The Basic Inductor
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When a length of wire is formed onto a coil, it becomes a basic inductor.
Magnetic lines of force around each loop in the winding of the coil effectively add to the lines of
force around the adjoining loops, forming a strong electromagnetic field within and around the coil.
The unit of inductance is the henry (H), defined as the inductance when one ampere per second
through the coil, induces one volt across the coil.
Inductance
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Inductance is a measure of a coil’s ability to establish an induced voltage as a result of a change in
its current, and that induced voltage is in a direction to oppose that change in current.
Induced voltage is directly proportional to L and di/dt
 di 
vL = L 
 dt 
Physical Characteristics
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Inductance is directly proportional to the permeability of the core material.
Inductance is directly proportional to the cross-sectional area of the core.
Inductance is directly proportional to the square of the number of turns of wire.
Inductance is inversely proportional to the length of the core material.
L=
N 2 µ r µ0 A
l
Example: determine the inductance of the coil in the following figure. The permeability of the core is
0.25 x10 −3 H / m .
Ans: 40 mH
Winding Resistance and Capacitance
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When many turns of wire a re used to construct a coil, the total resistance may be significant.
The inherent resistance is called the dc resistance or the winding resistance (RW).
When two conductors are place side by side, there is always some capacitance between them.
When many turns of wire are placed close together in a coil, there is a winding capacitance (CW).
CW becomes significant at high frequencies.
Faraday’s and Lenz’s Laws
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Recall Faraday’s law:
– The amount of voltage induced in a coil is directly proportional to the rate of change of the
magnetic field with respect to the coil.
Recall Lenz’s law:
– When the current through a coil changes, an induced voltage is created as a result of the
changing electromagnetic field, and the direction of the induced voltage is such that it always
opposes the change in current.
Inductor Types
Inductors are typically classified according to the type of core material used. The most common types are air
core, iron core, and ferrite core. Inductors may be either fixed, or variable.
Inductors in Series
• When inductors are connected in series, the total inductance increases.
LT = L1 + L2 + L3 + … + Ln
Example: determine the total inductance for each of the following circuits:
Ans: 9.5H and 18mH
Inductors in Parallel
• When inductors are connected in parallel, the total inductance is less than the smallest inductance.
1/LT = 1/L1 + 1/L2 + 1/L3 + … + 1/Ln
Example: find the total inductance in the following circuit:
Ans: 1.25mH
Inductors in DC Circuits
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When there is constant current in an inductor, there is no induced voltage.
There is a voltage drop in the circuit due to the winding resistance of the coil.
Inductance itself appears as a short to dc.
RL Time Constant
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Because the inductor’s basic action opposes a change in its current, it follows that current cannot
change instantaneously in an inductor.
τ = L/R
where: τ is in seconds
L is in henries (H)
R is in ohms
Exponential Formulas
The exponential formulas used to find the voltage and current in an RL circuit are identical to
those used in RC circuits:
v(t ) = VF + (VI − VF )e
i (t ) = I F + ( I I − I F )e
−t
τ
−t
τ
Energizing Current in an Inductor
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In a series RL circuit, the current will increase to approximately 63% of its full value in one timeconstant interval after the switch is closed.
The current reaches its final value in approximately 5τ.
Example: calculate the time constant in the following circuit. Also find the current in the circuit 1.2ms after
the switch is closed.
Ans: τ = 500µs; i = 182mA
De-energizing Current in an Inductor
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In a series RL circuit, the current will decrease to approximately 63% of its fully charged value one
time-constant interval after the switch is closed.
The current reaches 1% of its initial value in approximately 5τ. Considered to be equal to 0.
example: in the following figure, sw1 is opened at the instant that sw2 is closed. Assuming that steady-state
was reached prior to the switch change, calculate:
(i)
the time constant
(ii)
the initial coil current at the instant of switching
(iii) the current in the coil after 53us
Ans: τ = 20µs; i = 500mA; i = 35.3mA
Induced Voltage in a Series RL Circuit
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At the instant of switch closure, the inductor effectively acts as an open with all the applied voltage
across it.
During the first 5 time constants, the current is building up exponentially, and the induced coil
voltage is decreasing.
The resistor voltage increases with current.
After 5 time constants, all of the applied voltage is dropped across the resistor and none across the
coil.
Example: for the following figures, determine:
(i) vL at the instant the switch(es) operate as shown*, and
(ii) vL after 5τ
*assume that in figure (b) steady state has been reached before sw1 is opened
Ans: (a) 25V,0V; (b) 208V, 0V
Example: assuming the inductor in part (b) of the previous example has a value of 100mH, find:
(i)
the voltage across the inductor and the current in the circuit after 18ms assuming sw2 is open and
sw1 is closed at t=0.
(ii)
Ans: 2.88V; 1.843A
The voltage across the inductor and the current in the circuit 2.8 ms after the switches operate as
shown (assume the circuit parameters calculated in part (i) i.e. the switches are operated after
18ms has passed).
Ans: 11.2V; 112mA
Capacitor – Inductor Comparison
Capacitors and Inductors have many similarities, and many of the principles discussed apply to both. It is
therefore often useful to compare the characteristics of both in order to better understand their behaviour.
CAPACITORS
C=
INDUCTORS
Aε r ε 0
d
Energy stored in electric field: W =
N 2 µ r µ0 A
L=
l
1
CV 2
2
Energy stored in magnetic field: W =
1 2
LI
2
A capacitor will oppose a change in voltage – you
CANNOT instantaneously change the voltage on a
capacitor
You CAN instantaneously change the current in a
capacitive circuit
An inductor will oppose a change in current – you
CANNOT instantaneously change the current
flowing through a capacitor
You CAN instantaneously change the voltage in an
inductive circuit
 dv 
iC = C  
 dt 
 di 
vL = L 
 dt 
At t=0:
A neutral capacitor appears as a short to Vs
At t=0:
A neutral inductor appears as an open to Vs
At t = ∞ :
A capacitor appears as an open circuit to Vs
At t = ∞ :
An inductor appears as a short circuit to Vs