July 1934 Radio News & ShortWave 
[Table
of Contents]
Wax nostalgic about and learn from the history of early
electronics. See articles from
Radio & Television News, published 19191959. All copyrights hereby
acknowledged.

Here is a brief primer on
mutual inductance between inductors (aka coils), compliments of the July 1934 issue
of Radio News and the Shortwave magazine. It is lesson #31 in a series. Mutual inductance is your circuit's
friend if you want it to occur, as with a transformer, or it can be your circuit's
mortal enemy if you don't want it to occur, as when two inductors "talk" to each
other unintentionally because of proximity and relative orientation. One form of
mutual inductance not mentioned here but of utmost importance (and necessity) to radio is that existing
between elements in a directional antenna like a Yagi or log periodic configuration.
Radio Physics Course  Mutual Inductance
Figure 1  Inductors may be connected and placed so their
magnetic fields either buck each other or aid each other.
Alfred A. Ghirardi
Lesson 31
The electromagnetic induction due to two independent electric circuits reacting
upon each other, is called mutualinduction (see Figure 1). The previous examples
of the induction of voltage in the secondary winding of a transformer due to the
current flowing through the primary is an excellent illustration of mutualinduction.
Parallel conductors carrying independent alternating currents react upon each other
by reason of the mutual inductive influence between them. Mutual induction between
wires in radio transmitters, and in radio receivers, is often the cause of howling,
hum, etc., and certain steps may be taken to prevent this.
It is not necessary to again go into a detailed study of the actions taking place
during mutualinduction, as this has already been covered during our study of the
transformer. It should be remembered that induced voltage is produced in the secondary
circuit whenever current in the primary starts to flow, ceases to flow, changes
its rate of flow, or changes its direction of flow. The intensity of the induced
voltage depends upon, and is proportional to, the rate at which current changes
take place in the primary. The higher the frequency, the more rapid is the change
of current, and so the greater will be the induced voltage. The greater the amplitude,
or rise and fall, of current in the primary with a given frequency, the greater
is its rate of change, and the higher will be its induced voltage. The primary and
secondary circuits may be simply straight wires near each other, solenoid coils,
etc.
The total inductance depends upon the connections and the spacing and placing
of the coils.
From the point of view of the electron theory, the effects of mutualinduction
may be explained simply. Electrons are flowing around the primary winding when current
is sent through. While this stream of electrons is increasing, it causes electrons
in the secondary to flow around in the direction opposite to those in the primary.
The secondary electron streams by their movement, produce magnetic forces which
exert a backward push on those in the primary, and try to stop their flow. If the
primary circuit is opened, the stream of electrons in the primary comes to rest,
and those in the secondary reverse their direction of flow and tend to make the
electrons in the secondary keep on moving. Whatever change takes place in the stream
of electrons in the primary, the electrons in the secondary oppose the change by
means of the magnetic forces set up by their motion. The student should check up
these forces by applying the righthand rule to find the directions of the fields
in each case, remembering that the righthand rule refers to the direction of the
current flow  which is opposite to the direction of the electron flow.
Selfinduction can be easily understood by comparing it with the case of mutualinduction
explained above. If a coil is connected to a source of alternating current a stream
of electrons flows along from one turn to the next. The action between any two turns
is the same as if they were two separate coils. As the stream of electrons flow
through say the top turn of the coil, they set up a magnetic force which tends to
push all the electrons along in the other portion of the coil, that is, tend to
increase the current.
Two coils may be placed with reference to each other so that a part of the electromagnetic
field of one coil passes or cuts through the conductors forming the other coil.
Then there is a mutual inductive effect between the coils and they are said to be
coupled. The closer together the coils are, the greater are the number of lines
of force due to the primary current that link with the turns of the secondary, and
the closer or tighter the coupling is said to be. Also the better the permeability
of the magnetic circuit, the better is the coupling.
The induced voltage across the secondary of such a twocoil arrangement depends
upon the sizes of both coils, their relative positions and distance apart, the permeability
of the magnetic circuit, and the rate of change of the primary current. All of these
physical factors, except the rate of change of the primary current, are collectively
called the mutual inductance (M) of the circuit. The larger the coils are, the closer
they are to each other, and the more nearly their axes coincide, the greater is
their mutual inductance M. Since the mutual inductance possible between two coils
is affected by so many variable things, and since the design of radio apparatus
is almost entirely tied up with mutual inductances and variations thereof, it is
important that we study this subject in detail.
In many applications, inductors are connected in series, and are also placed
near each other so that magnetic coupling exists between them. The inductance of
a coil depends, among other factors, upon the square of the number of turns of wire
of which it is composed. Doubling the number of turns makes the inductance 4 times
as large, etc. Suppose we have two coils, built exactly alike, as shown in (A) of
Figure 1, and having the same inductance. If they are connected together in series
but kept apart to prevent magnetic interaction, the total inductance will simply
be equal to the sum of the two. However, if they are connected in series and brought
close together, we can have many conditions. If they are placed so the direction
of current flow and hence the lines of force of one are exactly opposite in direction
to the lines of force of the other as shown at (A) of. Figure 1, the total inductance
will be zero. This is called the "series opposing" position. If they are connected
together in series, with the currents flowing in the same direction and are brought
up to each other so that every line of force of the primary links with every turn
of wire of the secondary, and every line of force of the secondary links with every
turn of the primary, and the fields of each are in the same direction, the result
is the same as though we had a single coil made up of the two coils together, that
is, a single coil having twice as many turns as each of these coils. This condition
is shown at (B) of Figure 1. Since the inductance is proportional to the square
of the number of turns, it is evident that this combined inductance is equal to
2X2 or 4 times that of either coil alone. Therefore the combined inductance of two
similar coils connected and placed so as to be "series aiding" is four times that
the self inductance of either single coil.
In the case of seriesaiding coils, the total inductance is made up of the selfinductances
of coil 1 and coil 2, the mutual inductance due to the lines of force from coil
1 linking with coil 2, and the mutual inductance associated with the lines from
coil 2 which link with coil 1. These two latter mutual inductances (M) are equal,
since the coils are the same.
Therefore L = L_{1 }+ L_{1}+ 2M.
Since L_{1} = L_{2} and M = L_{1} if we substitute these
values for L in the above formula, we have L= L_{1}+L_{1}+2L_{1}
from which L = 4L_{1}
where L is the total inductance. If some of the lines of force from one coil
do not link with the otheras is the case especially if air forms the core  the
total inductance will be less than four times the inductance of one coil in this
case. In the series opposing case it will be less than zero. In any general case
the total inductance of two coils of any inductance value, connected so as to be
seriesaiding, will be:
L = L_{1}+ L_{2} + 2M
If they are connected in seriesopposing, the total inductance is:
L = L_{1} + L_{2}  2M
In order to know then just what the total inductance will be, the degree of coupling
must be known. The term "coefficient of coupling" enables us to predict just what
the total circuit inductance will be if the amount of coupling is known. Of course
the coefficient of coupling depends upon the total inductance in the primary and
secondary circuits as well as upon the mutual inductance between the inductances.
The coefficient of coupling is really a measure of the ease with which energy may
be transferred from one circuit to the other. The coefficient may be found from
K = M √L_{1} L_{2} all units being in henries, microhenries
or millihenries.
The maximum possible value of K is of course 1.0. This is called unity coupling.
The value of 1.0 is only approached in well designed ironcore transformers where
there is very little magnetic leakage. In aircore transformers the coupling may
be very "weak" since a large portion of the lines of force of the primary may never
reach the secondary. A low value of coupling for this type of coil would be about
0.1, and a high value 0.7. In a well designed ironcore transformer, coupling as
high as 98 or 99 % (K = 0.98) may be obtained, depending upon the design and the
amount of magnetic leakage present.
The mutual inductance depends only upon the two coils, and the coupling between
them or M = K √L_{1}, L_{2}. The coefficient of coupling K,
between any two circuits depends upon the total inductance in each circuit. Thus
if one of the two circuits had two inductors in series, the total combined value
of the two series inductances in this circuit would be substituted for L_{1}
in the above formula for K.
*Radio Technical Pub. Co. Publishers, Radio Physics Course.
Posted June 23, 2022 (updated from original post on
5/18/2013)
