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What is RLC Series Circuit? Circuit Diagram, Phasor Diagram, Derivation & Formula

RLC Series Circuit

Figure (1): RLC Series Circuit.

An RLC series circuit consisting of a resistor ‘R’ Ohm (Ω) in series with an inductance of ‘L’ Henrys and capacitance of ‘ C’ Farads connected to an A.C supply as shown in figure (1).

Applying Kirchhoff’s voltage law to the above circuit,

V={{V}_{R}}+{{V}_{L}}+{{V}_{C}}=IR+I{{X}_{L}}+I{{X}_{C}}

Where,

V = Source voltage

VR = Voltage drop across resistor = IR

VL = Voltage drop across inductor = IXL

VC = Voltage drop across capacitor = IXC

RLC Series Circuit Phasor Diagram

Figure (2): RLC Series Circuit Phasor Diagram when VL > VC.

The phasor diagram is drawn taking the current as the reference phasor as shown in figure (2).

From the phasors diagram

O{{E}^{2}}=O{{A}^{2}}+O{{D}^{2}}

{{V}^{2}}=V_{R}^{2}+{{\left( {{V}_{L}}-{{V}_{C}} \right)}^{2}}

V=\sqrt{V_{R}^{2}+{{\left( {{V}_{L}}-{{V}_{C}} \right)}^{2}}}

V=\sqrt{{{\left( IR \right)}^{2}}+{{\left( I{{X}_{L}}-I{{X}_{C}} \right)}^{2}}}

V=I\sqrt{{{R}^{2}}+{{\left( {{X}_{L}}-{{X}_{C}} \right)}^{2}}}

V=IZ

Where,

Z=\sqrt{{{R}^{2}}+{{X}^{2}}}\Omega

= Impedance of the circuit.

From the phasor diagram here, three cases arise.

  1. XL > XC i.e., XL – XC is positive
  2. XC > XL i.e., XL – XC is negative
  3. XL = XC i.e., XL – XC is zero.

Case (i): XL > XC

From the phasor diagram shown in figure (2), the net reactance is inductive. Therefore, for the condition XL > XC the circuit behaves like an inductive circuit. The current lags behind the voltage by 90º.

Net reactance,

X={{X}_{L}}-{{X}_{C}}

Impedance,

Z=\sqrt{{{R}^{2}}+{{X}^{2}}}\Omega

The phase difference between voltage and current,

\phi ={{\tan }^{-1}}\left( \frac{X}{R} \right)={{\tan }^{-1}}\left( \frac{{{X}_{L}}-{{X}_{C}}}{R} \right)

The power taken by the circuit,

P=VI\cos \phi \text{    watts}

Since pure inductance and capacitance do not consume power, power factor of the given circuit is \cos \phi lagging.

Case (ii): XC > XL

What is RLC Series Circuit Circuit Diagram, Phasor Diagram, Derivation & Formula

Figure (3): RLC Series Circuit Phasor Diagram when XC > XL.

From the phasor diagram shown in figure (3), the net reactance is capacitive. Therefore, for the condition XC > XL the circuit behaves like a capacitive circuit. The current leads voltage by 90º.

Net reactance,

X={{X}_{L}}-{{X}_{C}}

Impedance,

Z=\sqrt{{{R}^{2}}+{{X}^{2}}}

Z=\sqrt{{{R}^{2}}+{{\left( {{X}_{C}}-{{X}_{L}} \right)}^{2}}}\Omega

The phase difference between voltage and current,

\phi ={{\tan }^{-1}}\left[ \frac{X}{R} \right]

{{\tan }^{-1}}\left[ \frac{{{X}_{L}}-{{X}_{C}}}{R} \right]

The power taken by the circuit,

P=VI\cos \phi \text{    Watts}

Power factor of the given circuit is \cos \phi leading.

Case (iii): XL= XC

RLC Series Circuit Circuit Diagram, Phasor Diagram

Figure (4): RLC Series Circuit Phasor Diagram when XL = XC.

From the phasor diagram shown in figure (4), the net reactance of the circuit for the given condition XL= XC is zero. The circuit behaves like a resistive circuit.

Net reactance,

X={{X}_{L}}-{{X}_{C}}

X=0

Impedance,

Z=\sqrt{{{R}^{2}}+{{X}^{2}}}

Z=\sqrt{{{R}^{2}}+0}=R\Omega

Power taken by the circuit,

P=VI\cos \phi \text{    Watts}

The phase difference between voltage and current is zero. Thus,

\phi = 0

Since the current is in phase with voltage, the power factor \cos \phi is unity.

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