As described before the inductor is a special case of a more general set of circumstances. For the simple inductor the current producing the magnetic effect (‘field’) flows in the same conductor on which the magnetic field reacts.
This is self-inductance, and the volt-ampere relation is V
1
= L
11
dI
1
/dT; the subscripts are added for the sake of the discussion to indicate it describes the effect of I
1
on itself. However the magnetic field produced by one current also can act on an entirely different conductor. This is described as mutual inductance and the volt-ampere relation is written as V
2
= L
21
dI
1
/dt. Here L
21
describes the effect of I
1
on the other inductor. The selfand mutual inductance coefficients both are functions of geometry and material properties
Actually the situation can be somewhat more involved. Thus the inductively coupled coils both could be carrying current, so that each current provides a self-inductive and a mutually inductive component:
V
1
= L
11 dI1/dt + L
12 dI
2
V
2
= L
21 dI1/dt + L
22 dI
2
/dt
/dt
Thus V
1
is produced partially by I
1
reacting on itself, and partially by
I
2
reacting on I
1
. Similarly V
2
is produced partially by I
1
reacting on
I
2
, and partially by I
2
reacting on itself.
Alas there is still more complication. For self-inductance the volt-ampere relation is described with specific voltage and current polarities (to obtain a positive inductance coefficient). A similar specification must be given for the mutual inductance. Thus, in the figure above, if I
1
is positive it must be made clear whether V
2 is positive or negative. There is no implicit means of determining this from the circuit diagram; the circuit diagram shows topology but the proper polarity for the mutual induction is a function of the geometry (i.e.
orientation) of the inductors. Accordingly a convention is needed to convey information determined from the actual circuit. A commonly used convention that we describe is the 'dot' convention.
First select one coil, say coil #1, and place a dot (or a triangle or a square, or whatever) on one end or the other to distinguish one end. Then from an actual measurement or equivalent (by someone, sometime) determine the polarity of the voltage drop induced in coil #2 by a current in coil #1 flowing into the dot.
Indicate this polarity by placing a dot on the + end of the other coil. This convention then means a current flowing into the dot on one coil (either one) produces a voltage drop from the dotted to the undotted end of the other coil. (Or, if you prefer to be contrary, a current flowing into the undotted end of one coil produces a voltage drop from the undotted to the dotted end of the other coil.)
For linear circuits it can be shown that L
12
= L
21
, and commonly the letter M (for 'mutual') is used for the common value. Also L
11
and L
22
often are simplified to L
1
and L
2
.
Here is a simple illustration. Two coupled inductors are connected is series as shown on the left. The question considered is what is the value of a single uncoupled inductor that produces the same terminal behavior, i.e., what is the equivalent inductance to the two coupled inductors connected in series.? If the coils were not coupled that equivalent inductor would be L
1
+L
2
. Not so here.
Suppose a current I flows into L1 and so through both inductors (source not shown); calculate the terminal voltage drop and from this the equivalent inductance. The voltage drop across L1 is produced by two terms; there is the self-induced voltage L
1 dI/dt, and there is the contribution of the current flowing through L
2 because the current flows into the dotted end of L
2
; this is –MdI/dt. The minus sign applies
, and so produces a voltage drop from the dotted to the undotted end. Similarly there are two terms for the voltage drop across L2. The self-induced voltage is
L
2 dI/dt, and the coupled voltage is –MdI/dt. Hence terminal voltage = (L
1 equivalent inductance is L
1
+L
2
-2M.
+L
2
-2M) dI/dt, and the
Circuits Mutual Inductance 1 M H Miller

What happens if one of the dots is shifted to the other end of its coil? The equivalent inductance becomes
L
1
+L
2
-2M ( in effect replace –M by M).
Some example problems follow.
13.1 Calculate vo.
13.4 Keep in mind that this illustration involves a straightforward process of writing loop equations, using the appropriate constant of proportionality for the voltampere relation of each element. Be careful with the mutual inductance. Two things to note about the mutual inductance term: a) the current in each coil produces a voltage drop across the other; don't account for the current in L1 (for example) producing a voltage drop across L3, and then forget that the current in L3 produces a voltage drop across L1, and b) remember the polarity convention; a current into (out of) a 'dot' on one coil produces a voltage drop from (to) the 'dotted' to (from) the 'undotted' end of the other coil
Circuits Mutual Inductance 2 M H Miller

13.5
This is similar to 13.4. The 'tricky' part is to account for all the inductive couplings. For example L3 is coupled to L1, and so there is a contribution from I3 to the I1 loop equation. The difficulty is not really the mathematics but rather keeping track of all the terms.
13.17 Calculate I1 and I2. (Note that j1 and j2 are reactances and not inductances.
Circuits Mutual Inductance 3 M H Miller