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# RLC Series circuit

Table of Contents

## 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.

- Series Circuit
- 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=V_{0}sin⍵t. ——–(1)

Now, come to the RMS voltage across R, L and C are respectively V_{R}, V_{L} and V_{C}.

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

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

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

V_{C}=(-Q/C)

Now the usable voltage is V_{e}= V_{0}sin⍵t-L(dI/dt)-Q/C.

So, according to Ohm’s Law,

V_{0}sin⍵t-L(dI/dt)-Q/C=IR

⇒L(dI/dt)+Q/C+IR= V_{0}sin⍵t ———(2)

Now let assume, Q=Q_{0}sin(⍵t+⍺)

∴ I =dQ/dt=⍵Q_{0}cos(⍵t+⍺)

∴ dI/dt= -⍵^{2}Q_{0}sin(⍵t+⍺)

Now the (2) equation will look like –

-L⍵^{2}Q_{0}sin(⍵t+⍺) + Q_{0}sin(⍵t+⍺)/C + ⍵RQ_{0}cos(⍵t+⍺)= V_{0}sin⍵t

⇒ ⍵Q_{0}[-{⍵L-1/⍵C}sin(⍵t+⍺) + Rcos(⍵t+⍺)]= V_{0}sin⍵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= √{R^{2}+(⍵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 X_{L} and X_{C} respectively.

Where X_{L}= ⍵L and X_{C}=1/⍵C

∴ Z= √{R^{2}+(X_{L}-X_{C})^{2}}

Now, from (3) equation we will get,

⍵ZQ_{0}[-{⍵L-1/⍵C}sin(⍵t+⍺)/Z + Rcos(⍵t+⍺)/Z]= V_{0}sin⍵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.

∴ ⍵ZQ_{0}[-sinθsin(⍵t+⍺) + cosθcos(⍵t+⍺)]= V_{0}sin⍵t

⇒ ⍵ZQ_{0}cos(⍵t+⍺+θ)= V_{0}sin⍵t

⇒ ⍵ZQ_{0}sin(⍵t+⍺+θ+90°)= V_{0}sin⍵t

Comparing both sides we get,

⍵t+⍺+θ+90°=⍵t

⇒⍺+θ+90°=0

⇒⍺=-(θ+90°)

And ⍵ZQ_{0}= V_{0},

∴ Q_{0}= V_{0}/⍵Z

∴ Q=(V_{0}/⍵Z)sin(⍵t+⍺)

∴ I = (V_{0}/Z)cos(⍵t+⍺), i.e I_{0}= (V_{0}/Z) = V_{0}/√{R^{2}+(X_{L}-X_{C})^{2}}

∴ I = I_{0}cos(⍵t+⍺) = I_{0}sin(⍵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.

- No change in the frequency of the AC voltage and current.
- Current is lagging the voltage by an angle θ less than 90°.
Here, θ=tan

^{-1}{(⍵L-1/⍵C)/R}

Now, let’s talk about –

#### 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.