Transcript CSCI 2980: Introduction to Circuits, CAD, and Instrumentation

EENG 2610: Circuit Analysis
Class 10: Capacitors and Inductors
Oluwayomi Adamo
Department of Electrical Engineering
College of Engineering, University of North Texas
Capacitors and Inductors

Both are linear elements


Their terminal characteristics are described by linear differential
equations.
Both are storage elements

Able to absorb energy, store energy temporarily, and later supply
energy to circuit.

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Capacitors are capable of storing energy when a voltage is present
across the element. Energy is stored in electric field.
Inductors are capable of storing energy when a current is passing
through them. Energy is stored in magnetic field.
Important application: op-amp integrator


It produces an output voltage that is proportional to the integral of
the input voltage.
It can be used to simulate complex systems.
Capacitors



A capacitor is a circuit element that consists of two conducting
surfaces separated by non-conducting, or dielectric, material.
Capacitors are categorized by the type of dielectric material
used between the conducting plates. Each type is more
suitable for particular applications.
Unit of capacitance is coulombs per volt, or Farad (F).

Typical values range from thousands of micro-farads to a few
pico-farads.
Capacitance of two parallel plates of area A
Separated by distance d:
 A
C 0 r
d
 0 : electrostatic permitivity of free space,
 r : dielectric constant or relative permitivity
of the isulator in between.
The charge on the capacitor is proportional
to the voltage across it: q  Cv
C : capacitance of thecapacitorin farads
dq (t ) d
dv (t )
 {Cv(t )}  C
dt
dt
dt
1 t
1 t
v(t )   i ( x)dx  v(t0 )   i ( x)dx
C 
C t0
i (t ) 
Power: p(t )  v(t )i (t )  Cv (t ) dv (t )
dt
Energy:
wC (t )  
t

p( x)dx
1 2
1 q 2 (t )
 Cv (t ) 
2
2 C
i (t )  C
i (t )  C


dv (t )
dt
dv (t )
dt
Capacitors only store and release electrostatic
energy; they don’t create energy.
The capacitor is passive element and follows
passive sign convention.
i (t )  C
dv (t )
dt
Some Characteristics of Capacitor

Capacitor blocks DC current or capacitor is an open
circuit to DC current.
dv



iC
dt
Capacitor is often used to filter out unwanted DC
voltage/current.
When analyzing a circuit containing only DC
voltage/current source, we can replace capacitors with an
open circuit before analysis.
Capacitor has continuity of voltage

Voltage across capacitor is always continuous.
p(t )  v(t )i (t )  Cv (t )
dv (t )
dt
dv (t )

dt
v(t) is continuous!
Example 6.1: If the charge accumulated on two parallel conductors charged to
12V is 600 pC, what is the capacitance of the parallel conductors?
Example 6.2: The voltage across a 5 μF capacitor has the waveform shown in (a).
Determine the current waveform.
Example 6.3: Determine the energy stored in the electric
field of the capacitor in Example 6.2 at t = 6 ms.
Example 6.4: The current in an initially uncharged 4 μF capacitor is shown below. Derive the
waveforms for the voltage, power, and energy and compute the energy stored at t = 2 ms.
Inductors
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An inductor is a circuit element that consists of a
conducting wire usually in the form of a coil.
Inductors are typically categorized by the type of
core on which they are wound. Each type is more
suitable for particular applications.
The unit of inductance is volt-second per ampere, or
Henry (H).
di (t )
v(t )  L
dt
1 t
v( x)dx



L
1 t
 i (t0 )   v( x)dx
L t0
i (t ) 
 di(t ) 
p(t )  v(t )i(t )   L
i(t )

 dt 
t
1 2
Energy: wL (t ) 
p
(
x
)
dx

Li (t )
 
2
Power:

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Inductors only store and release electromagnetic
energy; they don’t create energy.
The inductor is passive element and follows
passive sign convention.
v (t )   L
di (t )
dt
Some Characteristics of Inductors

Inductor is a short circuit to DC current.


v(t )  L
di (t )
dt
In analyzing a circuit containing only DC voltage/current
source, we can replace inductors with a short circuit before
analysis.
Inductor has continuity of current

Current flowing through an inductor is always continuous.
 di(t ) 
p(t )  v(t )i(t )   L
i(t )

 dt 
di (t )

dt
i(t) is continuous!
Example 6.5: Find the total energy stored in the circuit.
V1
V2
V3
Example 6.7: The current in a 2 mH inductor is i(t )  2 sin 377t A
Determine the voltage across the inductor and the energy stored in the inductor.
Example 6.8: The voltage across a 200 mH inductor is given by the following
expression. Derive the waveforms for the current, energy, and power.
(1  3t )e 3t
v (t )  
0
t0
t0
Capacitor and Inductor Specifications

Capacitors: capacitance, working voltage, and
tolerance

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The working voltage is specified to keep the applied
voltage below the breakdown point of the dielectric.
Inductors: inductance, resistance, tolerance, current
rating

The major difference between wire-wound resistors and
inductors is the wire material. Low resistance materials are
used in inductors.