IEEE Solid-States Circuits Magazine - Fall 2020 - 88

µ >µ0

N1 Turns S1

N2 Turns

V1

S2

+

V2

-

I1

<

C2

r
ds
r
dl

r
ds
r
dl

r
B1
-

C1

r
B2

µ >µ0

S1

+

-

+

V1
I1

I2

r
ds

r
ds
r
dl

r
B1

r

N2 Turns S2

N1 Turns

(a)

r
dl

r1
-
(b)

r
B2

r2

+

V2
I2

FIGURE 2: The coupled inductors: (a) coils with the same section and (b) coils with different sections.

	

true, and we have some flux leak-
age. As sketched in Figure 2(b) in a
somewhat simplified and idealized
fashion, only some of the flux den-
sity Bv 2 generated by the coil on the
right-hand side links to the surface
S 1 of the coil on the left-hand side.
As a matter of fact, only the smaller
coil surface (between S 1 and S 2)
crossing the flux of Bv 1 and Bv 2 is
relevant for the mutual inductance.
Applying (4) and (5) to Figure 2(b), we
have L 1 = nN 21 rr 21 /,, L 2 = nN 22 rr 22 /,,
and M = nN 1 N 2 rr 21 /,, which implies
M 2 # L 1 L 2 . The latter result can be
proven true in general [15], [16], such
that we can define the magnetic cou-
pling, k , as

V2 = M dI 1 + L 2 dI 2 .(8)
dt
dt

A schematic symbol is introduced
for the coupled inductors, as depicted
in Figure 3(a). The dots marked on
the symbol have the following mean-
ing. When the current flows into
the dotted terminal of one coil, the
induced voltage has positive polarity
at the dotted terminal of the other
coil. This corresponds to a positive
mutual inductance (M 2 0 ). However,
the polarity of one electrical port can
be arbitrarily swapped with respect
to the dot convention, as illustrated by
(V2l, I 2l ) in Figure 3(a). In this case, a
180° phase shift is introduced in the
circuit, which is taken into account by
the mutual inductance being negative
(M 1 0 ). In summary, while the selfinductances L 1 and L 2 are always
positive, the mutual inductance can
be both positive and negative.
So far, we assumed that all of the
flux density generated by one coil
completely links to the other coupled
coil. However, this is, in general, not

Dot Convention
l1
M
+
L1
L2
V1
-

	

-
V2′
+
I2′

M (9)
L1 L2

and have | k | # 1.
We described the coupled induc-
tors as a two-port network. We can
also describe the device using an
equivalent circuit. There are various
possible equivalent circuits [16], [17];
the one displayed in Figure 3(b) is a

Swapped Port
Polarity
l2
+
V2
-

k=

l1
+
V1
-
Magnetizing
Inductance

(a)

Leakage
Inductance
(1-k 2)L2
I1′ 1:nk
l2
+
+
L1
V2
V2′
-
-
Ideal
Transformer
(b)

FIGURE 3: The (a) schematic symbol of coupled inductors and (b) equivalent circuit.

88	

FA L L 2 0 2 0	

IEEE SOLID-STATE CIRCUITS MAGAZINE	

particularly insightful one. It is made
of three components. The first is an
inductor shunting the primary coil.
It is called magnetizing inductance
and models the need to establish
magnetic flux in the device for it to
operate. The second component is an
inductor in series with the secondary
winding, called leakage inductance.
This inductance models the limited
coupling between the coils, which
results in flux leakage. The third
component is an ideal transformer.
An ideal transformer is a passive
lossless two-port network whose
behavior is described by the ratio of
the voltages at its ports, in this case
equal to V2l /V1 = n·k. The parameter
n is defined as n = L 2 /L 1 . From
(4), we know that, for a solenoid, the
inductance is proportional to the
square of the number of turns of the
coil. As a consequence, for two cou-
pled inductors as in Figure 2, we have
n ? N 2 /N 1 . Hence, n is called the
turn ratio. Because of the passivity
and the lack of losses, the ideal trans-
former enjoys an intrinsic imped-
ance transformation feature. With
reference to Figure 3(b), we have
V2l /I 2 = (- V1 /I 1l ) n 2 k 2; that is, if the
primary (secondary) port is termi-
nated onto impedance Z, the imped-
ance seen looking into the secondary
(primary) port is Z scaled up (down)
by the square of the ideal transformer
voltage gain. The coupled inductors
make a magnetic transformer, which
approximates an ideal transformer if
| k | " 1 and L 1 " 3. Hence, materials
with high permeability and windings
IEEE Solid-States Circuits Magazine - Fall 2020

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