Magnetically Coupled Circuits (Book Notes)

course” ECE-1109 Introduction to ECE

1. Fundamental Concepts

Mutual Inductance ( $M$ )

Definition: Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor due to a time-varying current flowing in the first inductor.
Mechanism: It occurs when two coils are in close proximity, and the magnetic flux caused by current in one coil links with the other coil.
Unit: Henrys (H).
Reciprocity: The mutual inductance is the same regardless of which coil is the source and which is the load: $M_{12} = M_{21} = M$ .

The Dot Convention To determine the polarity of the mutually induced voltage, the dot convention is used:

Current Entering: If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil.
Current Leaving: If a current leaves the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is negative at the dotted terminal of the second coil.

Coupling Coefficient ( $k$ )

This is a measure of the magnetic coupling between two coils, ranging from 0 to 1.
Formula: $k = \frac{M}{L _{1} L _{2}}$ or $M = k L_{1} L_{2}$ .
Types:
- $k = 1$ : Perfectly coupled (e.g., iron-core transformers).
- $k < 0.5$ : Loosely coupled (e.g., air-core transformers).

2. Key Formulas

Induced Voltage (Time Domain) For two coupled coils with currents $i_{1}$ and $i_{2}$ :

Voltage in coil 1: $v_{1} = L_{1} \frac{d i _{1}}{d t} \pm M \frac{d i _{2}}{d t}$ .
Voltage in coil 2: $v_{2} = L_{2} \frac{d i _{2}}{d t} \pm M \frac{d i _{1}}{d t}$ . (Note: The $\pm$ sign depends on the dot convention).

Frequency Domain (Phasors)

$V_{1} = (R_{1} + jω L_{1}) I_{1} \pm jω M I_{2}$ .
$V_{2} = \pm jω M I_{1} + (R_{2} + jω L_{2}) I_{2}$ .

Energy in Coupled Circuits The total energy stored in a pair of magnetically coupled coils is: $w = \frac{1}{2} L_{1} i_{1}^{2} + \frac{1}{2} L_{2} i_{2}^{2} \pm M i_{1} i_{2}$

Use the plus sign if both currents enter or leave the dotted terminals.
Use the minus sign if one current enters and the other leaves the dotted terminals.

Series Connection of Coupled Coils

Series-Aiding: $L = L_{1} + L_{2} + 2 M$ .
Series-Opposing: $L = L_{1} + L_{2} - 2 M$ .

3. Transformers

Transformers are four-terminal devices used to change current, voltage, or impedance levels.

A. Linear Transformers (Air-Core)

These are wound on magnetically linear materials (like air or plastic) and usually have loose coupling ( $k < 1$ ).
Reflected Impedance ( $Z_{R}$ ): The input impedance ( $Z_{in}$ ) seen at the primary source includes the primary impedance plus a term reflected from the secondary circuit.
- $Z_{in} = (R_{1} + jω L_{1}) + Z_{R}$ .
- $Z_{R} = \frac{ω ^{2} M ^{2}}{R _{2} + jω L _{2} + Z _{L}}$ .
- Note: $Z_{R}$ is not affected by the dot locations (polarity of $M$ ).

B. Ideal Transformers (Iron-Core)

Properties: Unity coupling ( $k = 1$ ), infinite inductances ( $L_{1}, L_{2}, M \to \infty$ ), and lossless coils ( $R_{1} = R_{2} = 0$ ).
Turns Ratio ( $n$ ): $n = \frac{N _{2}}{N _{1}}$ (Secondary turns / Primary turns).
Voltage Relationship: $V_{2} = n V_{1}$ (Step-up if $n > 1$ , Step-down if $n < 1$ ),.
Current Relationship: $I_{2} = \frac{I _{1}}{n}$ .
Impedance Matching: An ideal transformer reflects the load impedance $Z_{L}$ to the primary side as $Z_{in}$ : $Z_{in} = \frac{Z _{L}}{n ^{2}}$ .
Complex Power: $S_{1} = S_{2}$ (Power supplied to primary equals power delivered to secondary).

C. Autotransformers

A transformer with a single continuous winding common to both primary and secondary circuits.
Advantage: Smaller, lighter, and capable of transferring larger apparent power than two-winding transformers.
Disadvantage: Lack of electrical isolation between primary and secondary circuits.

4. Important Applications

Isolation: Transformers electrically isolate one part of a circuit from another (e.g., separating ac supply from a rectifier to reduce shock hazard).
Impedance Matching: Used to maximize power transfer between a source and a load when their impedances differ (e.g., matching a loudspeaker to an amplifier),.
Power Distribution: Step-up transformers increase voltage for efficient long-distance transmission (lowering $I^{2} R$ losses), and step-down transformers reduce voltage for safe residential use.

5. Exam-Specific Topics (from Term Questions)

Based on historical exam questions, pay attention to the following derivations and definitions:

Definitions: Be able to define “magnetic coupling” and “coupled circuits”.
Differences: Differentiate between electric and magnetic circuits.
Derivations:
- Derive the expression for the coefficient of magnetic coupling ( $k$ ) and show that its maximum value is unity ( $1$ ).
- Derive the expression for energy stored in a toroid ( $W = \frac{1}{2} \frac{B ^{2} l}{μ _{0}}$ ).
Laws: State and explain Ampere’s circuital law and Biot-Savart law in the context of magnetic circuits.

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