8.6 Mutual inductance

Mutual inductance occurs when changing current in one inductor creates an EMF in another nearby inductor. This phenomenon is crucial for understanding coupled inductors and their applications in electrical systems. It's measured in henries and represented by the symbol M.

Coupled inductors are used in transformers, wireless charging, and power transfer systems. The coupling coefficient measures the strength of magnetic coupling between inductors. Leakage inductance, which doesn't contribute to mutual coupling, can be minimized through proper design and placement.

Mutual inductance

• Mutual inductance is a phenomenon where a change in current through one inductor induces an electromotive force (EMF) in another nearby inductor
• It is a key concept in understanding the behavior of coupled inductors and their applications in various electrical and electronic systems
• Mutual inductance is represented by the symbol $M$ and is measured in henries (H)

Coupled inductors

• Coupled inductors are two or more inductors that are physically close to each other, allowing their magnetic fields to interact
• The interaction between the magnetic fields of coupled inductors leads to mutual inductance, where a change in current in one inductor induces an EMF in the other inductor(s)
• Coupled inductors are used in various applications, such as transformers, inductive power transfer systems, and wireless charging devices

Coupling coefficient

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• The coupling coefficient, denoted as $k$, is a dimensionless quantity that represents the degree of magnetic coupling between two inductors
• It ranges from 0 to 1, where 0 indicates no coupling and 1 indicates perfect coupling
• The coupling coefficient is calculated as $k = \frac{M}{\sqrt{L_1 L_2}}$, where $M$ is the mutual inductance and $L_1$ and $L_2$ are the self-inductances of the two inductors

Leakage inductance

• Leakage inductance refers to the portion of the total inductance that does not contribute to the mutual coupling between inductors
• It is caused by the magnetic flux that does not link with the other inductor(s) in a coupled system
• Leakage inductance can be minimized by proper design and placement of the inductors to maximize the magnetic coupling between them

Mutual inductance calculation

• Mutual inductance can be calculated using various methods, depending on the geometry and arrangement of the inductors
• The calculation of mutual inductance involves determining the magnetic flux linkage between the inductors and applying the appropriate formulas

Neumann formula

• The Neumann formula is a general expression for calculating the mutual inductance between two current-carrying conductors
• It is given by $M = \frac{\mu_0}{4\pi} \int_{C_1} \int_{C_2} \frac{d\vec{l_1} \cdot d\vec{l_2}}{r}$, where $\mu_0$ is the permeability of free space, $C_1$ and $C_2$ are the contours of the two conductors, $d\vec{l_1}$ and $d\vec{l_2}$ are the infinitesimal length elements along the conductors, and $r$ is the distance between the elements
• The Neumann formula can be applied to various geometries, but it often requires complex integration techniques

Coaxial coils

• Coaxial coils are a common arrangement of coupled inductors, where one coil is placed inside the other, sharing the same axis
• The mutual inductance between coaxial coils can be calculated using the formula $M = \frac{\mu_0 N_1 N_2 \pi r^2}{l}$, where $N_1$ and $N_2$ are the number of turns in each coil, $r$ is the radius of the inner coil, and $l$ is the length of the coils
• Coaxial coils are often used in transformers and inductive coupling applications

Coplanar coils

• Coplanar coils are another arrangement of coupled inductors, where the coils are placed side by side in the same plane
• The mutual inductance between coplanar coils can be approximated using the formula $M = \frac{\mu_0 N_1 N_2 \pi r_1^2 r_2^2}{2(r_1^2 + r_2^2)^{3/2}}$, where $N_1$ and $N_2$ are the number of turns in each coil, and $r_1$ and $r_2$ are the radii of the coils
• Coplanar coils are used in applications such as wireless power transfer systems and RFID tags

Energy in coupled inductors

• The presence of mutual inductance between coupled inductors affects the energy storage and transfer in the system
• Understanding the energy relationships in coupled inductors is essential for analyzing and designing systems that utilize mutual inductance

Stored energy

• The total energy stored in a system of coupled inductors is given by $W = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M I_1 I_2$, where $L_1$ and $L_2$ are the self-inductances of the inductors, $I_1$ and $I_2$ are the currents flowing through them, and $M$ is the mutual inductance
• The first two terms represent the energy stored in the individual inductors, while the third term represents the energy associated with the mutual inductance
• The sign of the mutual inductance term depends on the direction of the currents in the inductors

Energy transfer

• Mutual inductance enables the transfer of energy between coupled inductors
• When the current in one inductor changes, it induces an EMF in the other inductor, causing a current to flow and energy to be transferred
• The rate of energy transfer between coupled inductors is proportional to the mutual inductance and the rate of change of the currents
• Energy transfer through mutual inductance is the basis for many applications, such as transformers and wireless power transfer systems

Mutual inductance applications

• Mutual inductance is utilized in various electrical and electronic applications, where the coupling between inductors is exploited for energy transfer, isolation, or signal conditioning

Transformers

• Transformers are devices that use mutual inductance to transfer electrical energy between two or more circuits while providing electrical isolation
• They consist of two or more coils wound on a common magnetic core, with the primary coil connected to the input circuit and the secondary coil(s) connected to the output circuit(s)
• Transformers are used for voltage and current transformation, impedance matching, and electrical isolation in power systems, electronics, and communication devices

Ideal vs real transformers

• An ideal transformer is a theoretical model that assumes perfect coupling between the primary and secondary coils, no energy losses, and no leakage inductance
• In an ideal transformer, the voltage ratio is equal to the turns ratio, and the power is conserved between the primary and secondary sides
• Real transformers, however, have imperfect coupling, energy losses (core losses and copper losses), and leakage inductance, which affect their performance and efficiency
• The design of real transformers involves minimizing these non-ideal effects to approach the behavior of an ideal transformer

Inductive power transfer

• Inductive power transfer (IPT) is a method of wirelessly transmitting electrical power using the principle of mutual inductance
• In an IPT system, a primary coil is energized by an alternating current, creating a varying magnetic field
• A secondary coil placed in close proximity to the primary coil captures the magnetic field and induces an EMF, which can be used to power a load or charge a battery
• IPT is used in various applications, such as charging electric vehicles, powering implantable medical devices, and supplying power to consumer electronics

Wireless charging

• Wireless charging is a specific application of inductive power transfer, where mutual inductance is used to charge batteries in portable devices without the need for a physical connection
• A charging pad contains a primary coil that generates a varying magnetic field, while the device to be charged contains a secondary coil that captures the magnetic field and converts it into electrical energy to charge the battery
• Wireless charging standards, such as Qi and PMA, define the communication protocols and coil designs to ensure compatibility between different devices and charging pads

Mutual inductance vs self-inductance

• Mutual inductance and self-inductance are two related but distinct concepts in electromagnetism

Similarities

• Both mutual inductance and self-inductance are measured in henries (H)
• They both involve the interaction between electric currents and magnetic fields
• Both concepts are important in understanding the behavior of inductors and their applications in electrical and electronic systems

Key differences

• Self-inductance is the property of a single inductor, representing the ratio of the induced EMF to the rate of change of current in the inductor itself
• Mutual inductance, on the other hand, is the property of two or more inductors, representing the ratio of the induced EMF in one inductor to the rate of change of current in another inductor
• Self-inductance is always positive, while mutual inductance can be positive, negative, or zero, depending on the relative orientation of the inductors and the direction of the currents
• The energy stored in an inductor due to self-inductance is given by $W = \frac{1}{2} L I^2$, while the energy associated with mutual inductance is given by $W = M I_1 I_2$

Mutual inductance in AC circuits

• Mutual inductance plays a significant role in the behavior of AC circuits containing coupled inductors
• The presence of mutual inductance affects the impedance, voltage, and current relationships in the circuit

Impedance matrix

• In AC circuits with coupled inductors, the impedance matrix is used to represent the relationship between the voltages and currents in the system
• The impedance matrix includes the self-impedances of the inductors (which depend on their self-inductances and resistances) and the mutual impedances (which depend on the mutual inductances)
• The off-diagonal elements of the impedance matrix represent the mutual impedances, while the diagonal elements represent the self-impedances
• The impedance matrix allows for the analysis of voltage and current distributions in the circuit using matrix algebra

Equivalent circuits

• Coupled inductors in AC circuits can be represented by equivalent circuits that model their behavior
• The most common equivalent circuit for coupled inductors is the T-network, which consists of three inductors: two self-inductances (L1 and L2) and a mutual inductance (M)
• The T-network equivalent circuit allows for the analysis of the circuit using standard circuit analysis techniques, such as Kirchhoff's laws and impedance calculations
• Other equivalent circuits, such as the Pi-network and the coupled inductors with an ideal transformer, can also be used to model coupled inductors in AC circuits

Mutual inductance measurement

• Measuring mutual inductance is essential for characterizing coupled inductors and verifying their performance in various applications
• There are several methods for measuring mutual inductance, which can be broadly classified into direct measurement methods and indirect calculation techniques

Direct measurement methods

• Direct measurement methods involve physically connecting the coupled inductors to measurement instruments and observing the voltage or current responses
• One common direct measurement method is the voltage divider method, where a known voltage is applied to one inductor, and the voltage across the other inductor is measured
• Another direct method is the current measurement method, where a known current is passed through one inductor, and the induced current in the other inductor is measured
• Direct measurement methods provide accurate results but may require specialized equipment and careful setup to minimize measurement errors

Indirect calculation techniques

• Indirect calculation techniques involve measuring other parameters of the coupled inductors and using mathematical relationships to calculate the mutual inductance
• One indirect technique is the self-inductance measurement method, where the self-inductances of the individual inductors are measured, and the mutual inductance is calculated using the coupling coefficient formula
• Another indirect technique is the resonance method, where the coupled inductors are connected in a resonant circuit, and the resonant frequency is measured to determine the mutual inductance
• Indirect calculation techniques can be more convenient than direct methods, as they often require only standard LCR meters or impedance analyzers, but their accuracy depends on the accuracy of the measured parameters and the validity of the mathematical models used