RLC Series Circuit • Scientyfic World

# RLC Series circuit

## What is an RLC circuit?

This is a electrical circuit with a resistor(R), capacitor(C) and inductor(L).

The R,L & C are connected with each other in series and parallel.

So RLC circuits are of two types.

1. Series Circuit
2. Parallel Circuit

In today’s article we’ll specially study the series circuit.

## RLC series circuit:

SO, here in this picture, you can see the RLC circuit containing a resistor (R), an inductor (L), and a capacitor(C) in series. And a voltage source is connected with them which’s supply voltage is V=V0sin⍵t. ——–(1)

Now, come to the RMS voltage across R, L and C are respectively VR, VL and VC.

‘I’ is the r.m.s value of the current flow in the circuit.

For the inductor in the circuit,  the induced emf VL= -L(dI/dt)

And for the capacitor, there will be an opposite voltage of Q/C.

VC=(-Q/C)

Now the usable voltage is Ve= V0sin⍵t-L(dI/dt)-Q/C.

So, according to Ohm’s Law,

V0sin⍵t-L(dI/dt)-Q/C=IR

⇒L(dI/dt)+Q/C+IR= V0sin⍵t ———(2)

Now let assume, Q=Q0sin(⍵t+⍺)

∴ I =dQ/dt=⍵Q0cos(⍵t+⍺)

∴ dI/dt= -⍵2Q0sin(⍵t+⍺)

Now the (2) equation will look like –

-L⍵2Q0sin(⍵t+⍺) + Q0sin(⍵t+⍺)/C + ⍵RQ0cos(⍵t+⍺)= V0sin⍵t

⇒ ⍵Q0[-{⍵L-1/⍵C}sin(⍵t+⍺) + Rcos(⍵t+⍺)]= V0sin⍵t —————-(3)

Now let us assume,

But,

### What is Z?

Z is the impendence of the given circuit.

Impedence is as similar as resistance.
It is the effective resistance of an electric circuit or component to alternating current, arising from the combined effects of ohmic resistance and reactance.

### Impedance:

Basically, the Impedance of a circuit is the effective resistance of the circuit which is combined effects of ohmic resistance and reactance. And its value is given as-

Z= √{R2+(⍵L-1/⍵C)2}

Ok now, what is reactance?

Reactance is just like resistance. It also opposes the flow of the current throughout the circuit. The difference is that it exists only on the non-resistive component of the circuit.

So, Reactance is the opposition of a circuit to the flow of the current due to the inductance and capacitance.

We denote the reactance of both inductor and capacitor as XL and XC respectively.

Where XL= ⍵L and XC=1/⍵C

∴ Z= √{R2+(XL-XC)2}

Now, from (3) equation we will get,

⍵ZQ0[-{⍵L-1/⍵C}sin(⍵t+⍺)/Z + Rcos(⍵t+⍺)/Z]= V0sin⍵t                        ——(4)

Again from the given triangle, we can simplify this equation.

As {⍵L-1/⍵C}/Z= sinθ and R/Z=cosθ, we can substitute the value in the (4) equation.

∴  ⍵ZQ0[-sinθsin(⍵t+⍺) + cosθcos(⍵t+⍺)]= V0sin⍵t

⇒ ⍵ZQ0cos(⍵t+⍺+θ)= V0sin⍵t
⇒ ⍵ZQ0sin(⍵t+⍺+θ+90°)= V0sin⍵t
Comparing both sides we get,
⍵t+⍺+θ+90°=⍵t
⇒⍺+θ+90°=0
⇒⍺=-(θ+90°)
And ⍵ZQ0= V0,
∴ Q0= V0/⍵Z
∴ Q=(V0/⍵Z)sin(⍵t+⍺)

∴  I = (V0/Z)cos(⍵t+⍺), i.e I0= (V0/Z) = V0/√{R2+(XL-XC)2}

∴ I =  I0cos(⍵t+⍺) = I0sin(⍵t-θ) ————-(5)

### Conclusion:

After doing all calculations, we finally get the value of the AC current “I”. Now comparing the AC voltage and current we can conclude this total calculation.

1. No change in the frequency of the AC voltage and current.
2. Current is lagging the voltage by an angle θ less than 90°.

Here, θ=tan-1{(⍵L-1/⍵C)/R}

#### Resonance :

Resonance is a phenomenon that occurs when both reactances are the same. It simply indicates that the ac current and applied voltage are in the same phase.  Now, let’s understand this phenomenon with a small video.

Hope you understood the concept clearly from this video.