Mutual inductance and transformers

m — mutual inductance

m names two inductors, not two nodes:

m1,l1,l2,0.01

Those are the names of two l elements already in the circuit, and the last field is M in henries. It may be a symbol, and it may carry an SI prefix. A book that gives the coupling as a coefficient rather than an inductance is written as it states it:

m1,l1,l2,k=0.5

and Symbulator works out M = k√(L₁L₂) itself when it reads the description. k may be a symbol too, in which case M stays symbolic.

The two coils may instead be written as impedances in AC, r elements with positive imaginary values, coupled by a positive imaginary M in ohms, m1,r1,r2,2j; k= works there as well, giving jk√(|Z₁||Z₂|).

The line is checked before anything runs. Both names must be elements of the circuit and of one kind, two inductors in henries or two impedances in ohms; one of each is refused, where it used to be accepted and answered wrongly. A numeric value on either coil or on the coupling must be real and positive in henries, or positive imaginary in ohms, with no resistive or capacitive part; a symbol passes. With everything numeric the coupling may not exceed √(L₁L₂), which is a coefficient of 1, and a k= must lie between 0 and 1. And a pair written in ohms is an AC description: in DC, TR and FD it is refused, and the coils are to be written in henries.

An m has no answers of its own. It is a statement about two other elements; its effect turns up in their currents.

The dots are the node order

Symbulator cannot draw a dot, so the first node you name in each coupled inductor is its dotted end. That is the whole convention, and it is the one most likely to hand you a correct-looking wrong answer.

Two 30 mH coils coupled by 10 mH, the first across a 10 V source and the second across a 30 Ω load, at 1000 rad/s. We name the coils l1 and l2 and the load r2, and we write each coil's dotted end first, node 1 for the primary and node 2 for the secondary:

e,1,0,10
l1,1,0,0.03
l2,2,0,0.03
m1,l1,l2,0.01
r2,2,0,30

Run in AC at ω = 1000, the current in the load is

Now write the second coil the other way round, l2,0,2,0.03, the same coil with its dot at the other end, and run again:

Exactly the negative. Same magnitude, opposite sign, no error message. Nothing will warn you.

So: read the schematic, decide which end of each coil carries the dot, and name that node first in both coupled elements.

t — the ideal transformer

t1,n1,n2,turns1,turns2

The two-node form names the top terminal of each side and grounds the other two. A 120 V source on the primary, a 5 Ω load on the secondary, ten turns to one; we name the transformer t1 and the load rl, and call the two tops node 1 and node 2:

e,1,0,120
t1,1,2,10,1
rl,2,0,5

Run in DC, the load's voltage and current are

Ten to one, so a tenth of the 120 V across the load.

An ideal transformer has no leakage, no magnetising current and no losses. For a real one, model it as coupled inductors with m instead — that is the same choice you make on paper.

Four terminals

When neither side sits on ground, name all four with bracketed pairs:

t1,[tl,bl],[tr,br],[10,1]

[tl,bl] is the left port, top terminal first; [tr,br] the right. The turns must then be bracketed too. The two-node form is exactly this with both bottoms on 0.

A transformer reports a current at each live terminal rather than one current, since the two sides carry different ones. The names are the element's name followed by the node's: for t1 with ports at nodes 1 and 2, they are it11 and it12.

The same bracketed-pair form works on two-port blocks; see Two-ports.