Mutual inductance and ideal transformers

Two coils that share a magnetic field are not two separate elements, and a circuit description has to say so. This lesson covers the two ways of saying it: m for a mutual inductance between a pair of elements, and t for an ideal transformer.

10.1How to describe mutual inductance

Textbooks give a coupled pair one of two ways, and Symbulator takes both.

• Two inductors in henries, coupled by a mutual inductance also in henries.
• Two impedances in jΩ, coupled by a mutual impedance also in jΩ. Written this way the coils are r elements with imaginary values.

Either way the coupling itself is an m element with four fields: a name starting with m, the names of the two coupled elements, and the value of the coupling. For example: m1,l1,l2,1.5

That is a mutual inductance of 1.5 H between inductors l1 and l2. In jΩ it looks the same, naming two r elements instead: m1,r1,r2,2j

The dots are the node order

Every coupled-inductor problem in a textbook has a dot on each coil, marking the ends that share polarity. Symbulator has no way to draw a dot, so the first node you name in each coupled element is its dotted end.

That is the whole convention, and it is the thing most likely to give you a correct-looking wrong answer. If you list the nodes of one coil the other way round, the sign of the coupling flips and every current comes back wrong.

The value of a mutual inductance can use SI prefixes.

What answers do you get? None. An m element is a statement about two other elements, not a component with its own voltage and current — its effect shows up in the currents of the elements it couples.

10.2Instructive m problems

AS7's Example 13.1

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Calculate the phasor currents I₁ and I₂ in the circuit.

Solution

Three things to notice in the description. The bottom node of both halves is ground. The coils are given in jΩ, so they are r elements with imaginary values, and the mutual is imaginary too, to match. And the dotted node of each coupled element is named first.

Circuit Description

e,1,0,12
r1,1,2,-4j
r2,2,0,5j
m,r2,r3,3j
r3,3,0,6j
r4,3,0,12

AC, and leave RMS phasors unticked. Then read the two currents with Mini-Tools set to aa:

aa(ir2) reads 13.02∠-49.4° A and aa(ir4) reads 2.910∠14.04° A.

Both are correct.

AS7's Practice Problem 13.1

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Determine the voltage V_o in the circuit.

Solution

Again the dotted node of the coupled inductor comes first.

Circuit Description

e,1,0,(200∠45°)
r1,1,2,4
r2,2,0,8j
m,r2,r3,1j
r3,0,o,5j
r4,o,0,10

aa(vo) reads 20.00∠-134.43° V.

Correct.

AS7's Example 13.2

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Calculate the mesh currents in the circuit.

Solution

Notice the node order again, matching the dots.

Circuit Description

e,1,0,100
r1,1,2,4-3j
r2,2,0,6j
m,r2,r3,2j
r3,3,2,8j
r4,3,0,5

aa(ir1) reads 20.30∠3.5° A and aa(ir4) reads 8.693∠19.03° A.

Correct.

10.3How to describe ideal transformers

An ideal transformer has four terminals — two a side. Symbulator simulates the case where one terminal on each side is grounded, which covers the textbook problems; you name only the two live nodes.

A t element takes five fields: a name starting with t, the live node on the first side, the live node on the second side, and the turns on each side. For example: t1,1,2,80,800

That is 80 turns on the side connected to node 1 and 800 on the side connected to node 2, wound the same way. Only the ratio matters, so 1,10 would describe the same transformer.

Polarity is a minus sign. If the dots are on opposite ends, make one of the two turn counts negative: t,a,b,1,-5

It does not matter which of the two you make negative, as long as one of them is.

What answers do you get? The voltages at the two live nodes, as for any node in the circuit, and the current entering the transformer, named with the transformer and the node — for a transformer t on node 2, that is it2.

10.4Instructive t problems

AS7's Figure 13.33

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Obtain V_Th and Z_Eq for the part of the circuit to the right of nodes a and b.

Solution

Everything left of a and b is ignored. Note that this one is entirely symbolic — the turns ratio, the impedance and the source are all names.

Circuit Description

t,2,3,1,n
r2,3,4,z2
e2,4,0,vs2

Find equivalent, Thévenin / Norton, nodes 2 and 0, in DC.

vth is \(v_{s2}/n\) and req is \(z_2/n^2\).

Correct — the transformer refers the source by the turns ratio and the impedance by its square, which is exactly what an ideal transformer does.

AS7's Example 13.8

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For the ideal transformer circuit, find the source current I₁, the output voltage V_o, and the complex power supplied by the source.

Solution

One of the turn counts is negative here, because the dots are of opposite polarity.

Circuit Description

e,1,0,120
r1,1,2,4-6j
t,2,3,1,-2
ro,3,0,20

AC, with RMS phasors ticked this time — the question asks for complex power, and that is the convention it wants.

Three readings from Mini-Tools with aa: ir1 is 11.09∠33.69° A, vro is 110.9∠-146.31° V, and -se — the complex power supplied, so the opposite of the power consumed by the source — is 1331∠-33.69° VA.

Three conversions in a row is the point at which the setting is less work than the tool: tick Show AC answers as polar phasors in Settings and the first two are already in that form when the circuit solves. The third still wants aa, because -se is an expression rather than an answer.

All three are correct.