## Inductance

When
the current changes in a circuit containing inductance, the magnetic linkage
changes and induces a voltage in the inductance:

dy/dt
= e = L di/dt

Note that the induced voltage has a polarity which opposes the rate of
change of current.

Alternatively, by integration with
respect to time:

Y =
òedt
= LI

**
The inductance L of a circuit is
equal to the induced voltage divided by the rate of change of current:
L = e / di/dt = dy/dt
/ di/dt = dy/di
**

**
Alternatively, the inductance L of a
circuit is equal to the magnetic linkage divided by the current:
L =
Y / I
**

Note that the magnetic linkage
Y
is equal to the product of the number of turns N and the magnetic flux
F:

Y = NF
= LI

**
**

####
Mutual
Inductance

The
mutual inductance M of two coupled inductances L_{1} and L_{2}
is equal to the mutually induced voltage in one inductance divided by the
rate of change of current in the other inductance:

M = E_{2m} / (di_{1}/dt)

M = E_{1m} / (di_{2}/dt)

If the self
induced voltages of the inductances L_{1} and L_{2} are
respectively E_{1s} and E_{2s} for the same rates of change
of the current that produced the mutually induced voltages E_{1m}
and E_{2m}, then:

M = (E_{2m} / E_{1s})L_{1}

M = (E_{1m} / E_{2s})L_{2}

Combining these two equations:

M = (E_{1m}E_{2m} / E_{1s}E_{2s})^{½}
(L_{1}L_{2})^{½} = k_{M}(L_{1}L_{2})^{½}

where k_{M} is the mutual coupling coefficient of the two
inductances L_{1} and L_{2}.

**
If the
coupling between the two inductances L _{1} and L_{2} is
perfect, then the mutual inductance M is:**

M = (L_{1}L_{2})^{½}

**
**

####
Inductances
in Series

When
uncoupled inductances L_{1}, L_{2}, L_{3}, ... are
connected in series, the total inductance L_{S} is:

L_{S} = L_{1} + L_{2} + L_{3} + ...

When two
coupled inductances L_{1} and L_{2} with mutual inductance M
are connected in series, the total inductance L_{S} is:

L_{S} = L_{1} + L_{2} ± 2M

The plus or minus sign indicates that the coupling is either additive or
subtractive, depending on the connection polarity.

**
**

####
Inductances
in Parallel

When
uncoupled inductances L_{1}, L_{2}, L_{3}, ... are
connected in parallel, the total inductance L_{P} is:

1 / L_{P} = 1 / L_{1} + 1 / L_{2} + 1 / L_{3}
+ ...