Inductance and Energy Storage
4. Mutual Inductance
Learning outcomes
- I can define mutual inductance.
- I can explain how energy is transferred between coils.
- I can identify factors affecting mutual inductance.
- I can compare mutual inductance with self-inductance.
- I can explain how mutual inductance is used in electrical devices.
Mutual Inductance
Mutual inductance describes the ability of a changing current in one circuit to produce an induced EMF in another nearby circuit.
When current in the first coil changes, its magnetic field changes. If some of this changing magnetic field passes through a second coil, the magnetic flux through the second coil changes. According to Faraday's law, this changing flux produces an induced EMF.
The process can be summarized as:
changing current in Coil 1 → changing magnetic field → changing flux through Coil 2 → induced EMF in Coil 2
Mutual inductance allows electrical energy and signals to be transferred between circuits without requiring a direct electrical connection.
Defining Mutual Inductance
The strength of the electromagnetic interaction between two circuits is measured by their mutual inductance.
The symbol is:
M
The SI unit is the:
henry (H)
For two coils, the magnitude of the induced EMF in Coil 2 can be written:
|ε₂| = M|ΔI₁/Δt|
where:
- ε₂ = induced EMF in Coil 2
- M = mutual inductance
- ΔI₁ = change in current in Coil 1
- Δt = time interval
Including direction:
ε₂ = −M ΔI₁/Δt
The negative sign represents Lenz's law.
Meaning of Mutual Inductance
A large value of M means the coils are strongly coupled.
A relatively small change in primary current can produce a significant induced EMF in the second coil.
A small value of M means the coils are weakly coupled.
For the same rate of current change:
larger M → larger induced EMF
Meaning of One Henry
Suppose two coils have:
M = 1 H
If current in the first coil changes at:
1 A/s
the magnitude of the induced EMF in the second coil is:
1 V
Therefore:
1 H = 1 V·s/A
The henry is used for both:
- self-inductance
- mutual inductance
How Mutual Induction Occurs
Consider two nearby coils.
Coil 1 carries a changing current.
This changing current produces a changing magnetic field.
Some of that field passes through Coil 2.
Therefore, the magnetic flux through Coil 2 changes.
Faraday's law states:
ε = −N ΔΦ/Δt
so an EMF appears in Coil 2.
If Coil 2 forms part of a closed circuit, this EMF can produce a current.
EMF Does Not Always Mean Current
An important distinction is:
changing magnetic flux produces an EMF
but current flows only if there is a complete conducting path.
Therefore, an open secondary coil can have an induced voltage across its terminals even though no current flows.
If the secondary circuit is closed:
induced EMF → induced current
Energy Transfer Between Coils
Mutual inductance allows energy to move from one circuit to another through a magnetic field.
The basic energy pathway is:
electrical energy in Coil 1 → changing magnetic field → electrical energy in Coil 2
The coils do not need to touch.
They also do not need a direct electrical connection.
Instead, they must be magnetically coupled.
Primary and Secondary Coils
In many mutual-induction systems, the coils are called:
Primary coil — receives the original electrical input.
Secondary coil — receives energy through electromagnetic induction.
The changing current in the primary creates the changing magnetic flux responsible for the induced EMF in the secondary.
These terms are especially common when discussing transformers.
Energy Is Conserved
Mutual inductance does not create energy.
Energy transferred to the secondary ultimately comes from the source connected to the primary.
For an ideal system:
input energy = output energy
and:
input power = output power
Real devices lose some energy through:
- electrical resistance
- eddy currents
- magnetic hysteresis
- flux leakage
- heating
Therefore:
P_output < P_input
in real systems.
Why Changing Current Is Required
Suppose the primary carries a constant current.
A constant current produces a constant magnetic field.
A constant magnetic field produces constant magnetic flux through the secondary.
Therefore:
ΔΦ/Δt = 0
and:
ε₂ = 0
So a steady current does not produce continuous mutual induction.
What matters is:
changing current
rather than simply the presence of current.
Switching DC On and Off
A DC circuit can still produce mutual induction briefly.
When the primary is switched on:
- current increases
- magnetic field increases
- secondary flux increases
- EMF is induced
Once the current becomes steady:
induced EMF → 0
When the primary is switched off:
- current decreases
- magnetic field collapses
- secondary flux decreases
- another EMF is induced
The induced EMF during switch-off has the opposite direction from that during switch-on.
Alternating Current
Alternating current continually changes.
Therefore:
AC → changing magnetic field → changing magnetic flux → continuous induced EMF
This makes AC especially useful for devices based on mutual induction.
Transformers are the most important example.
Factors Affecting Mutual Inductance
Mutual inductance depends on how effectively the magnetic field from one coil links with the other.
Important factors include:
- number of turns
- distance between coils
- orientation of coils
- amount of overlap
- coil size and geometry
- magnetic core
- magnetic permeability
- leakage flux
Together, these determine the strength of magnetic coupling.
Number of Turns
Increasing the number of turns can increase mutual inductance.
More turns in the primary can strengthen its magnetic effect.
More turns in the secondary provide greater total flux linkage.
Therefore:
more turns → potentially greater mutual inductance
The exact relationship depends on the geometry of the two coils.
Distance Between the Coils
Mutual inductance generally becomes smaller as the coils are moved farther apart.
At greater distances:
- less primary magnetic field reaches the secondary
- less flux links the secondary
- coupling becomes weaker
Therefore:
greater separation → smaller M
in typical arrangements.
Coil Orientation
The relative orientation of the coils is extremely important.
If the coils are aligned so that much of the primary field passes through the secondary:
mutual inductance is relatively large
If one coil is rotated:
- less magnetic flux may pass through it
- flux linkage decreases
- mutual inductance decreases
Why Orientation Matters
Magnetic flux is:
Φ = BA cos θ
where θ is the angle between the magnetic field and the normal to the coil.
When the magnetic field is perpendicular to the plane of the coil:
θ = 0°
and:
Φ = BA
The flux is maximum.
If the field lies parallel to the plane:
θ = 90°
and:
Φ = 0
Therefore, changing the orientation of the secondary can dramatically change the magnetic flux linking it.
Coil Overlap
Coils that strongly overlap can share more magnetic flux.
If the coils are moved sideways so that their overlap decreases, less magnetic flux may link them.
Therefore:
greater effective overlap → stronger coupling
This principle is important in wireless power-transfer systems.
Magnetic Cores
A suitable magnetic core can greatly increase mutual inductance.
The core provides a path that guides magnetic flux from one coil through the other.
This can:
- increase magnetic flux
- reduce leakage
- improve coupling
- increase mutual inductance
- improve energy transfer
Transformer cores are designed for exactly this purpose.
Flux Leakage
Not all magnetic flux produced by the primary necessarily passes through the secondary.
The portion that does not link effectively with the other coil is called leakage flux.
Greater leakage means:
- weaker magnetic coupling
- smaller mutual inductance
- less effective energy transfer
Reducing leakage is therefore important when strong coupling is required.
Coupling Coefficient
The degree of magnetic coupling can also be described by the coupling coefficient, k.
For two coils:
M = k√(L₁L₂)
where:
- M = mutual inductance
- L₁ = self-inductance of Coil 1
- L₂ = self-inductance of Coil 2
- k = coupling coefficient
For idealized systems:
0 ≤ k ≤ 1
Meaning of the Coupling Coefficient
If:
k ≈ 1
the coils are very strongly coupled.
If:
k ≈ 0
the coils are very weakly coupled.
The equation:
M = k√(L₁L₂)
shows that mutual inductance depends on both:
- the inductances of the individual coils
- how effectively they are magnetically coupled
Worked Example: Coupling Coefficient
Two coils have:
L₁ = 4.0 H
L₂ = 9.0 H
and:
k = 0.50
Use:
M = k√(L₁L₂)
M = 0.50√[(4.0)(9.0)]
M = 0.50√36
M = 0.50(6.0)
M = 3.0 H
Worked Example: Induced EMF
Two coils have mutual inductance:
M = 0.40 H
The current in Coil 1 changes from:
2.0 A to 8.0 A
in:
0.30 s
Calculate:
ΔI₁ = 8.0 − 2.0
ΔI₁ = 6.0 A
Rate of current change:
ΔI₁/Δt = 6.0/0.30
ΔI₁/Δt = 20 A/s
Then:
|ε₂| = M|ΔI₁/Δt|
|ε₂| = (0.40)(20)
|ε₂| = 8.0 V
Worked Example: Finding Mutual Inductance
A changing current in Coil 1 produces an induced EMF of:
12 V
in Coil 2.
The current changes at:
30 A/s
Use:
|ε₂| = M|ΔI₁/Δt|
Rearrange:
M = |ε₂| / |ΔI₁/Δt|
M = 12/30
M = 0.40 H
Worked Example: Finding Rate of Current Change
Two coils have:
M = 0.25 H
The secondary experiences:
|ε₂| = 5.0 V
Use:
|ΔI₁/Δt| = |ε₂|/M
|ΔI₁/Δt| = 5.0/0.25
|ΔI₁/Δt| = 20 A/s
Self-Inductance
Self-inductance occurs when a changing current produces an induced EMF in the same circuit.
The relationship is:
ε = −L ΔI/Δt
where L is the self-inductance.
The coil responds to changes in its own current.
Mutual Inductance
Mutual inductance occurs when changing current in one circuit induces an EMF in another circuit.
The relationship is:
ε₂ = −M ΔI₁/Δt
where M is the mutual inductance.
The second coil responds to the changing magnetic field produced by the first.
Comparing Self-Inductance and Mutual Inductance
| Self-Inductance | Mutual Inductance |
|---|---|
| Occurs within the same circuit | Occurs between two circuits |
| Uses L | Uses M |
| ε = −L ΔI/Δt | ε₂ = −M ΔI₁/Δt |
| Circuit responds to its own changing current | Second circuit responds to changing current in first |
| Important in inductors and RL circuits | Important in transformers and coupled coils |
| Measured in henries | Measured in henries |
Both are consequences of Faraday's law and Lenz's law.
Both Effects Can Occur Together
Suppose two coils are placed around the same magnetic core.
When current changes in Coil 1:
- Coil 1 experiences self-induction
- Coil 2 experiences mutual induction
Therefore, self-inductance and mutual inductance can occur simultaneously.
This happens in transformers.
Transformers
A transformer is one of the most important applications of mutual inductance.
It contains:
- a primary winding
- a secondary winding
- usually a magnetic core
Alternating current in the primary produces a continuously changing magnetic field.
The core guides much of this changing magnetic flux through the secondary.
An alternating EMF is then induced in the secondary.
Energy Transfer in a Transformer
The transformer energy pathway can be represented as:
electrical energy in primary
↓
changing magnetic field
↓
changing magnetic flux through secondary
↓
electrical energy in secondary
The windings can remain electrically isolated even though energy is transferred between them.
Transformer Voltage Ratio
For an ideal transformer:
V_s/V_p = N_s/N_p
where:
- V_s = secondary voltage
- V_p = primary voltage
- N_s = number of secondary turns
- N_p = number of primary turns
Changing the turns ratio allows the transformer to change voltage.
Step-Up Transformers
For a step-up transformer:
N_s > N_p
so:
V_s > V_p
The transformer increases voltage.
For an ideal transformer, this increase in voltage is accompanied by a decrease in current.
Step-Down Transformers
For a step-down transformer:
N_s < N_p
so:
V_s < V_p
The transformer decreases voltage.
For an ideal transformer, the secondary current can then be greater than the primary current.
Wireless Charging
Wireless charging is another important application of mutual inductance.
A transmitter coil carries alternating current.
This creates a changing magnetic field.
A receiver coil inside the device experiences changing magnetic flux.
An EMF is induced in the receiver.
The resulting electrical energy can be used to charge a battery.
Why Wireless Charger Position Matters
If the transmitter and receiver coils are well aligned:
- magnetic coupling is stronger
- mutual inductance is greater
- more flux links the receiver
If the device is badly aligned:
- coupling becomes weaker
- less flux links the receiver
- energy transfer becomes less effective
This is why the position of a device on a wireless charging pad matters.
Electric Toothbrush Charging
Inductive charging is useful in devices where exposed electrical contacts may be undesirable.
Some electric toothbrush chargers contain:
- a transmitting coil in the base
- a receiving coil in the toothbrush
Changing current in the base produces a changing magnetic field.
Mutual induction transfers energy to the toothbrush without requiring exposed electrical contacts between the charger and device.
Inductive Sensors
Mutual inductance is also used in sensing systems.
A changing magnetic field produced by one coil can interact with:
- another coil
- conducting objects
- magnetic materials
Changes in coupling can then be detected electrically.
Applications include:
- position sensors
- proximity sensors
- metal detectors
- industrial measurement systems
Signal Transfer
Mutual inductance can transfer changing electrical signals between circuits.
Small transformers are used in some electronic systems for:
- signal coupling
- electrical isolation
- impedance matching
- noise control
The circuits can exchange energy and information without sharing a direct conductive path.
Ignition Systems
Some ignition systems use strongly coupled coils.
Energy is first stored in a magnetic field.
When current changes rapidly, a large change in magnetic flux occurs.
A secondary winding with many turns can develop a very large voltage.
This high voltage can then be used to create an electrical spark.
Electrical Isolation
Transformers can provide galvanic isolation.
The primary and secondary may have no direct conducting connection.
Instead:
energy crosses through magnetic coupling
This can help:
- improve safety
- isolate sensitive circuits
- reduce unwanted ground connections
- protect equipment
Strong and Weak Coupling
Consider two identical pairs of coils.
System A
The coils are:
- close together
- aligned
- wrapped around the same magnetic core
This system has relatively strong coupling and high mutual inductance.
System B
The coils are:
- far apart
- poorly aligned
- surrounded mainly by air
This system has much weaker coupling and lower mutual inductance.
Comparing Coil Arrangements
Suppose the same changing primary current is applied to several arrangements.
The largest induced secondary voltage would generally occur when the coils have:
- many turns
- close spacing
- strong alignment
- large useful overlap
- a suitable shared magnetic core
The smallest induced voltage would generally occur when the coils are:
- far apart
- poorly aligned
- weakly coupled
Energy Transfer Efficiency
Strong mutual inductance does not automatically mean a device is perfectly efficient.
Real systems can lose energy because of:
- wire resistance
- eddy-current heating
- hysteresis
- leakage flux
- electromagnetic radiation
- other circuit losses
Designers therefore try to achieve strong useful coupling while minimizing losses.
Mutual Inductance and Eddy Currents
Mutual induction involves intended coupling between circuits or coils.
However, changing magnetic fields can also induce unwanted circulating currents in nearby conducting materials.
These are eddy currents.
For example, transformer cores can develop eddy currents.
Laminated cores help reduce these unwanted currents and their associated heating.
Mutual Inductance and RL Circuits
The primary winding of a transformer has self-inductance.
Therefore, its current does not necessarily change instantaneously.
At the same time, its changing magnetic field can induce an EMF in another winding.
So the concepts studied in:
- self-induction
- inductance
- RL circuits
- magnetic energy
- mutual induction
are closely connected.
A Complete Calculation
Two coils have:
M = 0.60 H
The primary current decreases from:
8.0 A to 2.0 A
in:
0.20 s
Change in current:
ΔI₁ = 2.0 − 8.0
ΔI₁ = −6.0 A
Magnitude of the rate:
|ΔI₁/Δt| = 6.0/0.20
|ΔI₁/Δt| = 30 A/s
Induced EMF:
|ε₂| = M|ΔI₁/Δt|
|ε₂| = (0.60)(30)
|ε₂| = 18 V
The induced effect acts according to Lenz's law to oppose the decrease in magnetic flux.
Complete Coupling Problem
Two coils have:
L₁ = 0.80 H
L₂ = 0.20 H
and coupling coefficient:
k = 0.75
Calculate their mutual inductance.
Use:
M = k√(L₁L₂)
M = 0.75√[(0.80)(0.20)]
M = 0.75√0.16
M = 0.75(0.40)
M = 0.30 H
If current in Coil 1 changes at:
10 A/s
then:
|ε₂| = M|ΔI₁/Δt|
|ε₂| = (0.30)(10)
|ε₂| = 3.0 V
A Reliable Strategy for Mutual-Inductance Problems
First identify:
- which coil has the changing current
- which coil experiences the induced EMF
- whether the coils are magnetically coupled
For a simple induced-EMF problem use:
|ε₂| = M|ΔI₁/Δt|
For coupling problems use:
M = k√(L₁L₂)
For transformer problems, the turns ratio may be needed:
V_s/V_p = N_s/N_p
Then consider the physical arrangement:
- distance
- orientation
- turns
- core material
- overlap
- leakage flux
Finally, use Lenz's law if the direction of the induced effect is required.
Common Mistakes
Confusing mutual inductance with mutual induction
Mutual induction is the process. Mutual inductance M is the quantity describing the strength of coupling.
Thinking the coils must be electrically connected
They can transfer energy magnetically while remaining electrically isolated.
Thinking a steady current continuously induces an EMF
A steady current produces a steady field. Continuous induction requires changing magnetic flux.
Ignoring coil position
Distance and orientation strongly affect coupling.
Assuming all magnetic flux links both coils
Some flux can leak away.
Confusing L and M
L describes self-inductance. M describes mutual inductance.
Forgetting the rate of current change
Induced EMF depends on ΔI/Δt, not simply the amount of current.
Assuming energy is created in the secondary
Energy transferred to the secondary ultimately comes from the primary circuit.
Thinking stronger coupling means perfect efficiency
Resistance, core losses, leakage, and other effects can still reduce efficiency.
Key Terms
Mutual inductance — measure of how strongly changing current in one circuit induces an EMF in another circuit.
Mutual induction — process by which changing current in one circuit produces an induced EMF in another.
Self-inductance — ability of a circuit's changing current to induce an EMF in the same circuit.
Henry (H) — SI unit of self-inductance and mutual inductance.
Primary coil — coil carrying the changing current that produces the changing magnetic field.
Secondary coil — coil in which an EMF is induced by changing magnetic flux from another coil.
Magnetic coupling — degree to which magnetic flux produced by one circuit links another circuit.
Coupling coefficient — number k describing the strength of magnetic coupling between two inductors.
Flux linkage — total magnetic flux linked with the turns of a coil.
Leakage flux — magnetic flux that does not effectively link both coupled coils.
Faraday's law — relationship between changing magnetic flux and induced EMF.
Lenz's law — principle stating that an induced effect opposes the change producing it.
Transformer — device that transfers electrical energy between circuits using mutual induction.
Galvanic isolation — separation of circuits so there is no direct conductive connection between them.
Wireless power transfer — transfer of electrical energy without direct electrical contacts, often using magnetic coupling.
Key Takeaways
- Mutual inductance describes electromagnetic coupling between two circuits.
- Its symbol is M and its SI unit is the henry.
- Changing current in one coil produces a changing magnetic field.
- Changing magnetic flux through another coil produces an induced EMF.
- For simple situations, |ε₂| = M|ΔI₁/Δt|.
- Faster current changes produce larger induced EMFs.
- Greater mutual inductance produces larger induced EMFs for the same rate of current change.
- Mutual induction transfers energy through magnetic fields rather than through a direct electrical connection.
- A secondary circuit must be closed for an induced current to flow, although an induced EMF can exist in an open circuit.
- Mutual inductance depends on coil geometry, turns, distance, orientation, overlap, and core material.
- Moving coils closer together generally increases coupling.
- Better coil alignment generally increases coupling.
- A suitable common magnetic core can greatly increase mutual inductance.
- Leakage flux reduces magnetic coupling.
- The coupling coefficient can be used with M = k√(L₁L₂).
- Self-inductance involves one circuit responding to its own changing current.
- Mutual inductance involves one circuit responding to changing current in another circuit.
- Both self-inductance and mutual inductance follow Faraday's and Lenz's laws.
- Transformers use mutual inductance to transfer electrical energy between windings.
- Wireless chargers use magnetic coupling between transmitter and receiver coils.
- Mutual inductance is also important in sensors, ignition systems, signal transformers, and electrically isolated circuits.
Check Your Understanding
- Define mutual inductance and state its SI unit.
- Explain the difference between mutual induction and mutual inductance.
- Describe how changing current in one coil can produce an EMF in another coil.
- Why does a constant DC current in the primary not continuously induce an EMF in the secondary?
- Explain how energy can be transferred between two coils that have no direct electrical connection.
- Identify four factors that can affect mutual inductance between two coils.
- Explain why moving two coils farther apart generally decreases their mutual inductance.
- Explain why changing the orientation of one coil can affect the induced EMF.
- Two coils have M = 0.30 H. Current in Coil 1 changes at 20 A/s. Calculate the magnitude of the induced EMF in Coil 2.
- A secondary coil experiences an induced EMF of 12 V when primary current changes at 40 A/s. Calculate the mutual inductance.
- Two coils have L₁ = 2.0 H, L₂ = 8.0 H, and k = 0.50. Calculate their mutual inductance.
- Compare self-inductance and mutual inductance. Give at least two similarities and two differences.
- Explain how mutual inductance allows a transformer to transfer electrical energy from its primary winding to its secondary winding.
- Explain why correct alignment between transmitter and receiver coils is important in a wireless charging system.
- An engineer wants to increase the mutual inductance between two coils. Describe at least four changes that could be made to their design or arrangement and explain why each would help.