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 configuration.
July 1934 Radio News & Short-Wave
Wax nostalgic about and learn from the history of early electronics.
See articles from Radio &
Television News, published 1919 - 1959. All copyrights hereby acknowledged.
Radio Physics Course - Mutual Inductance
Alfred A. Ghirardi
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
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
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 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 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 same.
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:
L = L1 + L2
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.
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 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.
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