Circuits With Resistance And Inductance

Circuits with Resistance and Inductance - Resistive Circuits

In previous lessons, we learned about circuits with resistors only.
Today, we will explore circuits with both resistors and inductors.
This combination forms what is known as a resistive circuit.

Definition of Inductance

Inductance is the property of a conductor to oppose changes in current.
It is denoted by the symbol L and its unit is Henry (H).
Inductance depends on the geometry and material of the conductor.

Inductor in a Circuit

An inductor is represented by the symbol L and it consists of a coil of wire.
When a current passes through an inductor, it creates a magnetic field around it.
This magnetic field opposes any changes in current and stores energy.

Inductive Reactance

Inductive reactance (XL) is the opposition that an inductor offers to the flow of alternating current.
It is directly proportional to the frequency of the current.
The formula for inductive reactance is given by XL = 2πfL.

Impedance

Impedance (Z) is the total opposition to the flow of current in a circuit.
It is the combination of resistance (R) and reactance (X).
The formula for impedance in a resistive circuit with inductors is given by Z = √(R^2 + XL^2).

Time Constant

The time constant (τ) is a measure of how long it takes for the current in an RL circuit to reach its final value.
It is given by the formula τ = L/R, where L is the inductance and R is the resistance.
The time constant determines the rate at which the circuit settles to its final value.

Current in a Resistive Circuit with an Inductor

In a resistive circuit with an inductor, the current lags behind the voltage.
The magnitude of the current depends on the frequency of the voltage signal and the values of resistance and inductance.

Phasor Diagram

A phasor diagram is used to represent the relationship between voltage, current, and impedance in a circuit.
It is a graphical representation of vectors that show the magnitudes and phase differences between these quantities.
The phasor diagram helps us understand the behavior of the circuit at different frequencies.

Examples of Resistive Circuits with Inductors

Example 1: Calculate the impedance in a circuit with a resistor of 50 ohms and an inductor with a reactance of 30 ohms at a frequency of 60 Hz.
Example 2: Find the time constant of an RL circuit with an inductance of 0.5 H and a resistance of 10 ohms.

Properties of Inductance:

Inductance is a scalar quantity measured in Henrys (H).
It determines the ability of an inductor to store energy in the form of a magnetic field.
Inductance depends on the number of turns, the area of cross-section, and the length of the coil.
Inductance is always positive and can be increased by increasing the number of turns or the magnetic permeability.

Self-Inductance:

Self-inductance occurs when the changing magnetic field produced by the current in a coil induces an electromotive force (emf) in the same coil.
It is the ability of a coil to oppose changes in current flowing through it.
The magnitude of the self-induced emf is given by L dI/dt, where L is the self-inductance and dI/dt is the rate of change of current.

Mutual Inductance:

Mutual inductance occurs when a changing magnetic field in one coil induces an emf in another coil.
It is the ability of one coil to induce emf in the nearby coil.
The magnitude of the mutual-induced emf is given by M dI1/dt, where M is the mutual inductance and dI1/dt is the rate of change of current in the first coil.

Energy Stored in an Inductor:

An inductor stores energy in its magnetic field.
The energy stored in an inductor is given by the formula E = (1/2) L I^2, where E is the energy, L is the inductance, and I is the current.
The energy stored in an inductor is proportional to the square of the current flowing through it.

RL Series Circuit:

An RL series circuit consists of a resistor and an inductor connected in series.
In an RL series circuit, the current is common through both the resistor and the inductor.
The voltage across the resistor and the inductor can be calculated using Ohm’s law and the equations V_R = I R and V_L = I XL, respectively.

Phase Angle in RL Circuits:

In an RL circuit, the current lags behind the voltage due to the inductive reactance.
The phase angle (θ) is the angle difference between the voltage and the current in an RL circuit.
The phase angle can be calculated using the formula θ = arctan(XL/R), where XL is the inductive reactance and R is the resistance.

Power in RL Circuits:

The power in an RL circuit can be calculated using the equation P = I^2 R, where P is the power, I is the current, and R is the resistance.
The power is dissipative in the resistor and is not stored in the inductor.
The power factor in an RL circuit is given by cos(θ), where θ is the phase angle.

Resonance in RL Circuits:

Resonance in RL circuits occurs when the inductive reactance equals the resistance (XL = R).
At resonance, the phase angle is 45 degrees, and the impedance is at its minimum.
Resonance frequency can be calculated using the formula f = 1/2π√(LC), where L is the inductance and C is the capacitance.

RL Circuit Applications:

RL circuits are commonly used in transformers, inductors, power supplies, and motors.
Transformers use mutual inductance to transfer electric power from one circuit to another.
Inductors store energy and smooth out fluctuations in current in power supplies.
Motors use the interaction between magnetic fields of an inductor and a permanent magnet to convert electrical energy into mechanical work.

Real-Life Examples:

Inductance and RL circuits are prevalent in everyday life.
Examples include electric motors, power grids, telecommunication devices, and electronic circuitries.
Understanding RL circuits is crucial in designing and troubleshooting electrical systems and devices.
RL circuits play a vital role in various industries like automotive, aerospace, telecommunications, and power generation. Slide 21: Behavior of RL Circuits
In an RL circuit, the behavior of the current and voltage depends on the values of resistance, inductance, and the applied voltage.
When the applied voltage is constant (DC), the current in an RL circuit gradually increases towards its maximum value based on the time constant.
When the applied voltage is alternating (AC), the behavior of the current in an RL circuit depends on the frequency of the voltage signal. Slide 22: Time Constant of RL Circuits
The time constant (τ) is a measure of how quickly the current changes in an RL circuit.
In an RL circuit, the time constant is given by the formula τ = L/R, where L is the inductance and R is the resistance.
A larger τ indicates a slower change in current, while a smaller τ represents a faster change. Slide 23: RL Circuit Analysis - DC Voltage
In an RL circuit with a constant (DC) voltage source, the current increases gradually over time.
Initially, when the circuit is closed, the current is zero and gradually rises towards its maximum value.
The time it takes for the current to reach approximately 63% of its final value is equal to one time constant (τ). Slide 24: RL Circuit Analysis - AC Voltage
In an RL circuit driven by an alternating (AC) voltage source, the behavior of the current is more complex.
The magnitude and phase angle of the current depend on the frequency and the values of resistance and inductance.
The current lags behind the voltage by an angle determined by the ratio of inductive reactance to resistance. Slide 25: RL Circuit Examples - DC Voltage
Example 1: In an RL circuit with an inductance of 0.2 H and a resistance of 10 ohms, calculate the time constant and the current at t=2τ when connected to a constant voltage source of 12 V.
Example 2: In an RL circuit with an inductance of 0.5 H and a resistance of 20 ohms, calculate the time it takes for the current to reach 90% of its final value. Slide 26: RL Circuit Examples - AC Voltage
Example 1: In an RL circuit with 8 ohms of resistance and an inductive reactance of 12 ohms, calculate the current magnitude and phase angle when connected to an AC voltage source with a frequency of 50 Hz.
Example 2: In an RL circuit with 0.1 H of inductance and a resistance of 15 ohms, calculate the current amplitude and phase angle when connected to an AC voltage source with a frequency of 1000 Hz. Slide 27: Inductors in Series
When inductors are connected in series, the total inductance (L_total) is given by the sum of individual inductances: L_total = L1 + L2 + L3 + …
The current flowing through each inductor is the same, and the total voltage across the inductors is the sum of individual voltages. Slide 28: Inductors in Parallel
When inductors are connected in parallel, the total inductance (1/L_total) is given by the reciprocal of the sum of individual reciprocals: 1/L_total = 1/L1 + 1/L2 + 1/L3 + …
The voltage across each inductor is the same, and the total current through the inductors is the sum of individual currents. Slide 29: RL Filters
RL filters are used to pass or attenuate certain frequencies in a circuit.
Low-pass RL filters allow low frequencies to pass while attenuating high frequencies.
High-pass RL filters allow high frequencies to pass while attenuating low frequencies.
Band-pass RL filters allow a specific band of frequencies to pass while attenuating others. Slide 30: RL Circuit Applications
RL circuits have a wide range of applications in various fields.
They are used in power supply circuits, audio and video equipment, electric motors, transformers, and more.
Understanding RL circuits is crucial in designing and analyzing electrical systems and devices.
RL circuits play a vital role in industries such as telecommunications, power distribution, electronics, and automation.