is a brief primer on mutual inductance between inductors (aka coils).
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 to radio is that existing
between elements in a directional antenna like a Yagi or log periodic
July 1934 Radio News & Short-Wave
of Contents]These articles are scanned and OCRed from old editions of the Radio & Television News magazine.
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I have already posted. All copyrights (if any) are hereby
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vintage Radio News
Radio Physics Course - Mutual Inductance
Alfred A. Ghirardi
The electromagnetic induction
due to two independent electric circuits reacting upon each other, is
called mutual-induction (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 mutual-induction. 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
mutual-induction, 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.
Figure 1. Inductors may be connected and
placed so their magnetic fields either buck each other or aid each other.
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
mutual-induction 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 right-hand rule to find the directions of the fields
in each case, remembering that the right-hand rule refers to the direction
of the current flow - which is opposite to the direction of the electron
Self-induction can be easily understood by comparing it
with the case of mutual-induction 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 two-coil 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
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.
case of series-aiding coils, the total inductance is made up of the
self-inductances 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
Therefore L = L1
+ L1 + 2M.
Since L1 = L2
and M = L1 if we substitute
these values for L in the above formula, we have L= L1+L1+2L1
from which L = 4L1
where L is the total inductance. If some of the lines of force
from one coil do not link with the other-as 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 series-aiding,
L = L1+
L2 + 2M
are connected in series-opposing, the total inductance is:
= L1 + L2
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 √L1 L2
all units being in henries, microhenries or millihenries.
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 iron-core transformers
where there is very little magnetic leakage. In air-core 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 iron-core 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 √L1,
L2. 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 L1
in the above formula for K.
*Radio Technical Pub. Co. Publishers,
Radio Physics Course.
September 18, 2013