor for this circuit is

Cos θ = Z/R

Resonance Frequency:

When inductive reactance XL & capacitive reactance X­c of the circuit is equal.

Resonance Frequency of parallel RLC circuit

Where

  • L = Inductance of inductor
  • C = Capacitance of capacitor

Quality Factor:

It is the ratio of stored energy to the energy dissipated in the circuit.

Quality Factor of parallel RLC circuit

Bandwidth:

B.W = fr / Q

Resonant Circuit Current:

The total current through the circuit when the circuit is at resonance.

At resonance, the X= XC , so Z = R

IT = V/R

Current Magnification

Parallel resonance RLC circuit is also known current magnification circuit. Because, current flowing through the circuit is Q times the input current

Imag = Q IT

Characteristic Equation:

Characteristic equation of parallel RLC circuit

Neper Frequency For Parallel RLC Circuit:

Neper frequency for Parallel RLC circuit

Resonant Radian Frequency For Parallal RLC Circuit:

Resonant radian frequency for parallal RLC circuit

Voltage Response:

  • Over-Damped Response

When

ω0< α2

The roots s1 & s2 are real & distinct

  • Under-Damped Response

When

ω0> α2

The roots s1 & s2 are complex & conjugate of each other

  • Critically Damped Response

When

ω0= α2

The roots s1 & s2 are real & equal.

Series RLC Circuit:

 Impedance:

The total impedance of the series RLC circuit is;

total impedance of the series RLC circuit

Power Factor:

The power factor of Series RLC circuit;

Cos θ = R/Z

Resonance Frequency:

The frequency at which the inductive reactance XL = Capacitive reactance X­­c is known as resonance frequency.

Resonance Frequency of series RLC circuit

Where

  • L = Inductance of the inductor
  • C = Capacitance of the capacitor

Quality Factor:

Quality Factor of series RLC circuit

Bandwidth:

B.W = (fr / Q)

  • B.W = (R / L)              in rad/s
  • B.W = (R / 2πL)         in hz

Lower Cutoff Frequency & Upper Cutoff Frequency:

fh = fr + ½ B.W

fl = fr – ½ B.W

Characteristic Equation:

Characteristic equation of series RLC circuit

Neper Frequency For series RLC Circuit:

Neper Frequency For series RLC Circuit

Resonant Radian Frequency For series RLC Circuit:

Resonant Radian Frequency For series RLC Circuit

Voltage Response:

  • Over-Damped Response

When

ω0< α2

The roots s1 & s2 are real & distinct

  • Under-Damped Response

When

ω0> α2

The roots s1 & s2 are complex & conjugate of each other

  • Critically Damped Response

When

ω0= α2

The roots s1 and s2 are real & equal

Related Formulas and Equations Posts:

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