RL Circuit

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.

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

• Resistor( R) :The resistance of a material depends on its type and shape. When electrical current passes through it, the material converts some of the electrical energy into heat. Resistance quantifies the hindrance to the flow of current in an electrical circuit and is denoted in ohms, symbolized by the Greek letter omega (Ω).
• Inductor( L) An inductor is a two-terminal circuit element that stores energy in its magnetic field. Inductors are commonly made by wrapping wire into a coil. An inductor is an unresistant electrical element that stores energy in its own field when current overflows through it. A unit is the Henry ( H).

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

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 V_R and V_L.

So potential across the resistance:

V_R(t)=(Vo)_rsin(ωt)

and inductor-

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

So, resultant potential:

V_o=√(V_o)²_r+(V_o)_L²

=√(I_oR²)+(IoX_l)²

V_o=I_o √R²+X_L² where Z=impedence of RL circuit=√R²+X_L²

Then V_o = IoZ, where_o 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

[Tex]tan(\phi) [/Tex]=X_L/R

Power in RL Series Circuit

Alternating Voltage across the circuit is given as

[Tex]v=V_msin\omega t [/Tex]

Current can be given as

[Tex]i=I_msin(\omega t-\phi) [/Tex]

so ,the instantaneous power is given by the Equation

p=v i

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

[Tex]P=(V_m sin\omega t)* I_m sin(\omega t-\phi) [/Tex]

[Tex]p=\frac{V_mI_m}{2}*2sin(\omega t – \phi)sin\omega t [/Tex]

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

[Tex]\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) [/Tex]

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

[Tex]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) [/Tex]

so p can be calculated as

[Tex]p=V_(rms)*I_(rms)cos\phi = VI cos\phi [/Tex]

where [Tex]cos\phi [/Tex] is the power factor

[Tex]cos\phi [/Tex] can be calculated as

[Tex]cos\phi=\frac{V_R}{V}=\frac{IR}{IZ}=\frac{R}{Z} [/Tex]

Now substituting this Value we will get

[Tex]P=IZ*I*\frac{R}{Z}=I^{2}*R [/Tex]

Power Consume in the Circuit is [Tex]I^{2}*R [/Tex]

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.

[Tex]\Tau=L/R [/Tex] where L is inductor and R is resistance

Waveform and Power Curve of the RL Series 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 [Tex]\phi [/Tex] and during180 and [Tex](180+\phi) [/Tex].

There are two types of RL Circuits given below :

1. RL Series Circuit
2. RL Parallel 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:

[Tex]Z=\sqrt{R^2 + (X_L)^2}[/Tex]

where, [Tex]( X_L = \omega L )[/Tex] is the inductive reactance and [Tex]( \omega )[/Tex] is the angular frequency [Tex]( \omega = 2\pi f )[/Tex], with [Tex]( f ) [/Tex]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 [Tex](V_L)[/Tex] leads the current by 90 degrees.

The total voltage (V) is the phasor sum of [Tex]( V_R ) [/Tex]and [Tex]( V_L )[/Tex].

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:

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

where [Tex]( V )[/Tex] is the supply voltage.

Phase Angle (θ)

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

[Tex]\tan(\theta) = \frac{X_L}{R} = \frac{\omega L}{R}[/Tex]

This means the current lags the voltage by [Tex]( \theta )[/Tex] degrees.

Power

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

[Tex]P = I^2 R[/Tex]

The power factor (PF) is:

[Tex] \text{PF} = \cos(\theta) = \frac{R}{Z}[/Tex]

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:

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

where [Tex](X_L = \omega L )[/Tex] is the inductive reactance.

Voltage and Current Relationship

• The voltage across both the resistor and the inductor is the same.
• The current through the resistor [Tex](I_R)[/Tex] is in phase with the voltage.
• The current through the inductor [Tex](I_L)[/Tex] lags the voltage by 90 degrees.

Currents in the Circuit

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

[Tex] I = \sqrt{I_R^2 + I_L^2} [/Tex]

where [Tex]( I_R = \frac{V}{R} )\ and\ ( I_L = \frac{V}{X_L} )[/Tex]

Phase Angle (θ)

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

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

This means the total current lags the voltage by [Tex]( \theta )[/Tex] degrees.

Power

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

[Tex] P = V^2 \left( \frac{1}{R} \right) [/Tex]

The power factor (PF) is:

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

• Energy storehouse: Advantage Inductors in RL circuits store energy in their own fields. This energy can be released when demanded, making RL circuits useful in operations where energy storage and release are essential.
• Filtering Advantage: RL circuits can function as low-pass filters, permitting low-frequency signals to pass through while attenuating high-frequency signals. This is advantageous in applications where the removal of high-frequency noise is required.
• Smooth Current Transitions: Where high-frequency inductors repel changes in current, leading to smooth transitions in current inflow. This feature is crucial in situations where a smooth change in current is required.

• Complexity :RL circuits can be trickier to understand and construct than circuits with just resistors or capacitors. The addition of inductors brings in more intricate equations, which can make analysis more challenging.
• Back EMF: When the current flowing through an inductor changes, it produces a back electromotive force (EMF) that resists the change. This can cause a voltage spike and may necessitate additional circuitry for safeguarding.
• Power Losses: RL circuits can suffer power losses, particularly in the form of resistive losses in the coil winding. These losses can reduce the overall effectiveness of the circuit.

• RL circuits are used in magnetic detectors and inductive proximity detectors. Changes in the magnetic field, frequently caused by the presence of metallic objects, alter the inductance of the circuit.

• RL circuits are integral factors in electronic Filters, particularly low-pass and high-pass Filters. They control the frequency response of a system by allowing certain frequencies to pass while attenuating others.

• Inductors in RL circuits store energy in the form of a magnetic field when current overflows through them.

• RL circuits are employed in motor control systems, particularly in the environment of inductive loads similar to electric motors.

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.

What is the Power factor RL Circuit?

The power factor of an RL circuit is the ratio of real power dissipation to biases.

How does an RL circuit behave in DC (Direct Current)?

In direct current, the inductor acts as a short circuit and the circuit behaves similarly to a pure resistive circuit.

What is inductive reactance?

Inductive reactance (XL) is the resistance that an inductor offers to AC current. It varies in proportion to both the frequency of the alternating signal and the inductance of the coil.