RL Circuit (original) (raw)

Last Updated : 28 May, 2024

In this Article, we will see the characteristics of circuits consisting of a resistor and an inductor in series (RL circuits). The primary focus will be on the response of an RL circuit to a step voltage and a voltage square wave. An RL circuit, also referred to as a resistor-inductor circuit, plays a foundational role in electrical engineering and inductive elements.

In this Article, We will be going through the RL Circuit, We First go through What is the RL Circuit, and We will see RL circuit formulas, Waveforms, and Power curves. At last, we will conclude our Article with its Advantages, Disadvantages, and Some FAQs.

Table of Content

What is RL Circuit?

An RL circuit is a type of electrical circuit that consists of resistive( R) and inductive( L) rudiments. The crucial factors are a resistor( R) and an inductor( L) connected in series or parallel. These circuits are basic in electrical engineering and are essential for understanding the structure of electrical systems.

Then there is a brief explanation of the crucial rudiments in an RL circuit

RL circuits are generally encountered in diverse electronic systems, including pollutants, mills, and power inventories.

Relationships in the RL Circuit

resister and inductor in a series connection. In the circuit, current flows through the inductor and resistance.

let current flow **i in the circuit, and the potential difference resistance and inductor are VR and VL.

RL-Circuit-Diagram

RL Circuit

So potential across the resistance:

Phasor-Diagram

Phasor Diagram

VR(t)=(Vo)rsin(ωt)

and inductor-

VL(t)=(Vo)LRsin(ωt+π/2) :: (where π/2 is the voltage lead current)

So, resultant potential:

Vo=√(Vo)2r+(Vo)L2

=√(IoR2)+(IoXl)2

Vo=Io √R2+XL2 where Z=impedence of RL circuit=√R2+XL2

Then Vo = IoZ, whereo Io is the o current in amps.

Instantaneous current and voltage in an RL circuit

i=E/R(1-ϵ-tR/L)

Where i = instantaneous current in amperes at time t

E = supply voltage

R = series resistance, in ohms (including inductor winding resistance)

ε = exponential constant =2.718

t = time in seconds from current start

L =represents the inductance of the inductor, measured in Henry.

Frequency Response

The frequency response of an RL (resistor-inductor) circuit can be expressed in terms of impedance.

Z=√R2+(2*pai*f*L)2 where f=frequency in Ac signal

Phase Difference

tan(\phi)=XL/R

Power in RL Series Circuit

Alternating Voltage across the circuit is given as

v=V_msin\omega t

Current can be given as

i=I_msin(\omega t-\phi)

so ,the instantaneous power is given by the Equation

p=v i

now Substituting the value of v and I from the above equation,

P=(V_m sin\omega t)* I_m sin(\omega t-\phi)

p=\frac{V_mI_m}{2}*2sin(\omega t - \phi)sin\omega t

p=\frac{V_m}{\sqrt{2}}*\frac{I_m}{\sqrt{2}}[cos\phi -cos(2\omega t - \phi)]

\frac{V_m}{\sqrt{2}}*\frac{I_m}{\sqrt{2}} cos\phi-\frac{V_m}{\sqrt{2}}*\frac{I_m}{\sqrt{2}}cos(2\omega t-\phi)

so the average power consumed in one cycle in the circuit can be given by

p= average of \frac{V_m}{\sqrt{2}}*\frac{I_m}{\sqrt{2}} cos\phi- average of\frac{V_m}{\sqrt{2}}*\frac{I_m}{\sqrt{2}}cos(2\omega t-\phi)

so p can be calculated as

p=V_(rms)*I_(rms)cos\phi = VI cos\phi

where cos\phi is the power factor

cos\phi can be calculated as

cos\phi=\frac{V_R}{V}=\frac{IR}{IZ}=\frac{R}{Z}

Now substituting this Value we will get

P=IZ*I*\frac{R}{Z}=I^{2}*R

Power Consume in the Circuit is I^{2}*R

Time Constant Formula

T=L/R

Inductor current doesn't change instantaneously So transit responses are measured in terms of the ratio of the inductor and resistance.

RL-Circuit

RL Circuit

\Tau=L/R where L is inductor and R is resistance

Waveform and Power Curve of the RL Series Circuit

Waveform-and-power-Curve-of-Rl-circuit

Waveform and power Curve of Rl circuit

The Following Represents the Wave Form and power Curve of the RL Circuit. The power is positive in the Cycle Except between angle 0 and \phi and during180 and (180+\phi).

Types of RL Circuit

There are two types of RL Circuits given below :

  1. RL Series Circuit
  2. RL Parallel Circuit

**RL Series Circuit

It is a kind of circuit that consists of resistance R and Inductance L where resistance R connected in series with the coil which is having an inductance L.

**Impedance (Z)

The total impedance in a series RL circuit is given by:

Z=\sqrt{R^2 + (X_L)^2}

where, ( X_L = \omega L ) is the inductive reactance and ( \omega ) is the angular frequency ( \omega = 2\pi f ), with ( f ) being the frequency of the AC source).

Voltage and Current Relationship

The voltage across the resistor is in phase with the current.

The voltage across the inductor (V_L) leads the current by 90 degrees.

The total voltage (V) is the phasor sum of ( V_R ) and ( V_L ).

Current in the Circuit

The current (I) in a series RL circuit is the same through both the resistor and the inductor and is given by:

I = \frac{V}{Z} = \frac{V}{\sqrt{R^2 + (\omega L)^2}}

where ( V ) is the supply voltage.

Phase Angle (θ)

The phase angle between the total voltage and the current is given by:

\tan(\theta) = \frac{X_L}{R} = \frac{\omega L}{R}

This means the current lags the voltage by ( \theta ) degrees.

Power

The real power (P) consumed in the circuit is due to the resistor and is given by:

P = I^2 R

The power factor (PF) is:

\text{PF} = \cos(\theta) = \frac{R}{Z}

**RL Parallel Circuit

It is a kind of circuit that consists of resistance R and Inductance L where resistance R connected in parallel with the coil which is having an inductance L.

An RL parallel circuit consists of a resistor (R) and an inductor (L) connected in parallel with a voltage source. The key characteristics and behavior of this circuit can be summarized as follows:

**Impedance (Z)

The total impedance in a parallel RL circuit is given by:

\frac{1}{Z} = \sqrt{\left( \frac{1}{R} \right)^2 + \left( \frac{1}{X_L} \right)^2}

where (X_L = \omega L ) is the inductive reactance.

**Voltage and Current Relationship

**Currents in the Circuit

The total current (I) is the phasor sum of the currents through the resistor and the inductor.

I = \sqrt{I_R^2 + I_L^2}

where ( I_R = \frac{V}{R} )\ and\ ( I_L = \frac{V}{X_L} )

**Phase Angle (θ)

The phase angle between the total current and the voltage is given by:

\tan(\theta) = \frac{I_L}{I_R} = \frac{\frac{V}{X_L}}{\frac{V}{R}} = \frac{R}{\omega L}

This means the total current lags the voltage by ( \theta ) degrees.

**Power

The real power (P) consumed in the circuit is due to the resistor and is given by:

P = V^2 \left( \frac{1}{R} \right)

The power factor (PF) is:

\text{PF} = \cos(\theta) = \frac{R}{\sqrt{R^2 + (\omega L)^2}}

Advantages of RL Circuits

Disadvantages of RL Circuits

**Uses of RL Circuit

Conclusion

In conclusion, RL circuits, combining resistors and inductors, serve vital roles in electronic systems. They find operations in pollutants, mills, energy stores, detectors, and motor control. The interplay of resistance and inductance enables effective energy transfer, shapes signals, and contributes to the stability and functionality of different electrical biases.