Transcript ECE 3144 Lecture 4

ECE 3144 Lecture 26
Dr. Rose Q. Hu
Electrical and Computer Engineering Department
Mississippi State University
1
Chapter 5: Capacitance and Inductance
• Two more passive circuit elements are introduced in this chapter:
capacitors and inductors
• Unlike resistors, capacitors and inductors are able to absorb energy
from the circuit, store it temporarily, and later return it.
• Unlike resistors, their terminal characteristics are described by linear
differential equations.
• Capacitors are capable of storing energy when a voltage is present
across the element.
• Inductors are capable of storing energy when a current is passing
through them.
• In this chapter, we are particularly interested in the following
objectives:
–
–
–
–
Finding the voltage-current relationship of ideal capacitors
Find the voltage-current relationship of ideal inductors
Calculating the energy stored in inductors and capacitors
Methods for reducing series/parallel combinations of inductors and
capacitors
– Predicting the behavior of op-amp circuits with capacitors
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Ideal capacitors
•
•
•
•
A capacitor is a circuit that consists of two conducting surface separated by a nonconducing, or dielectric, material.
In the above figure, the positive charge is transferred to one plate and negative charges
to the other. The charge on the capacitor is proportional to the voltage across it => q =
Cv, where C is the proportionality factor known as the capacitance of the element.
Capacitance unit
– We now define the farad (F) as one coulomb per volt, and use this as our unit of
capacitance.
Due to the presence of dielectric, there is no conductance current flowing internally
between the two plates of the capacitors. However, the equal currents are entering ands
leaving the two terminals of the capacitor. The positive current entering one plate
represents positive charge accumulated on the plate; the positive current leaving one
plate represents negative charge accumulated on the plate. The current and the
increasing charge on one plate are related as the familiar equation:
i (t ) 
dq (t )
dt
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Ideal capacitors
• To complete KCL analysis in this case, we introduce one more concept:
displacement current.
– Via electromagnetic field theory, it can be shown that the conductance current
is equal to the displacement current that flows internally between the two
plates of the capacitors and is present any time that an electric field or voltage
varies with time.
• KCL is therefore satisfied if we include both conductance and
displacement currents.
• Like a resistor, a capacitor constructed of two parallel pales of area A,
separated by a distance d, has a capacitance C=A/d, where  is the
permitivity, a constant of the insulating material between the plates.
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Integral voltage-current relationship for a capacitor
We know that q(t)=Cv(t) and i(t) = dq(t)/dt =>
i (t )  C
dv (t )
dt
(1)
Equation (1) can be rewritten as
dv(t ) 
1
i (t )dt
C
(2)
Integrate (2) between times t0 and t
1 t
v(t ) 
i ( x) dx  v (t 0 )

t
0
C
Where v(t0) is the voltage due to the charge that accumulates on the capacitor from
time – to t0. Or we can express in another way
1
v(t ) 
C

t

i ( x)dx
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Energy Storage for a capacitor
The power delivered to a capacitor is
dv(t )
dt
The energy stored in the electric field is therefore
p (t )  v(t )i (t )  Cv(t )
t
wC (t )   Cv( x)


v (t )
dv( x)
dx  C  v( x)dv( x)
v (  )
dx
1
C{v 2 (t )  v 2 ()} ,
2
v(-) = 0
1
Cv 2 (t ) J
2
The energy stored can also be written as by using q=Cv

1 q 2 (t )
wC (t ) 
2 C
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Ideal inductors
•
•
•
•
An inductor is a circuit element that consists of a conducing wire usually in the form of
coil.
We know that a current-carrying conductor produces a magnetic field, which is linearly
related to the current that produces it.
A changing magnetic field can induce a voltage in the neighboring circuit, which is
proportional to the time rate change of the current producing the magnetic field.
The above statements indicate:
di (t )
dt
L, the constant of proportionality, is what we call the inductance. It is measured in the
unit of henry. 1 henry (H) is equal to 1 volt-second per ampere.
Like a capacitor, an inductor has an inductance of N2A/s, where A is the cross-sectional
area, s is the axial length of the helix, N is the number of complete turns of wire, and 
is the a constant of the material inside the helix, call permeability.
If the current is constant, then we have v=0. Thus we may view an inductor as a “short
circuit to dc”.
Another fact is that a sudden or discontinuous change in the current must be associated
with an infinite voltage across the inductor.
v (t )  L
•
•
•
•
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Integral voltage-current relationships for an inductor
We know that
v (t )  L
di (t )
dt
v (t )  L
=>
di (t )
dt
Following development of the mathematical equations for the capacitor, we find
1
i (t ) 
L

t

v ( x)dx
Which lead to another equation
1 t
i (t )   v( x)dx  i (t0 )
L t0
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Energy storage for an inductor
The power delivered to an inductor is given the current-voltage product as
p  vi  Li (t )
di (t )
dt
The energy wL accepted by the inductor is stored in the magnetic field
around the coil, and expressed as
t
t
1 2
wL (t )   pdt  L  i (t )di(t )  Li (t ) J


2
The energy accepted by the inductor between t0 and t is
wL (t )  wL (t 0 ) 
1 2
1
Li (t )  Li 2 (t 0 )
2
2
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Examples
• Provided during the class
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Homework for lecture 26
• Problems 5.4, 5.7, 5.17,5.18, 5.20, 5.23,
5.31
• Due March 25
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