Inductors
Energy Storage Devices
Objective of Lecture
• Describe
• The construction of an inductor
• How energy is stored in an inductor
• The electrical properties of an inductor
• Relationship between voltage, current, and inductance; power; and energy
• Equivalent inductance when a set of inductors are in series and in parallel
Inductors
• Generally - coil of conducting wire
• Usually wrapped around a solid core. If no core is used, then the inductor is
said to have an ‘air core’.
http://bzupages.com/f231/energy-stored-inductor-uzma-noreen-group6-part2-1464/
Symbols
http://www.allaboutcircuits.com/vol_1/chpt_15/1.html
Alternative Names for Inductors
Reactor- inductor in a power grid
Choke - designed to block a particular frequency while allowing
currents at lower frequencies or d.c. currents through
Commonly used in RF (radio frequency) circuitry
Coil - often coated with varnish and/or wrapped with insulating tape
to provide additional insulation and secure them in place
A winding is a coil with taps (terminals).
Solenoid – a three dimensional coil.
Also used to denote an electromagnet where the magnetic field is generated
by current flowing through a toroidal inductor.
Energy Storage
The flow of current through an inductor creates a magnetic field (right
hand rule).
B field
If the current flowing through the inductor drops, the magnetic field will
also decrease and energy is released through the generation of a current.
http://en.wikibooks.org/wiki/Circuit_Theory/Mutual_Inductance
Sign Convention
• The sign convention used with an
inductor is the same as for a power
dissipating device.
• When current flows into the positive side of
the voltage across the inductor, it is positive
and the inductor is dissipating power.
• When the inductor releases energy back into
the circuit, the sign of the current will be
negative.
Current and Voltage Relationships
• L , inductance, has the units of Henries (H)
1 H = 1 V-s/A
di
vL L
dt
t1
1
iL vL dt
L to
Power and Energy
t1
pL vL iL LiL iL dt
to
t1 t1
diL
w L iL dt L iL diL
to
dt to
Inductors
• Stores energy in an magnetic field created by the electric current
flowing through it.
• Inductor opposes change in current flowing through it.
• Current through an inductor is continuous; voltage can be discontinuous.
http://www.rfcafe.com/references/electrical/Electricity%20-
%20Basic%20Navy%20Training%20Courses/electricity%20-%20basic%20navy%20training%20courses%20-
Calculations of L
For a solenoid (toroidal inductor)
N mA N m r mo A
2 2
L
N is the number of turns of wire
A is the cross-sectional area of the toroid in m2.
mr is the relative permeability of the core material
mo is the vacuum permeability (4π × 10-7 H/m)
l is the length of the wire used to wrap the toroid in meters
Wire
Unfortunately, even bare wire
has inductance.
7
L ln 4 1 2 x10 H
d
d is the diameter of the wire in
meters.
Properties of an Inductor
• Acts like an short circuit at steady state when connected to a d.c.
voltage or current source.
• Current through an inductor must be continuous
• There are no abrupt changes to the current, but there can be abrupt changes in the
voltage across an inductor.
• An ideal inductor does not dissipate energy, it takes power from the
circuit when storing energy and returns it when discharging.
Properties of a Real Inductor
• Real inductors do dissipate energy due resistive losses in the length of
wire and capacitive coupling between turns of the wire.
Inductors in Series
Leq for Inductors in Series
vin v1 v2 v3 v4
di di
v1 L1 v2 L2
dt dt
i
di di
v3 L3 v4 L4
dt dt
di di di di
vin L1 L2 L3 L4
dt dt dt dt
di
vin Leq
dt
L eq L1 L2 L3 L4
Inductors in Parallel
Leq for Inductors in Parallel
iin i1 i2 i3 i4
t1 t1
1 1
i1
L1 vdt
to
i2
L2 vdt
to
t1 i t1
1 1
i3
L3 vdt
to
i4
L4 vdt
to
t1 t1 t1 t1
1 1 1 1
iin
L1 t vdt L2 t vdt L3 t vdt L4 vdt
to
o o o
t1
1
iin
Leq vdt
to
L eq 1 L1 1 L2 1 L3 1 L4
1
General Equations for Leq
Series Combination Parallel Combination
• If S inductors are in • If P inductors are in
series, then parallel, then:
1
S P 1
Leq Ls Leq
s 1 p 1 L p
Summary
• Inductors are energy storage devices.
• An ideal inductor act like a short circuit at steady state
when a DC voltage or current has been applied.
• The current through an inductor must be a
continuous function; the voltage across an inductor
can be discontinuous.
• The equation for equivalent inductance for
inductors in series inductors in parallel
1
S P
1
Leq Ls Leq
s 1 p 1 L p
