Energy Storage Elements
ELCITWO
Contents
Introduction
Capacitors
Energy Storage in a Capacitor
Series and Parallel Capacitors
Inductors
Energy Storage in an Inductor
Series and Parallel Inductors
Plotting Capacitor and Inductor Voltage and
Current (in MatLab)
Introduction
This lecture aims to discuss the nature of
capacitors and inductors. The analysis of their
characteristics involves either integration or
differentiation.
Electric circuits that contain capacitors and/or
inductors are represented by differential
equations. Circuits that do not contain them
are represented by algebraic equations.
Introduction
Circuits that contains capacitors and/or
inductors are dynamic circuits, while circuits
that do not contain them are static circuits.
Circuits that contain capacitors
inductors are able to store energy.
and/or
Circuits that contain
inductors have memory.
and/or
capacitors
Introduction
In a dc circuit, capacitors act like open circuits
and inductors act like short circuits.
A set of series or parallel capacitor/inductor
can be reduced to an equivalent one.
Integrating and differentiating circuits can be
accomplished by these components working with
an operational amplifier.
Capacitors
A capacitor is a two-terminal element that is a
model of a device consisting of two conducting
plates separated by a non-conducting material.
Electric energy is stored on the plates.
http://sdsu-physics.org/physics180/physics196/Topics/capacitance.html
Example
1. Calculate the capacitance of the 2mm2 plates
separated by 0.2mm.
2. For a charge of 30C, what would be the
potential difference for the capacitor in
problem 1?
3. What is the electric field that exist between
the plates?
Capacitance is a measure of the ability of a
device to store energy in the form of
separated charge or an electric field.
It has the unit C/V and in called Farad (in
honor of Michael Faraday)
Current flows while the
charge flow from one
plate to the other.
i = dq / dt
If differentiated,
i = C (dv/dt)
Capacitors use various dielectrics and are built in several forms.
Some common capacitors use impregnated paper for a
dielectric, whereas others use mica sheets, ceramics, and
organic and metal films.
The voltage across a capacitor cannot change
instantaneously.
Examples
The plots represent the current and voltage of
the capacitor in the circuit. Determine the
values of the constants a and b, used to label
the plot of the capacitor current.
The current i(t) = 3.75e-1.2t A for t>0 and the
output voltage of the capacitor is v(t) = 4
1.25e-1.2t V for t>0. Find the value of the
capacitance C.
Determine the current i(t) for t> 0 for the
given circuit when vs(t) is given in figure (a).
Energy Storage in a Capacitor
A current flows and a charge is stored on the plates of the
capacitor. Eventually, the voltage across the capacitor is a
constant, and the current through the capacitor is zero.
The capacitor has stored energy by virtue of the
separation of charges between the capacitor plates.
These charges have an electrical force acting on them.
The forces acting on the charges stored in a
capacitor are said to result from an electric
field. An electric field is defined as the force
acting on a unit positive charge in a specified
region.
The energy required originally to separate the
charges is now stored by the capacitor in the
electric field. The energy stored in the
capacitor is wc(t) = (1/2) Cv2(t) Joules
Example
A 10 mF is charged to 100V. Find the energy
stored by the capacitor and the voltage of the
capacitor at t = 0+ after the switched is
opened.
Series and Parallel Capacitors
Parallel: Use KCL and analyze the current in each
branch.
Cp = C1 + C2 + C3 + + CN
Series: Use KVL and analyze the voltage drops in
each capacitor.
1/Cs = 1/C1 + 1/C2 + 1/C3 + + 1/CN
Inductors
A wire maybe shaped as a multi-turn coil.
If we use a current source, we find that the voltage
across the coil is proportional to the rate of change of
the current. This proportionality may be expressed by
the equation v = L (di/dt).
Inductor a two-terminal element consisting of a winding of N-turns for
introducing inductance into an electric circuit.
Inductance the property of an electric device by which a time-varying
current through the device produces a voltage across it. It is a measure of
the ability of a device to store energy in the form of a magnetic field.
Single-layer coils wound in a helix are often
called solenoids. If the length of the coil is
greater than half of the diameter and the core
is of non-ferromagnetic material, the
inductance of the coil is given by
L = [(uoN2A) / (l + 0.45d)] Henry
Current equation for inductors.
1
i
L
vdt i (t
t0
Example
The input to the circuit shown is v(t)=4e-20t V
for t>0. The output current is i(t) = -1.2e-20t
1.5 A for t>0. The initial inductor current is
iL(0) = -3.5A. Find the value of L and R.
Energy Storage in an Inductor
The power in an inductor is p = vi = [L(di/dt)]i.
The energy stored in the inductor is stored in
its magnetic field. The energy stored in the
inductor during the interval t0 to t is given by
W = Li2
Example
Find the power and the energy stored in a
0.1H inductor when i=20te-2tA and v = 2e-2t (12t) V for t>0 and i = 0 for t < 0.
Series and Parallel Inductors
Series: Use KVL and analyze the voltage in each
inductor.
Ls = L1 + L2 + L3 + + LN
Parallel: Use KCL and analyze the current in each
branch.
1/LP = 1/L1 + 1/L2 + 1/L3 + + 1/LN
Reference
Introduction to Electric Circuits 7th Edition by
Dorf and Svoboda (2006)
