🔦Electrical Circuits and Systems II Unit 5 – Coupled Circuits & Mutual Inductance

Key Concepts and Definitions

• Coupled circuits consist of two or more inductors in close proximity, allowing magnetic fields to interact and influence each other
• Mutual inductance ($M$) quantifies the coupling between two inductors, measured in henries (H)
  • Determined by the physical characteristics of the inductors and their relative positions
• Coupling coefficient ($k$) represents the degree of magnetic coupling between inductors, ranging from 0 (no coupling) to 1 (perfect coupling)
  • Calculated as $k = \frac{M}{\sqrt{L_1 L_2}}$, where $L_1$ and $L_2$ are the self-inductances of the coupled inductors
• Dot convention indicates the relative winding directions of coupled inductors, ensuring consistent voltage and current relationships
• Leakage inductance refers to the portion of an inductor's magnetic flux that does not link to the other inductor in a coupled circuit
• Magnetizing inductance represents the inductance that produces the magnetizing flux in a transformer

Magnetic Fields and Flux

• Magnetic fields are created by moving charges or permanent magnets, characterized by field lines indicating the direction of the field
• Magnetic flux ($\Phi$) is the total number of magnetic field lines passing through a given area, measured in webers (Wb)
  • Calculated as $\Phi = \int \vec{B} \cdot d\vec{A}$, where $\vec{B}$ is the magnetic field and $d\vec{A}$ is the area element
• Faraday's law of induction states that a changing magnetic flux through a loop induces an electromotive force (EMF) in the loop
  • Induced EMF is proportional to the rate of change of magnetic flux, given by $\mathcal{E} = -\frac{d\Phi}{dt}$
• Lenz's law determines the direction of the induced EMF, opposing the change in magnetic flux that produced it
• Ampère's circuital law relates the magnetic field around a closed loop to the electric current passing through the loop
  • Expressed as $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$, where $\mu_0$ is the permeability of free space and $I_{enc}$ is the enclosed current

Self-Inductance Basics

• Self-inductance ($L$) is the property of an inductor that opposes changes in current flowing through it, measured in henries (H)
  • Defined as $L = \frac{\Phi}{I}$, where $\Phi$ is the magnetic flux and $I$ is the current
• Inductors store energy in their magnetic fields when current flows through them
  • Energy stored is given by $E = \frac{1}{2}LI^2$, where $L$ is the inductance and $I$ is the current
• Voltage across an inductor is proportional to the rate of change of current, expressed as $v_L = L\frac{di}{dt}$
• Inductors exhibit inductive reactance ($X_L$) in AC circuits, which opposes changes in current
  • Inductive reactance is calculated as $X_L = 2\pi fL$, where $f$ is the frequency and $L$ is the inductance
• Series and parallel combinations of inductors follow specific rules:
  • Series: $L_{total} = L_1 + L_2 + ... + L_n$
  • Parallel: $\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}$

Mutual Inductance Principles

• Mutual inductance occurs when the magnetic flux generated by one inductor links to another inductor
• Mutual inductance ($M$) is the ratio of the induced voltage in the secondary inductor to the rate of change of current in the primary inductor
  • Expressed as $M = \frac{v_2}{di_1/dt}$, where $v_2$ is the induced voltage in the secondary and $di_1/dt$ is the rate of change of current in the primary
• The dot convention is used to indicate the relative winding directions of coupled inductors
  • Dots on the same side indicate that currents entering the dotted terminals produce magnetic fields that add constructively
• Coupling coefficient ($k$) quantifies the degree of magnetic coupling between inductors
  • Perfect coupling ($k = 1$) occurs when all the magnetic flux generated by one inductor links to the other
  • No coupling ($k = 0$) occurs when none of the magnetic flux generated by one inductor links to the other
• Mutual inductance is always less than or equal to the geometric mean of the self-inductances of the coupled inductors
  • Expressed as $M \leq \sqrt{L_1 L_2}$, where $L_1$ and $L_2$ are the self-inductances of the coupled inductors

Coupled Circuit Analysis

• Coupled circuits can be analyzed using Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL)
  • KVL states that the sum of voltages around a closed loop is zero
  • KCL states that the sum of currents entering a node is equal to the sum of currents leaving the node
• Mutual inductance introduces additional voltage terms in the circuit equations
  • For a two-inductor coupled circuit: $v_1 = L_1 \frac{di_1}{dt} + M \frac{di_2}{dt}$ and $v_2 = L_2 \frac{di_2}{dt} + M \frac{di_1}{dt}$
• Coupled circuits can be simplified using equivalent circuits that replace the coupled inductors with a combination of self-inductances and controlled sources
  • T-equivalent circuit represents the coupled inductors using three inductors and a coupling factor
  • $\pi$-equivalent circuit represents the coupled inductors using two inductors and a coupling factor
• Phasor analysis can be used to solve coupled circuits in the frequency domain
  • Inductors are represented by their impedances: $Z_L = j\omega L$, where $\omega$ is the angular frequency
  • Mutual inductance introduces additional voltage phasors: $V_1 = j\omega L_1 I_1 + j\omega M I_2$ and $V_2 = j\omega L_2 I_2 + j\omega M I_1$

Transformer Theory

• Transformers are devices that use mutual inductance to transfer energy between circuits while providing electrical isolation
• Ideal transformers have perfect coupling ($k = 1$), no leakage inductance, and no losses
  • Voltage ratio is equal to the turns ratio: $\frac{V_1}{V_2} = \frac{N_1}{N_2}$, where $N_1$ and $N_2$ are the number of turns in the primary and secondary windings
  • Current ratio is inversely proportional to the turns ratio: $\frac{I_1}{I_2} = \frac{N_2}{N_1}$
• Practical transformers have imperfect coupling, leakage inductance, and losses (copper and core losses)
  • Leakage inductance results in voltage drops and affects the voltage regulation of the transformer
  • Copper losses are due to the resistance of the windings and cause power dissipation
  • Core losses are due to hysteresis and eddy currents in the magnetic core and also cause power dissipation
• Equivalent circuit models (T-equivalent and $\pi$-equivalent) are used to analyze practical transformers
  • Models include leakage inductances, winding resistances, and a magnetizing branch to represent core losses and magnetizing current

Applications and Real-World Examples

• Power transformers are used in electrical power systems to step up or step down voltages for efficient transmission and distribution
  • High-voltage transmission minimizes power losses over long distances
  • Distribution transformers step down the voltage to levels suitable for end-users (120/240 V in North America)
• Audio transformers are used in audio equipment to provide impedance matching and electrical isolation
  • Microphone transformers match the low impedance of microphones to the high impedance of preamplifiers
  • Output transformers in tube amplifiers match the high impedance of the output tubes to the low impedance of loudspeakers
• Coupled inductors are used in radio frequency (RF) circuits for impedance matching and filtering
  • Directional couplers use coupled transmission lines to sample a portion of the signal power while maintaining isolation between ports
  • Bandpass filters use coupled resonators to achieve sharp frequency selectivity and high Q factors
• Wireless power transfer systems use coupled coils to transfer energy between a transmitter and a receiver without physical contact
  • Inductive charging for smartphones and electric vehicles
  • Resonant coupling allows for increased transfer distance and efficiency compared to simple inductive coupling

Problem-Solving Techniques

• Identify the type of coupling (series or parallel) and the dot convention used in the circuit diagram
• Label the voltages, currents, and impedances in the circuit using consistent notation
• Write the voltage and current equations for each inductor using KVL and KCL, taking into account the mutual inductance terms
• Solve the system of equations using matrix methods, substitution, or other algebraic techniques
  • For sinusoidal steady-state analysis, use phasor notation and complex impedances
• Check the solution for consistency with the dot convention and the expected behavior of the circuit
• Simplify the circuit using equivalent models (T-equivalent or $\pi$-equivalent) if necessary
  • Replace the coupled inductors with the equivalent model and solve the resulting circuit
• Use symmetry and superposition to simplify the analysis of symmetric or multi-source circuits
  • Identify symmetric components and analyze them separately
  • Use superposition to find the total response by summing the responses to each source individually
• Verify the results using simulation tools (SPICE, MATLAB, etc.) or experimental measurements
  • Compare the analytical solution with the simulated or measured results
  • Investigate any discrepancies and refine the model or analysis as needed