What precisely is the numerical (or any) meaning of mutual inductance and, is it useful, representative (of something) or important?

Question

When we consider two inductors that are magnetically coupled, we sometimes talk about mutual inductance. As an example, consider two inductors (such as 9 μH and 4 μH) that are 100% coupled and produce magnetic fields that are aiding. We calculate the mutual inductance (M) like this: -

$$M = \sqrt{L_1\times L_2} \hspace{1cm}=\hspace{1cm} \sqrt{9\text{ }\mu H\times 4\text{ }\mu H} \hspace{1cm}=\hspace{1cm} 6\text{ }\mu H$$

So, a 100% coupled 9 μH and 4 μH inductor have a mutual inductance of 6 μH. But what does the 6 μH represent? I can see that 6 μH is the geometric mean of 9 μH and 4 μH but is that all?

Using "M", we can find the net inductance using this formula: -

$$L_{NET} = L_1 + L_2 + 2\sqrt{L_1\times L_2}$$

And, in the simple example above, we get 9 μH + 4 μH + (2 x 6) μH = 25 μH.

But we don't need M to calculate \$L_{NET}\$. We can calculate it more directly with this formula: -

$$L_{NET} = \left(\sqrt{L_1}+\sqrt{L_2} \right)^2\hspace{1cm}=\hspace{1cm} \left(3+2\right)^2\text{ }\mu H\hspace{1cm} = \hspace{1cm}25 \mu H$$

• So, does M (mutual inductance) have a numerical (or any other) useful meaning?
• Does it represent something that is important to us?

Is it only of (some) use when calculating the net inductance of two coupled inductors?

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Update Sept 10th

Given the answers so far, I'm sorry to say that I still find M to be fairly meaningless given that it translates into more useful terms like N (turns ratio) and k (coupling factor) in every situation the current answers have raised.

If someone can come up with an EE example where M is clearly a benefit over more pragmatic variables, it would be good.

Answer 1

If you are writing mesh equations for a circuit containing a transformer (or coupled inductors), use of mutual inductance leads to a simple formulation for the potential difference across a branch containing one of the inductors.

For example if \$V_1\$ is the voltage across the branch formed by inductor L1 and \$V_2\$ is the voltage across L2, with mutual inductance \$M\$ between the two inductors, then

$$V_1 = L_1\frac{dI_1}{dt}-M\frac{dI_2}{dt}$$ and $$V_2 = L_2\frac{dI_2}{dt}-M\frac{dI_1}{dt}$$

Or, in the frequency domain $$v_1 = j\omega L_1 i_1 - j\omega M i_2$$ and $$v_2 = j\omega L_2 i_2 - j\omega M i_1.$$

Of course there's no information lost if you substitute \$M=k\sqrt{L_1L_2}\$, but the equation has a simpler form if you condense that into one term instead of three terms with a square root to be calculated.

Also, the first three or four pages I found by googling (that was part of the question, right?) all define \$M\$ first, and then define \$k\$ in terms of \$M\$. Presumably this was the historic way the terms were developed as outlined in user4574's answer. So one benefit of defining the mutual inductance is to give you something from which to define the coupling factor \$k\$, if you prefer to formulate your equations in terms of \$k\$.

Answer 2

I have never found M to be useful and don't understand why it appears to have such prominence.

One reason M enjoys so much prominence is because it was part of Maxwell's original publication on electrodynamics. The first known appearance of M in literature that I am aware of is right in Maxwell's 1865 publication "A dynamical theory of the electromagnetic field" (See page 468).

Is it only of use when calculating the net inductance of two coupled inductors?

Maxwell lays it out pretty clearly in his publication (although his terminology is a bit archaic).

For two circuits (inductors) A and B. The electromagnetic momentum belonging to A is...

L*x + M * y

The electromagnetic momentum belonging to B is...

M*x + N * y

The equation for the current x in A is...

v = R * x + d/dt(L * x + M * y)

The equation for the current y in B is...

n = S * y + d/dt(M * x + N * y)

where v and n are the electromotive forces (voltages), x and y are the currents, and R and S are the resistances in A and B respectively.

The implication of the equation is that the voltage on one inductor consists of three components.

1. An I *R resistive part.
2. An inductance * di/dt part from itself
3. A mutual-inductance * di/dt from the current in the other inductor.

So basically the point of mutual inductance is that if I change the current in one inductor then it causes a voltage in the other one, which is proportional to M and the rate of change in current. This is of course the basis for a transformer.

Also, note that inductance is typically proportional to the square of the number of turns, which is why M is proportional to the square root of the inductance of each winding.

Answer 3

It may be true that M is not important in transformers or coupled inductors.

If someone can come up with an EE example where M is clearly a benefit over more pragmatic variables, it would be good.

1. Here is an example where M could be useful.

Consider an example of two multi-turn air-core coils is placed at a distance in a wireless charging application. If you need to design the wireless charging coils to have desired performance/characteristics you need to calculate M and it becomes useful. (Image sources: 1 and 2)

Let's say the gap between two coils varies between 4cm - 6cm, the maximum diameter of the coils should be less than 10 cm. The design goal is to optimize the coil parameters such as diameter, the number of turns, turn pitch, etc to achieve the best power transfer efficiency at a given output power. (Formula image source.)

Now the output power and efficiency are:

You need to first calculate mutual inductance using Elliptic integrals e.g. follows: (Formula image source.)

With this calculation of M (and of course known ways of calculating coil resistances), you can implement a complete optimization problem as a function of coil geometric parameters and distances.

In this example, you cannot use the coupling coefficient directly as the value of coupling depend on the distance and coil parameters in a non-linear manner.

2. Meaning of M

Although we write as \$M=k \sqrt{L_1 L_2}\$, in fact, the fundamental physical quantity is \$M\$ because: (Image source)

\$k\$ is a derived coefficient to get an understanding of coupling percentage. It would be more appropriate to express \$k=M/ \sqrt{L_1 L_2}\$, at least in the above example.

Answer 4

What precisely is the numerical (or any) meaning of mutual inductance

So, a 100% coupled 9 μH and 4 μH inductor have a mutual inductance of 6 μH. But what does the 6 μH represent?

So, does M (mutual inductance) have a numerical (or any other) useful meaning?

As we know, for a two-inductor system, the total voltage in each coil is:

\$v_1(t) = L_1 i_1'(t) \pm M i_2'(t) \tag 1\$

\$v_2(t) = L_2 i_2'(t) \pm M i_1'(t) \tag 2\$

Let's assume one coil is kept open-circuited while we apply a time-varying current to the other coil. Then, separately, the previous equations simplify to:

\$v_1(t) = \pm M i_2'(t) \tag 3\$

\$v_2(t) = \pm M i_1'(t) \tag 4\$

The plus-minus signs represent the fact that the voltage induced by mutual induction (i.e. on one coil due to the other coil) depends on the relative spatial orientation of the two windings, i.e. it depends on the position of the dots. Of course it also depends on the reference direction for both currents and the reference polarity for both voltages.

Now suppose the rate of change of the current in one coil is 1 amp per second. Then, ignoring units the previous equations separately simplify to:

\$v_1(t) = \pm M \tag 5\$

\$v_2(t) = \pm M \tag 6\$

From the two above equations (5) and (6), the take-away is: the mutual inductance between two inductors tells you the volts induced in one open-circuited coil when we (linearly, i.e. at a constant rate) change the current in the other coil by one ampere in one second. Of course, if such change in current is not of one amp, which is in general the case, then we must use eqs. (3) and (4).

So for the two-inductor system of your question, using eqs. (5) and (6), the \$M\$ = 6 μH means that 6 volts are induced in one (open-circuited) coil, when we linearly change the current in the other coil by 1 ampere in 1 second (even if the coupling coefficient is not 100% as your example). More generally (eqs. (3) and (4)), the \$M\$ = 6 μH means that, when we linearly change the current in one coil by \$\Delta I\$ amperes in 1 second, \$6 \cdot \Delta I\$ volts are induced in the other (open-circuited) coil.

For example, consider the following circuit, in which \$L_1\$ = 7 H, \$L_2\$ = 4 H and \$M\$ = 3 H. (I know those numbers are big, but it doesn't matter in this discussion.) Notice that from \$t_1\$ = 1 s to \$t_2\$ = 2 s (so the change in time is \$\Delta t\$ = 1 s), the current in \$L_1\$ changes from \$I_1\$ = 0.5 A to \$I_2\$ = 1 A, thus the change in current is \$\Delta I\$ = +0.5 A, therefore the voltage induced in \$L_2\$ is \$M \cdot \Delta I = 3 \cdot 0.5 = 1.5\$ volts.

Figure 1. Image source: own.

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[...] and, is it useful

It allows us to use circuit equations (equations involving currents and voltages only, which are scalar functions of scalar variables) instead of directly using Maxwell's equations (equations involving electric and magnetic fields, which are vector functions of vector and scalar variables, which are harder to solve).

Note that many circuit analysis textbooks (Sadiku, Dorf, Hayt, Irwin, Nilsson, Thomas, Van Valkenburg...) start by defining the mutual inductance, then introduce the coupling coefficient by defining it as a ratio of two quantities measured in units of henrys (mutual inductance and the geometric mean of the self-inductances, where the three inductances can be derived from Maxwell's equations). It's kind of similar to how power factor is defined as a ratio of two quantities measured in watts (active power and apparent power, both of which are independently defined). (Yes, I know apparent power is measured in volt-amperes, but that's exactly equivalent to watts; we use VA for apparent to not confuse it with active power.)

As you've shown in part 3 of your answer, we can formulate the circuit equations in terms of the mutual inductance and the self-inductances, or, in terms of the coupling coefficient (which depends on the mutual inductance and self-inductances) and the turns ratio (which depends on the self-inductances). Either way is just as useful in the sense that they allow us to still work with circuit variables (voltages and currents) only.

Perhaps one way (using \$k\$ and \$N\$) is more practical than the other (using \$M\$ and \$L_1\$ and \$L_2\$, as you showed) because it yields shorter equations, but that doesn't make the latter less useful. And for that matter, we could substitute \$k = M/\sqrt{L_1 L_2}\$ and \$N = \sqrt{L_2/L_1}\$, to see that our equations still depend on \$M\$, \$L_1\$ and \$L_2\$.

Here's another way I look at this. If we change the current in one coil, we find out a voltage is induced in the other open-circuited coil, in other words the induced voltage is directly proportional to the rate of change of current: \$v_2(t) \propto i_1'(t)\$. The mutual inductance simply allows us to write this proportionality as an equation: \$v_2(t) = M i_1'(t)\$. I would say this is analogous to how Georg Ohm found out that, in some devices under some circumstances, the voltage across the device is directly proportional to the current through the device (\$v(t) \propto i(t)\$); then he introduced (static/DC) resistance as the proportionality constant, to allow us to write the proportionality as an equation: \$v(t) = R i(t)\$. So the mutual inductance is what allows us to relate changing currents to induced voltages, that's why it's useful.

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representative (of something)

Qualitatively (in words only, without numbers), the mutual inductance between two coils is the ability of one inductor to induce a voltage across the other inductor, and vice versa.

For a quantitative explanation, I'll repeat it: the mutual inductance between two inductors tells you the volts induced in one open-circuited coil when we linearly change the current in the other coil by one ampere in one second.

The above interpretations of mutual inductance are analogous to self-inductance. Qualitatively, the self-inductance of a coil is the ability of a time-varying current in the coil to induce a voltage across the coil. Quantitatively, the self-inductance of a coil tells you the volts induced in the coil when we linearly change the current in the coil by one ampere in one second.

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or important?

\$M\$ (and \$L_1\$ and \$L_2\$) is just as important as \$k\$ (and \$N\$) for setting up the circuit equations. You can use one or the other.

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Does it represent something that is important to us?

What is important to you? Do you consider resistance important? Because we can write circuit equations in terms of conductance \$G\$ instead of resistance \$R\$, just like we can write the circuit equations in terms of \$k\$ instead of \$M\$.

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Is it only of (some) use when calculating the net inductance of two coupled inductors?

As far as I know, the only purpose of calculating the net inductance is to simplify inductors connected in series or in parallel. But we can use \$M\$ even when the inductors aren't neither in series nor in parallel, as shown in the following circuit.

Figure 2. Image source: own.

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Part 2 - you don't need to use M (coupling less than 100%)

You have said that using \$k\$ is simpler than \$M\$. Do you really think doing the procedure you showed in part 2 of your answer is simpler than just using \$M\$? In your procedure (shown as an image), you had to 1) manipulate the self-inductances and mutual inductance; 2) convert the inductances of the new coupled inductors into turns; 3) combine the turns; 4) convert the equivalent turns to an equivalent inductance. Don't you think it's easier to use \$M\$ instead, since you don't need to do any conversion nor manipulation?

Furthermore, even if you considered your method of part 2 to be simpler, I think you can't apply it for inductors that aren't in aiding or opposing series or parallel connections, such as the those in figure 2 of this answer.

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Edit

Here's another way some people (like me) look at this \$M\$-vs-\$k\$ dispute.

As we know, (static/DC) resistance is defined as the ratio \$R = V_{ab}/I\$. We can use electromagnetic theory to find an expression for the resistance in terms of the material of the conductor (using the resistivity \$\rho\$ or conductivity \$\sigma\$) and the geometry of the conductor (using the length \$L\$, the surface area \$S\$, etc.) For example, if we want to find the resistance of a cylindrical conductor, and if we assume the conductor has uniform electric field \$\mathbf E\$ and uniform current density \$\mathbf J\$, then we can find the resistance as follows. The voltage is:

\$\begin{align} V_{ab} &= \displaystyle\int_a^b {\mathbf E} \cdot {\mathrm d} {\mathrm l} \\ &= \displaystyle\int_a^b E {\mathrm d} l \cos{(\theta)} && \text{expand dot product} \\ &= \displaystyle\int_a^b E {\mathrm d} l && \text{electric field is parallel to length vector} \\ &= E \displaystyle\int_a^b {\mathrm d} l && \text{electric field is independent of length} \\ &= E L && \text{integrate the differential length} \end{align}\$

the current is:

\$\begin{align} I &= \int_S {\mathbf J} \cdot {\mathrm d} {\mathbf S} \\ &= \int_S J {\mathrm d} S \cos{(\theta)} && \text{expand dot product} \\ &= \int_S J {\mathrm d} S && \text{current density is parallel to surface vector} \\ &= J \int_S {\mathrm d} S && \text{current density is independent of area} \\ &= J S && \text{integrate the differential surface} \end{align}\$

and substituting the two previous results for the current and voltage into \$R = V_{ab}/I\$, we get:

\$\begin{align} R &= \dfrac{EL}{JS} \\ &= \dfrac{EL}{(\sigma E) S} && \text{substitute $J = \sigma E$} \\ &= \dfrac{L}{\sigma S} && \text{simplify} \\ &= \dfrac{\rho L}{S} && \text{substitute $\sigma = 1/\rho$, if desired} \end{align}\$

which is a well-known formula. (End of example.)

For capacitance, we define the capacitance between two separated conductors/plates as ratio of the electric charge stored on either conductor, to the voltage or potential difference across the two conductors (\$C = Q/V_{ab}\$); then we can use Gauss' law and the voltage equation \$V_{ab} = \int_a^b {\mathbf E} \cdot {\mathrm d} {\mathrm l}\$ to express the capacitance in terms of the dielectric and its geometry only; in this way we get formulas such as \$C = \epsilon S/d\$ for parallel-plate capacitor.

For self-inductance, we define the self-inductance of a conductor as the ratio of the total magnetic flux linkages to the current which it they link (\$L = \lambda/I\$); then we can use Ampère's law and the magnetic flux equation \$\Phi = \int_S {\mathbf B} \cdot {\mathrm d} {\mathrm S}\$ to express the inductance in terms of the material and its geometry only; in this way we get formulas such as \$L = \mu_0 N^2 S/(2 \pi R)\$ for toroidal inductor.

In this way we can find the resistances, capacitances and inductances of conductors. That's how it's done for power transmission lines (e.g. read the textbooks by Glover & Sarma [chapter 4] or Grainger & Steveson [chapters 4 and 5] on power systems analysis).

Similar to the above properties (resistance, capacitance, inductance), the mutual inductance can be derived from equations of electromagnetic theory, so what's why we say it's a "fundamental property". On the other hand, we conceive the coupling coefficient \$k\$ simply as a dimensionless factor defined in terms of fundamental properties, specifically as the ratio of mutual inductance to the geometric mean of the self-inductances.

Even though calculating \$R\$, \$C\$, \$L\$ and \$M\$ involves electromagnetic theory/Maxwell's equations, those constants are considered as given/known in circuit analysis, so that we don't have to ever use Maxwell's equations again.

There's nothing wrong on only using \$k\$ (and \$N\$) in the circuit equations. If you find ways to always express the circuit equations in terms of \$k\$ without \$M\$, that's fine, whether the resulting expressions are shorter or longer. Simply put, I and other people think of \$k\$ as a derived quantity from \$M\$, so that saying \$M\$ is useless is like saying \$k\$ is also useless (which obviously is not true).

Answer 5

Part 1 - you don't need to use M (coupling = 100%)

This statement came from the first posted answer (now deleted): -

Try analysing any circuit with coupled inductors without using M

If two inductors are coupled 100% series aiding, you can convert the inductances into "turns" by assuming that they share the same core (100% coupled). Consider this from the question: -

As an example, consider two inductors (such as 9 μH and 4 μH) that are 100% coupled and produce magnetic fields that are aiding.

You can arbitrarily assume that the inductance factor (\$A_L\$) is 1 μH per turn.

Then, you have this: -

• 9 μH has 3 turns
• 4 μH has 2 turns

And, the total number of turns is 5 hence, the net inductance is 5² μH = 25 μH.

In other words, you do this: -

$$L_{NET} = (\sqrt{L_1}+\sqrt{L_2})^2$$

Mister mutual inductance was not needed nor harmed!

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Part 2 - you don't need to use M (coupling less than 100%)

Another comment suggested that 100% coupling was a special case. Here's the same example with 60% coupling. I've done it pictorially because I think it's more meaningful to see the steps: -

If I were to calculate net inductance using the formula in the question I'd get the same answer.

To answer my own question; so far, M appears to be fairly meaningless, not particularly useful and not representative of anything significantly important. Coupling factor, k, on the other hand, appears to be much more fundamental and useful in the understanding coupled inductors.

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Part 3 - you don't need to use M (isolated windings with any degree of coupling)

In @user4574's answer this was said: -

So basically the point of mutual inductance is that if I change the current in one inductor then it causes a voltage in the other one, which is proportional to M and the rate of change in current.

The inference I take here is that M is useful, but I have my doubts when one considers the importance of k, the coupling factor...

In the answer supplied by @ThePhoton this formula was given "for a circuit containing a transformer (or coupled inductors)": -

$$V_2 = L_2\frac{dI_2}{dt}+M\frac{dI_1}{dt}$$

This was also mentioned in a comment by @P2000: -

@ThePhoton points out, if you attempt a similar example as the one above but where the current or voltage are NOT shared (i.e. only flux is coupled) you'll end up calculating an M to solve equations.

But this just doesn't appear to be true.

Consider the \$V_2\$ formula directly above. It contains two terms; the first term is when there is an output current on the "secondary" winding and there is a resultant volt-drop but, the main term to concentrate on is that one that includes M. So, in the case when there is no secondary current the \$V_2\$ formula becomes this: -

$$V_2 = M\frac{dI_1}{dt}$$

Knowing that \$L_1\frac{dI_1}{dt}\$ can be equated to the "primary" voltage \$V_1\$, we can manipulate the simplified \$V_2\$ equation to this: -

$$V_2 = M\cdot \frac{L_1}{L_1}\frac{dI_1}{dt}\hspace{1cm} =\hspace{1cm} M\frac{1}{L_1}\cdot V_1$$

Then, because \$M = k\sqrt{L_1 L_2}\$, the above formula can be rewritten: -

$$V_2 =k\sqrt{L_1 L_2}\frac{1}{L_1}\cdot V_1\hspace{1cm} =\hspace{1cm}k\cdot V_1\cdot\sqrt{\frac{L_2}{L_1}}$$

And, the square root of the inductance ratio is well-known to be the turns ratio, N, hence: -

$$V_2 =k\cdot V_1\cdot N$$

Can anyone explain why this derived formula isn't simpler, more useful and more intuitive compared to the formula that uses M (as proposed by @ThePhoton and @user4574)?

Anyone who knows stuff about transformers will recognize it as the basic primary-secondary voltage relationship.

Answer 6

Inductance tells you the voltage a current change induces in a given inductor. Mutual inductance tells you the voltage a current change in one inductor induces in the other inductor.

Answer 7

Mutual inductance can be computed (if not always analytically, numerically) from the sole geometry of the coils (which need not be solenoids, can have arbitrary shapes and orientation) using Neumann's formula:

Can you do the same with k from first principles? Can you compute k, given the geometry of the coils, without invoking the quantities used to define M (magnetic flux or vector potential)?

Answer 8

Andy's observation has good merit. It is clear to me that,

"\$M\$ is a single letter representation for \$k\sqrt{L_{1}L_{2}}\$ and really nothing more."

So why does \$M\$ for mutual inductance exist. I speculate that back in the day a place holder for an unknown inductive quantity was required until a solution was found.

Is \$M\$ useful or important? Well, I suppose that if using \$M\$ adds clarity or reduces clutter in how a problem is solved then the answer is yes. If not then No. I guess I am sidestepping the question. But in reality, \$M\$ is not necessary as Andy has clearly demonstrated in his answer.

Does \$M\$ or \$k\sqrt{L_{1}L_{2}}\$ have any sort of meaning?

I start with the meaning of inductance as derived from Faraday and Lenz. These folks gave us:

$$v_{\phi}=-\frac{d\phi}{dt}$$

If the magnetic flux \$\phi\$ is created by current in a wire (single loop), then the equation may be modified into:

$$v_{\phi}=-\frac{d\phi}{di}\frac{di}{dt}$$

For several loops (turns)\$N\$: $$v_{\phi}=-N\frac{d\phi}{di}\frac{di}{dt}$$ \$-N\frac{d\phi}{di}\$ is the ability for current to create the flux within its region of influence. It is this ability that is called inductance \$L\$. Really it is self-inductance. $$L=-N\frac{d\phi}{di}$$

If there is a second loop of wire nearby sitting in the changing flux of the first, then clearly by Faraday/Lenz, a voltage will be induced across the second loop of wire.

This is not new, EE folks should be familiar with this. So then consider \$M\$. The two loops of wire can be easily envisioned with a proportion (\$k\$) of the flux from the first loop flowing through the second loop. The inductance equation can be written: $$L_{12}=-N_{1}\frac{d\phi_{2}}{di_{1}}=-kN_{1}\frac{d\phi_{1}}{di_{1}}=kL_{1}$$

$$L_{21}=-N_{2}\frac{d\phi_{1}}{di_{2}}=-kN_{2}\frac{d\phi_{2}}{di_{2}}=kL_{2}$$

These coupling inductances represent the ability of a current in one loop to establish a magnetic flux in another loop. This is where the mutual inductance is defined as the geometric mean of the coupling inductance:

$$M=\sqrt{L_{12}L_{21}}=k\sqrt{L_{1}L_{2}}$$

So here is the meaning:

Mutual inductance is the geometric mean of the coupling inductances.

This averaging tends to hide the meaning of the coupling inductances.

So in the end I support Andy's answer to his own question.