Week 7 Circuit Theory 7

University of Southampton Electricity & Electronics Introduction Course

Course Title: Circuit Theory (7)Instructor: Dr. Josh RobertsonContact: J.J.Robertson@soton.ac.uk

Learning Objectives

• Understand the basics of an inductor

• Calculate the time constant for an inductor

• Understand the transient curves for an L-R circuit

Upcoming

• Circuit Theory Tests scheduled Tests 15-16 and marking 16-17

• Electronics I Thursday Test pick-up 13-15

Inductors

Definition and Function

An inductor is a passive circuit element that resists changes in current. Acting as a magnetic energy storage device, it plays a crucial role in the management of electrical energy in circuits. Inductors are commonly referred to as coils or chokes, indicated in circuit diagrams by the letter ‘L’.

Behavior of Inductors

Inductors work by storing kinetic energy from moving electrons in a magnetic field. Unlike resistors, which dissipate energy as heat, inductors conserve energy. Typically, they are made of insulated wire coiled around a core to amplify the magnetic field's strength. Their operation adheres to Faraday’s law of induction, which describes how a change in magnetic field can induce voltage, and Lenz’s law, which dictates the direction of induced voltage based on the change in current.

Voltage Characteristics

• Increasing Current: When current flows through an inductor, it generates a back electromotive force (EMF) opposing the current's direction, a phenomenon referred to as charging. This back EMF prevents the current from instantaneously reaching its maximum value.

• Decreasing Current: Conversely, if the current decreases, the inductor generates a voltage that aids the current flow, effectively acting as a power source, a process termed discharging. The energy stored in the magnetic field is released back into the circuit.

Inductance

Definition

Inductance measures the induced EMF in response to a change in current. Self-inductance occurs when the induced EMF is within the same circuit.

Measurement

Inductance is measured in Henrys (H), where 1 Henry induces a voltage of 1 Volt when the current changes at a rate of 1 Ampere per second. Typical inductance values range from 1 µH (microhenry, 10^-6 H) to 20 H (Henrys).

Energy Stored in Inductors

The energy stored in an inductor can be calculated using the equation:E_L = (1/2) L I²For instance, an 8H inductor carrying a current of 3A stores energy as follows:E = 1/2 * 8 * 3² = 36 Joules.

Factors Affecting Inductance

• Number of Turns: The more turns in the wire coil, the greater the inductance due to increased magnetic field strength.

• Cross-sectional Area: A larger cross-sectional area of the coil enhances the inductance by allowing more magnetic field lines to pass through.

• Magnetic Core: Utilizing a magnetic core, like iron, concentrates the magnetic field, thereby increasing inductance significantly.

• Coil Arrangement: The geometrical arrangement and physical dimensions of the coil play critical roles in determining inductance. A short, thick coil possesses higher inductance than a long, thin one.

Applications of Inductors

Inductors are employed for various purposes, including:

• Filtering and smoothing high-frequency noise in electric circuits

• Storing and transferring energy in power converters (DC-AC or AC-DC)

• Creating tuned oscillators or LC circuits for frequency modulation

• Matching impedances in RF applications to maximize power transfer

Combining Inductors

Inductors can be connected in series or parallel:

• In Series: The total inductance (Ls) is the sum of individual inductances: Ls = L1 + L2 + L3…

• In Parallel: The total inductance (Lp) is calculated using the reciprocal formula: Lp = 1/(1/L1 + 1/L2 + 1/L3)Care should be taken to shield inductors during calculations to account for interactions between their magnetic fields.

Mutual Inductance

Mutual inductance occurs when two inductors are placed close to each other, allowing one to induce EMF in the other due to the shared magnetic field generated by their respective current flow.

Transients in L-R Circuits

Transients occur when the circuit undergoes a change, such as switching from one DC configuration to another.

Current Growth in L-R Circuit

When the circuit switch is closed, the initial current begins to increase towards its final value, determined by Kirchhoff's Voltage Law (KVL). The inductor actively counters this change through back EMF, leading to a gradual rise in current.

Current Decay in L-R Circuit

When the switch is opened, current rapidly decreases, leading to a collapsing magnetic field that induces a back EMF, which momentarily continues to drive current. The voltage across both the resistor and the inductor decays exponentially to zero over time.

Time Constant for Inductors

The time constant (τ) is critical in defining how quickly current reaches its steady state after a switch is closed. τ is calculated as the ratio of inductance (L) to resistance (R) in the circuit and is often approximated to five time constants for the current to stabilize fully.

Thevenin's Theorem

Thevenin's theorem is used to simplify complex circuit analysis by transforming a complicated circuit into a single equivalent voltage source in series with a resistance. This approach facilitates easier calculations, particularly in power or battery systems.

Steps to Use Thevenin's Theorem

1. Remove the load resistance from the circuit.

2. Replace EMF sources with their internal resistances and calculate the Thevenin resistance (Rth).

3. Find the open-circuit voltage (Vth) at the terminals.

4. Draw the equivalent circuit and calculate other currents/voltages easily from there.

Practice Problems

1. Calculate the resistance of a coil with an inductance of 6H connected in series with a 10 Ohm resistor to a 120 V DC supply.

2. Determine the current flow immediately after a short circuit occurs.

3. Find the time required for the current to fall to 10% of its initial value after opening the switch.