lecture_EE1C2_lec5_magnetically_coupled_circuits_2025-12-15

📌 Overview

This lecture introduces Magnetically Coupled Circuits, where a changing magnetic flux in one coil induces a voltage in a nearby coil (Mutual Inductance). We define the Dot Convention for analyzing polarity and explore both Linear and Ideal Transformers, providing methods to simplify these circuits into decoupled equivalent circuits.

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🎯 Learning Objectives

• Mutual Inductance: Understand how time-varying current in one coil induces voltage in another.
• Dot Convention: Apply the rule to determine the polarity of induced voltages.
• Coupling Coefficient ($k$): Calculate $k$ and understand energy in coupled coils.
• Circuit Analysis: Analyze circuits by replacing magnetic coupling with dependent sources.
• Transformers:
  • Analyze Linear Transformers using T/$\Pi$ equivalent networks or Reflected Impedance.
  • Analyze Ideal Transformers using turns ratio ($n$) and impedance transformation.

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💡 Key Concepts & Definitions

1. Self and Mutual Inductance

• Self-Inductance ($L$): Voltage induced in a coil by a time-varying current in the same coil. $v(t)=L\frac{di(t)}{dt}\stackrel{\text{Phasor}}{\to }\mathbf{V}=j\omega L\mathbf{I}$
• Mutual Inductance ($M$): Voltage induced in a coil by a time-varying current in a nearby coil. $v_2(t)=M_{21}\frac{d{i}_1(t)}{dt}\stackrel{\text{Phasor}}{\to }{\mathbf{V}}_2=j\omega M{\mathbf{I}}_1$
  • Note: For linear media, $M_{12}=M_{21}=M$.
  • $M$ is always positive.

2. The Dot Convention (Crucial!)

The physical winding direction determines the polarity of the induced voltage. We use dots to abstract this.

The Rule:

1. Current ENTERS dot: The induced voltage in the other coil is POSITIVE (+) at its dot.
2. Current LEAVES dot: The induced voltage in the other coil is NEGATIVE (-) at its dot.

3. Coupling Coefficient ($k$) & Energy

• Energy: Total stored energy in a pair of coupled coils: $w=\frac{1}{2}{L}_1{i}_1^2+\frac{1}{2}{L}_2{i}_2^2\pm M{i}_1{i}_2$
  • Use $+$ if currents both enter (or both leave) the dots.
  • Use $-$ if one enters and one leaves.
• Coupling Coefficient ($k$): Measures magnetic coupling tightness ($0\le k\le 1$). $k=\frac{M}{\sqrt{L_1{L}_2}}$
  • $k=1$: Perfectly coupled (Ideal Transformer).
  • $k=0$: Uncoupled.

4. Series Connected Inductors

When two coupled coils are connected in series, their mutual inductance affects the total equivalent inductance.

• Series-Aiding: Current enters both dotted terminals (or leaves both). Magnetic fluxes reinforce. $L_{eq}=L_1+L_2+2M$
• Series-Opposing: Current enters one dotted terminal and leaves the other. Magnetic fluxes oppose. $L_{eq}=L_1+L_2-2M$

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⚙️ Circuit Analysis Methods

Method 1: Mesh Analysis with Dependent Sources

This is the most robust method for general coupled circuits.

Step-by-Step Plan:

1. Phasor Domain: Convert all sources and components to phasors ($j\omega L$, $1/j\omega C$).
2. Self-Inductance: Write the impedance for the self-inductance normally ($Z_L=j\omega L$).
3. Mutual Inductance (The Trick):
   • Insert a dependent voltage source in series with each coil.
   • Value: Magnitude is $j\omega M{\mathbf{I}}_{other}$.
   • Polarity: Use the Dot Convention based on the direction of the current in the other coil.
4. Solve: The circuit is now “decoupled” (magnetically). Solve using standard KVL/Mesh Analysis.

Method 2: Linear Transformer Equivalent Circuits

Instead of dependent sources, replace the T-junction of coupled inductors with an equivalent T-Network of uncoupled inductors.

• When to use: When the two coils share a common node (ground).
• The T-Network:
  • $L_a=L_1-M$
  • $L_b=L_2-M$
  • $L_c=M$
  • (If dots are opposing relative to the common node, $M$ becomes $-M$).

Method 3: Reflected Impedance (Linear Transformer)

We can view the secondary side impedance as if it were on the primary side. $Z_{in}=Z_{11}+\frac{(\omega M{)}^2}{Z_{22}+Z_L}$

• $Z_{11}$: Total impedance of primary loop (source + $L_1$).
• $Z_{22}$: Total self-impedance of secondary loop ($L_2$).
• $Z_L$: Load impedance.

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➗ Ideal Transformers

An idealization where $k=1$, $L_1,L_2\to \infty$, and coils are lossless.

Key Properties

1. Turns Ratio: $n=\frac{N_2}{N_1}$ (Secondary / Primary).
2. Voltage Ratio: $\frac{{\mathbf{V}}_2}{{\mathbf{V}}_1}=n$
   • Sign: Positive $n$ if both $+$ at dots (or both non-dots).
3. Current Ratio: $\frac{{\mathbf{I}}_2}{{\mathbf{I}}_1}=\frac{1}{n}$
   • Sign: Negative $-1/n$ if both currents enter (or leave) dots.
   • Mnemonic: Power in = Power out ($V_1{I}_1=V_2{I}_2$).

Impedance Transformation

The impedance $Z_L$ on the secondary side “looks like” a different impedance from the primary side.

$Z_{in}=\frac{Z_L}{n^2}$

• Step-down ($n<1$): High voltage side sees High impedance.
• Step-up ($n>1$): Low voltage side sees Low impedance.

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✍️ Example Exercise

Problem: Find the equivalent resistance $R_{eq}$ seen by the source at the far left. The circuit consists of a source, a resistor, a 1:2 transformer, a middle resistor, a 3:4 transformer, and a load.

Circuit Description:

1. Left: Source + $10\Omega$ resistor $\to$ Primary of T1 ($n_1=2$).
2. Middle: Secondary of T1 $\to$ $2.8\Omega$ resistor $\to$ Primary of T2 ($n_2=4/3$).
3. Right: Secondary of T2 $\to$ $32\Omega$ Load.

Solution Plan (Working Backwards):

Step 1: Reflect the Load ($32\Omega$) to the Middle Stage We look from the middle stage into T2.

• Transformer T2 has ratio $3:4\Rightarrow n_2=\frac{4}{3}$.
• Formula: $Z_{in}=\frac{Z_L}{n^2}$. $R_{load}^{\prime }=\frac{32}{(4/3{)}^2}=\frac{32}{16/9}=32\cdot \frac{9}{16}=18\Omega$

Step 2: Combine Resistances in Middle Stage The reflected load is in series with the middle resistor. $R_{mid\_total}=2.8\Omega +18\Omega =20.8\Omega$

Step 3: Reflect Middle Stage to Source Side We look from the source side into T1.

• Transformer T1 has ratio $1:2\Rightarrow n_1=\frac{2}{1}=2$. $R_{mid}^{\prime }=\frac{R_{mid\_total}}{n_1^2}=\frac{20.8}{2^2}=\frac{20.8}{4}=5.2\Omega$

Step 4: Total Equivalent Resistance Add the source’s series resistance. $R_{eq}=10\Omega +5.2\Omega =15.2\mathbf{\Omega }$

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🔗 Resources

• Presentation:

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❓ Post lecture

• Common Pitfall: Forgetting to square the turns ratio $n$ in impedance reflection ($Z/n^2$).
• Common Pitfall: In the “Dependent Source Method”, the dependent source in Loop 1 depends on Current 2 ($j\omega M{I}_2$), and vice versa.

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📖 Homework

• Practice Problem 13.1: Mesh analysis with Dot Convention.
• Practice Problem 13.3: Calculating Energy & Coupling Coefficient.
• Practice Problem 13.4: Linear Transformer Analysis.
• SGH4: Complete the assigned tasks.