RLC Circuit in Series with an AC Source

When the inductance L, resistance R and capacitor C are connected in series to an alternating source of voltage, then the circuit is called an RLC circuit. As they are connected in series, all of them will have the same amount of current flowing through them, but the voltage will vary. Here the name of this circuit can be LCR, RLC or LRC. 

Let us quickly take a look over individual elements before jumping into combination circuits. 

• If only resistance is connected to an AC source, then it has a current in phase with the potential, represented by the formula I = Iosinwt, and voltage is V = Vosinwt. Hence the equation for current becomes I = Vo/R.

• If only an inductor is connected to an AC source, then the current lags the potential by 900, which is represented by the formula I = Iosin(wt– π/2) and voltage is V = Vosin(wt-π/2). Hence the equation for current becomes I = Vo/XL where XL is the inductive reactance calculated by 

XL = 2πfL and its unit is ohm (Ω).

• If only a capacitor is connected to an AC source, then the current leads the potential by 900, which is represented by the formula I = Iosin(wt+ π/2) and voltage is V = Vosin(wt-π/2), hence the equation for current becomes I = Vo/XC where XC is the capacitive reactance calculated by 

XC =1/ 2πfC and its unit is ohm (Ω).

What is RLC Circuit

In a series RLC circuit, the resistance, inductance and capacitance are connected in a series to an AC source. Refer to the following circuit diagram

Here the current flowing through the circuit before entering the resistance is i=i0sinwt.

When the current enters the resistance, the potential difference across resistance VR , the potential difference across the inductor VL and the potential difference across the capacitor VC will be represented with time as 

VR (t) = (V0)R sinwt as current is in phase with the potential in resistance. 

VL(t) = (V0)L sin(wt+π/2) as in inductor the current lags the potential by 900

VC(t) = (V0)C sin(wt-π/2) as in capacitor the current leads the potential by 900

Where (V0)L is the peak value of potential in inductance, (V0)R is the peak value of potential in resistance and (V0)C is the peak value of potential in capacitance

This can be shown by phasor diagram as follows:

(i) Phasor diagram of Voltages (ii) Phasor diagram of Impedance

Now, let us consider different scenarios.

In the capacitor, current leads the potential by 900, in an inductor, the current lags the potential by 900  and in the resistor, it’s in phase with the current.

As we have seen in the phasor diagram, the potential VL of the inductor leads the current by 900, the potential VC of the capacitor lags the current by 900 and the potential VR of the resistor is in phase with the current. Hence, the vector follows the direction of the potential, which is more in magnitude.

For instance,

i) If VL > VC , Where VL = IXL and VC = IXC

V = VL – VC , where the voltage will lead the current by 900

V2 = VR2 + (VL-VC)2

V = √(I2 (R2 +( XL-XC)2))

V = I √ (R2 + (XL-XC)2)

Where V = V0sin(wt+ф) as here, the inductive potential is more in magnitude.

√ (R2 + (XL-XC)2) is the collective resistance offered by the inductance, capacitance and resistance known as Impedance shown by Z 

Hence,

V = IZ 

Z = √ (R2 + (XL-XC)2)

Hence the angle ф by which the current lags potential can be found out by

Tanф =( XL-XC)/R 

ii) If VC > VL , Where VL = IXL and VC = IXC

V = VC – VL , where the voltage will lag the current by 900

V2 = VR2 + (VC-VL)2

V = √(I2 (R2 +( XC-XL)2))

V = I √ (R2 + (XC-XL)2)

Where V = V0sin(wt-ф) as here, the capacitive potential is more in magnitude. 

√ (R2 + (XC-XL)2) is the collective resistance offered by the inductance, capacitance and resistance known as Impedance shown by Z 

Hence,

V = IZ 

Z = √ (R2 + (XC-XL)2)

Hence the angle ф by which the current lags potential can be found out by 

Tanф =( XC-XL)/R

iii)If VC = VL , Where VL = IXL and VC = IXC 

The potentials of the inductor and capacitor will cancel out, making this a pure resistor circuit.

V = IR 

As R = Z

Where V = V0sinwt

This is also known as the Resonant Circuit.

Thus collectively, we can write the value of V as 

V = VL ~ VC   where ~ is the difference between the two values.

   = IXL ~ IXC

The capacitor absorbs or stores the energy, and the inductor tends to oppose the charge going through it. Hence the collective resistance or impedance of capacitor and inductor is

 XC ~ XL

And the angle ф by which the current lags/leads potential can be found out by 

Tanф =( XC~XL)/R

Conclusion

Thus, with the help of simple formulas and definitions, we can solve an LCR circuit easily. The impedance of the LCR circuit plays an important role in deciding the total lag/lead of the current from the potential. Various aspects such as Impedance, the magnitude of the potential, phasing angle together for the efficient functioning of an LCR circuit.