Chapter 06.ppt Capacitance and inductance

Chapter Six Capacitance and Inductance
Chair Professor Rui-Xiang Yin (South China University of Technology)
6.1 Capacitance
6.2 Inductance
6.3 Step Response in RC and LR Networks
6.4 Circuit Applications
Contents of this Chapter:

6.0 Introduction
Up to now we have dealt with resistive networks which responses to excitations
depend only on the instantaneous value of the excitation.
This chapter initiates the discussion of a much broader class of networks in which
responses to a time-varying source depend on the past behavior of the source as
well as on its instantaneous value. In circuit structure the networks have two
kinds of new elements, capacitance and inductance.
6.1 Capacitance电容
6.1.1 The Ideal Capacitance理想电容
The ideal capacitance is an element in which the terminal voltage is proportional to
charges stored in it rather than to current through it. It is an ideal model of the plate
capacitor.

According to the physics, there is an uniform electric field between two charged
plates. The electric filed intensity is proportional to the charges stored in the plates.
1
E k Q
 
where k1 is a proportional coefficient (constant depending on the structure of the plate
capacitor). As we know from the electric theory, the voltage between two points in an
uniform electric field is proportional to the electric field intensity and the distance
between the points.
2
v k d E
  
Therefore, the voltage is also proportional to the charges stored in the plates.
1
v Q
C
  The proportional coefficient C is called the capacitance of the plate
capacitor, which is a constant related to the structure.

Generally, if an element has a v-q characteristic, that is to say, the charges
stored in it is depends the voltage between its terminals, the element is called a
capacitance element. If the v-q characteristic is a direct line, the element is
called linear (ideal) capacitance, the slope C is its parameter, also named
capacitance. The unit of capacitance is farad (abbreviated F).
According to the definition of current in chapter 2, we have
Q dQ dv
i C
t dt dt

   


This is the v-i characteristic of an ideal capacitance. Since current flows only
when v is changing (so we call capacitance a dynamic element动态元件), we
conclude that the ideal capacitance is equivalent to an open circuit for DC
current 直流状态电容呈现开路(because the DC voltage does not change with
time), a result that nicely matches our intuitive view that the insulating gap
between the plates should not permit a steady flow of charge.
Based on the mathematics, the v-i characteristic of an ideal capacitance may be
converted to following form
Q dQ dv
i C
t dt dt

   

0
0 0
1 1 1 1
( ) ( ) ( ) ( ) (0) ( )
t t t
v t i d i d i d v i d
C C C C
       
 
    
   
It is shown that the voltage at time t depends on not only the instantaneous
charge current (from the initial time 0 to time t) but also the initial voltage.
Therefore we call the capacitance a memorial element记忆元件.

6.1.2 Linearity of a capacitance
The ideal capacitance is linear element, which satisfies the linearity.
Additivity
According to the v-i characteristic of ideal capacitance,
1 2
1 2
dv dv
i C i C
dt dt
   
Now the terminal voltage of the capacitance is a
combination of two voltages, then
1 2 1 2
1 2
( )
dv d v v dv dv
i C C C C i i
dt dt dt dt

         
Multiplicity
C
C
C
v v
dv
i C
dt
dv
C
dt

 
 
C
C
C
v A v
dv
i C
dt
dv
A C
dt
 
 
 

6.1.3 Energy Storage
Assume the initial voltage of a capacitance is zero at time t=0. At time t=T, the
capacitance is charged to voltage vC. How much electric energy is absorbed by
the capacitance?
According the definition of electric power, with relational reference the power
of an ideal capacitance at time t is
2
1
( ) ( ) ( ) ( ) ( )
2
C
C C C C C
dv d
p t v t i t v t C C v t
dt dt
 
     
 
 
Therefore, total electrical energy absorbed by the capacitance when it charges
from 0 to vC volts is
( )
2 2
0
0
1 1
( ) ( ) ( )
2 2
C
v t
t
C C C C C
d
W t p d C v dv Cv t
dt
 
 
   
 
 
 
This energy is stored between the capacitance plates in electric field form (electric
stored energy). When a load is connected to the capacitance, the capacitance will
transfer energy to the load. The total energy delivered to the load during discharging
放电 is the same as the total energy absorbed by the capacitance during the charging
充电. No energy is lost in the capacitance itself. We say, therefore, that an ideal
capacitance is a lossless energy storage element无损储能元件.

6.1.4 Capacitors
Actual capacitance elements are called capacitors. They are made in many
shapes and sizes, but they always consist of two conductors separated by some
kind of insulator. Among the Insulating materials used are mica云母, paper.
or simply an air gap. The capacitance values for actual capacitors are usually
much less than 1 farad. Physically convenient units of capacitance are the
microfarad (mF = l0-6 farad) and the picofarad (pF = 10-12 farad). The ideal
capacitance is a model of actual capacitance elements.
The insulators in actual capacitors are very good open circuits, but they are
not perfect. At sufficiently high voltages, the insulators can suffer electrical
breakdown电场击穿 and a sudden discharge can take place. A less dramatic
imperfection is sometimes important even at low voltages. Tiny amounts of
current called the leakage current漏电流can flow through the insulator or
through oily or moist films on the surface of the capacitor. A charged capacitor,
therefore, will discharge spontaneously. This discharge is usually very slow,
with discharge times measured in minutes or even days.

6.1.5 Parallel and Series Combinations
1. Parallel combination of capacitances
If N capacitance C1, C2, …, CN are
connected in parallel, the voltage is
the same for them. The total charges
restored is all parallel capacitances is
 
1 2
1 2
1 2
N
N
N
Q Q Q Q
C v C v C v
C C C v
C v
   
   
    
 
Parallel
Equivalent

2. Series combination of capacitances
N capacitance C1, C2, …, CN are connected in series. Since
the charging current is the same for all capacitance, the
charges stored in their plates are also the same. Therefore
the voltage crossing individual capacitance is
1 2
1 2
N
N
Q Q Q
v v v
C C C
  
Recall KVL we have the terminal voltage
1 2
1 2 1 2
1 1 1
( ) ( )
N
N N
Q Q Q Q
v v v v Q
C C C C C C C
            

6.2 Inductance电感
6.2.1 The Ideal Inductance理想电感
The ideal inductance is an element with a v-i characteristic complementary to
that of the ideal capacitance. Its circuit symbol is shown as
In an ideal inductance, the voltage is proportional to the rate of change of current
( )
( )
di t
v t L
dt
 
The constant of proportionality, L, is called the inductance.
Its unit is henry亨利 (abbreviated H).

Since the voltage exists nonzero only when i is changing (so we also call
inductance a dynamic element动态元件),
For DC currents, di(t)/dt=0 (current does not change with time), therefore,
the terminal voltage of inductance must be zero and ideal inductance
behaves as a short circuit.
Based on the mathematics, the v-i characteristic of an ideal inductance may
be converted to following form
0
0 0
1 1 1 1
( ) ( ) ( ) ( ) (0) ( )
t t t
i t v d v d v d i v d
L L L L
       
 
        
   
It is shown that the current through the inductance at time t depends on not only
the instantaneous voltage (from the initial time 0 to time t) but also the initial
current. Therefore we also call the inductance a memorial element记忆元件.

6.2.2 Linearity of the inductance
Similar to the ideal capacitance, the ideal inductance is also linear element.
1
1
di
v L
dt
 
2
2
di
v L
dt
 
Additivity
Multiplicity
1 2
1 2
( )
di d i i
v L L v v
dt dt

     
v
v
2 2
2
( )
d Ai di
v L A L Av
dt dt
 
    
 
 

6.2.3 Energy Storage
Capacitance stores energy in an electric field (electric stored energy). However
the inductance stores energy in a magnetic field (magnetic stored energy). If the
reference polarity is relative, at time t the power absorbed by the inductance is
Assume the initial current of an
inductance is zero at time t =0. At time
t =T, there is a current iL(T) flowing
through the inductance. How much
electric energy is absorbed by the
inductance?
( )
2 2 2
0 0 0
1 1 1
( ) ( ) ( ( )) ( ) ( )
2 2 2
L
i T
T T
L L L L L
d
W T p t dt L i t dt L d i L i T
dt
      
  
This energy only depends the final current and isn't consumed by the inductance.
The energy is stored in the inductance in magnetic field form. When a load is
connected to the inductance, the inductance will transfer energy to the load (like
the capacitance). So the inductance is also called a lossless energy storage element.

6.2.4 Series and Parallel Inductances
Series combination of inductances

Parallel combination of inductances

6.2.5 Inductors
Actual inductance elements are called inductors. They usually consist of a coil.
Often this coil is wound in a core of magnetic material磁性材料骨架 (iron or
ferrite) that enhances the strength of the magnetic field in the coil.
The behavior of actual inductors departs from the ideal more than the
corresponding behavior of actual capacitors. One reason is that all real coils
and core materials have some resistance. This resistance is an inherent part of
the actual element. (An important exception is a coil made out of super-
conducting wire, which behaves like a perfect inductance.)
An actual inductor is often modeled as an ideal inductance in series with a
small resistance (coil resistance), as shown follows

6.2.6 Mutual Inductance and Transforms 互感和变压器
Mutual inductance 互感
If there are two inductances (coils) L1 and L2 in a circuit, the current through L1
produces magnetic field not only in L1 but also partially in L2，and vice versa
We name the magnetic coupling between coils mutual inductance.

Transformer 变压器
A transformer consists of two or more coils, usually with different number of
turns, wound around a common core. There is no connection between the coils,
but for time-varying waveforms the mutual inductance between the coils
enables currents in one of the coils to produce voltages in the other coils.
Ideal transformer 理想变压器
The ideal transformer is a model of transformers. It has four terminals.
(1) Circuit symbol (2) Turns ratio 匝数比
The turns ratio between two windings is the parameter
of an ideal transformer, which is labeled in the circuit
symbol as
1 2
1: :
n N N

(3) Same polarity terminal-pair同名端
If the currents of two windings are entered in the doted
terminals, the magnetic fields produced by the two
currents are strengthened mutually. The doted
terminals are called same polarity terminal-pair.

(4) Characteristics of ideal transformer
a. Voltage transform电压变换
If v2>v1 (n>1), the transformer is said a step-up transformer升压变压器.
If v2<v1 (n<1) the transformer is said a step-down transformer降压变压器.
2 1
v n v
 
b. Current transform电流变换 2 1
1
i i
n
  
c. Power transform功率变换
Power of port 1: 1 1 1
P v i
 
Power of port 2: 2 2 2 1 1 1 1
1
( ) ( )
P v i nv i v i
n
      
Therefore, the ideal transformer is a lossless element. It only transfers power
from source to load. The winding connected to the source is called Primary
Winding (一次侧绕组), and the one connected to the load is called Secondary
Winding (二次侧绕组).
1 2 0
P P
 
Energy conservation

d. Resistance transform阻抗变换
2
1 2
2 2
1 2 2
1 1
T L
v
v v
n
R R
i n i n i n
 
    
 
  
 

Model of actual transformer 实际变压器模型
Actual transformers differ from the ideal transformer in several ways.
(1) Leakage inductance
There is some magnetic field that links only one of the coils, and that
contributes to a simple inductance of one winding or the other rather than to
the mutual inductance between the windings. This effect can be modeled as a
leakage inductance Le1 and Le2. Because there is leakage inductance, the
response of the transformer to rapidly varying waveforms will be reduced.
(2) Magnetizing inductance
The inductance of two windings is not infinite. So there is DC current in the
Primary Winding to set up the magnetic field. This effect is modeled as a
magnetizing inductance Lm, which is in parallel with the primary of the ideal
transformer, and thus shorts out any dc component of i1 . The effect of Lm,
therefore, is to prevent DC components of the current from being coupled
between primary and secondary.

(3) Coil resistances
R1 and R2 are the coil resistances of primary and secondary windings.
(4) Coupling capacitance
There is no electric connection between two windings. In other words, an
insulator separates them. This structure behaves a capacitance, which permits
the coupling of common-mode signals from the primary to the secondary.
Summarily, the model of actual transformer is shown as

6.3 Step Response in RC and LR Networks
RC、LR电路的阶跃响应
This section deals with the determination of the response of RC and LR networks
to step sources. The step function of unit height at t = 0 is written u-1(t), the
amplitude factor A indicating the height of the step in vs.

DC (direct current) stable state
6.3.1 Initial and Final States
If all voltages and currents in a network do not vary with time and are constant,
the network is said to be in direct current (DC) stable state 直流稳态.
In the DC stable state, the capacitance is equivalent to an open-circuit, and the
inductance is equivalent to a short-circuit.
The initial and Final responses of RC and RL networks
To determine a response to a step source vs it is useful to divide the total time
axis into three intervals
(1) Initial: an initial time interval before the step occurs during which us can
be considered a DC source (zero),
(2) Middle: a time interval just after the step occurs during which the
response to the sudden change in vs takes place,
(3) Final: a final time interval beginning long enough after the step so that
one can once again think of vs as a DC source.

The division between the initial and middle intervals obviously occurs at the
instant of the step, at t = 0 in above figure. The division between the middle
and final time intervals is less clear at the moment, and we shall delay being
more precise until Section 6.3.3. For now let us concentrate on the initial and
final intervals, the times during which the network has settled down to its
appropriate DC response.
Because the initial and final responses are DC stable states, they can be
obtained from the DC equivalent circuits substituting a short-circuit for
inductance and an open-circuit for capacitance.

6.3.2 Continuity Conditions连续性条件
Impulse function冲激函数
The impulse function is labeled by its integral over all time instead of by its
amplitude (which is infinite!). The label is called impulse strength冲激强度.
For voltage impulse function v(t)=F u0(t), the impulse strength F has the
dimension of magnetic flux磁通量 (unit is Weber韦伯, W).
For current impulse function i(t)=Q u0(t), the impulse strength Q has the
dimension of charge (unit is coulomb库仑, C).

Risetime 上升时间
In practice, a voltage or current cannot change its value from one to another
instantaneously. Define the time for a voltage or current to go from 10% to
90% of its final value as Risetime, labeled as tr .
The step function is an ideal model of real step source having a short enough
risetime. The "short enough" means that risetime of step source is much smaller
than the time constant of the circuit电路的时间常数.

Charge continuity 电荷连续性
If there is not an infinite current following to the capacitance, the
charge stored in the capacitance can not change suddenly.
In other words, as a time function, the charge stored in a capacitance should
be a continuous function.
Magnetic flux continuity 磁通连续性
If there is not an infinite voltage applied to the inductance, the
magnetic flux in the inductance can not change suddenly.
In other words, as a time function, the magnetic flux in a inductance should
be a continuous function.

Continuity conditions 连续性条件（中文教材中多称换路定律）
If there isn't any infinite current and voltage in a circuit or network, the voltage
of a capacitance and the current of an inductance cannot suddenly change.
That is to say, in any time t (including the step time) the equation should be tenable
0 0
0 0
lim ( ) lim ( ) ( ) ( )
lim ( ) lim ( ) ( ) ( )
C C C C
L L L L
v t v t v t v t
i t i t i t i t
 
 
 
 
      
      
Where, vc indicates the terminal voltage of a capacitance, and iL indicates the
current through an inductance.
Apply the continuity condition to the networks exited by step source, we have
(0 ) (0 ); (0 ) (0 )
C C L L
v v i i
   
 

6.3.3 The Natural Response自由响应
Networks that contain energy storage elements (C or L) differ in one very
important way from purely resistive networks.
In the purely resistive networks, whenever the independent sources are set
to zero, all voltages and currents are zero instantaneously.
By contrast, in networks with energy storage elements, after all independent
sources are set to zero, because there are energies stored in energy storage
elements, the voltages and currents in the circuit can be nonzero values.
When all independent sources in a circuit are set to zero, the response (voltage
or current) produced by only the energies stored in the energy storage
elements is called Natural Response, which persists until all of the stored
energy is dissipated in the resistances.

Now let's look an example for Natural Response.
Assume the initial voltage of the capacitance
is vc(0). According to KVL and Ohm's law,
we have
( )
( ) ( ) ( ) C
C
dv t
v t R i t and i t C
dt
    
We yield a differential equation about the terminal voltage of the capacitance:
( )
( )
C
di t
v t RC
dt
  
Based on the mathematics, we can solve this differential equation
( )
t
RC
C
v t A e

 
This is an exponential function, in which A is a constants.

Because vC(t =0) = vC(0), therefore
0
( 0) (0)
RC
C C
v t A e A v

    
Then we conclude the responses as (for t0)
( ) (0)
( ) (0) ( )
t t
C C
RC RC
C C
v t v
v t v e i t e
R R
 
    
In the responses, τ=RC is called the time constant, which indicates the speed of
responses' changing with time and depends only on the structure of the circuit.
Following figure draws the responses' changing with time.
time
t
 2 3 4 5
vC(t) 0.368vC(0) 0.135 vC(0) 0.050 vC(0) 0.018 vC(0) 0.007 vC(0)

After three time constants (t = 3τ), the exponential is 95% gone. After five
time constants (t = 5τ), the exponential is 99.3% gone, and is usually
considered to be negligible thereafter. In engineering, it is considered that the
circuit has reached final stable state after 3~5 time constants.

6.3.4 Single-Time-Constant Circuits
单时间常数电路（中文书籍一般称一阶电路）
The networks that contain either one capacitor or one inductor, along with a
arbitrary combination of sources and resistors, are called single-time-constant
networks. In the single-time-constant network, the network equation describing
the relation between response and excitation should be a first-order differential
equation. And the natural response has an exponential behavior.
RC single-time-constant circuit RC一阶电路
(1) circuit structure

(2) Circuit Equations
Basic relationships : :
R C S R C
KVL v v v KCL i i
  
Characteristic of elements ; C
R R C C
dv
v R i R i i C
dt
    
The circuit equations for the step source
for the capacitance voltage vC
( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) 0
C C
C S C S
C
C S C
dv t dv t
v t RC v t v t v t
dt dt
dv t
v t RC V v for t
dt

    
    
for the capacitance current iC
( ) ( ) ( )
( ) ( ) ( )
( )
( ) 0 ( ) 0
C S C
C C S
C
C C
di t dv t di t
i t RC C i t i t
dt dt dt
di t
i t RC i for t
dt

    
    

for the resistance voltage vR
( )
( ) ( )
( ) ( ) ' ( )
( )
( ) 0 ( ) 0
S
R R
R R S
R
R R
dv t
dv t dv t
v t RC RC v t v t
dt dt dt
dv t
v t v for t
dt


    
    
Where  = RC is time constant, vC(∞)=VS, vR(∞)=0, iC(∞)=0 are the final
responses in the circuit. It is noticed that all the equations are the same form.
( )
( ) ( ) 0
df t
f t f for t
dt

   
where f indicates the response. The circuit equation depends only on time
constant and final stable state response, f(∞).

(3) Responses
Solving the circuit equations with the initial conditions, f(0), we yield the
responses. The initial condition is the initial energy storage in the capacitor
C, depended by vC(0+).
In our circuit,
the initial responses (t=0+) are
(0 ) (0 )
(0 )
(continuity condition)
(
(0 )
(0 )
(0 )
)
( ' )
C C
R S C
S C
C
v v
KVL
v V v
V v
Ohm s L w
i a
R
 
 



 


Solving the differential
equations we yield
   
1
( ) (0 ) ( ) (0 ) 0
t t
C S C S S C S
v t V v V e u t V v V e t
 
 
 

 
         
 
 
1
( ) (0 ) ( ) (0 ) 0
t t
R R R
v t v e u t v e t
 
 
 

     
1
( ) (0 ) ( ) (0 ) 0
t t
C C C
i t i e u t i e t
 
 
 

     

All the responses are determined by three factors:
   
   
1
1
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
t t
t t
i t i i i e u t i i i e t
v t v v v e u t v v v e t
 
 
 
 

 
 

 
             
 
 
 
             
 
 
Therefore, for the RC single-time-constant circuit with step source, it is
sufficient to calculate the three factors in order to determine responses.

LR Single-Time-Constant Circuit LR一阶电路
(1) Circuit Structure
Similar to the RC single-time-
constant circuit, the circuit
containing one inductor can also
be separated into two parts,
inductor and all other circuits
connected to the inductor. By
Thévenin equivalent, the later
may be equivalent to a voltage
source and a resistor in series.
So the LR single-time-constant
circuit has following general
structure

(2) Circuit Equations
Basic relationships : :
R L S R L
KVL v v v KCL i i
  
Characteristic of elements ; L
R R L L
di
v R i R i v L
dt
    
The circuit equations for the step source
( )
( ) ( )
( ) ( ) ( )
( )
( ) ( ) 0
S
L L
L L S
S
L
L L
v t
L di t di t
i t i t i t
R dt R dt
V
di t
i t i for t
dt R


    
    
( )
( ) ( )
( ) ( ) ' ( )
( )
( ) 0 ( ) 0
S
L L
L L S
L
L L
dv t
L dv t L dv t
v t v t v t
R dt R dt dt
dv t
v t v for t
dt


    
    

( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) 0
R R
R S R S
R
R S R
L dv t dv t
v t v t v t v t
R dt dt
dv t
v t V v for t
dt


    
    
Where  = L/R is the time constant of LR circuit, vL(∞)=0, vR(∞)=VS,
iL(∞)=VS/R are the final responses in the circuit. It is noticed that all the
equations are the same form
( )
( ) ( ) 0
df t
f t f for t
dt

   
where f indicates the response. The circuit equation depends only on time
constant and final stable state response, f(∞).

(3) Responses
Solving the circuit equations with the initial conditions, f(0), we yield the
responses. The initial condition is the initial energy storage in the inductor
L, depended by iL(0+).
In our circuit, the initial
responses (t=0+) are
(0 ) (0 )
(0 )
(continuity cond
(0 )
(0 ) (0 )
ition)
( ' )
( )
L L
R L
L S R
i i
v R i
v
Ohm s Law
K
V v VL
 
 
 

 
 
Solving the differential
equations we yield
   
1
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
t t
L L L L L L L
 
 
 

 
             
 
 
   
1
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
t t
R R R R R R R
 
 
 

 
             
 
 
1
( ) (0 ) ( ) (0 ) 0
t t
L L L
v t v e u t v e t
 
 
 

     

All the responses are determined by three factors, time
constant, initial response and final response
   
   
1
1
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
t t
t t
 
 
 
 

 
 

 
             
 
 
 
             
 
 
Therefore, for the LR single-time-constant circuit with step source,
it is also sufficient to calculate the three factors in order to
determine responses.

The Three-Factor Method for single-time-constant circuits三要素法
From above analysis, we can conclude that the circuit equations and responses of
single-time-constant circuit with dc or no source or step source are determined
perfectly by three factors, time constant, initial stable response, and final stable
response.
   
   
1
1
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
( ) ( ) (0 ) ( ) ( ) ( ) (0 ) ( ) 0
t t
t t
 
 
 
 

 
 

 
             
 
 
 
             
 
 
This is called three-factor method for single-time-constant circuit analysis.
Time constant τ is defined as (unit second, s):
RC circuit τ =RC
LR circuit τ = L/R
where R is the Thévenin equivalent resistance.

Final stable response, f(∞), may be determined in final stable state of the
circuit (see section 6.3.1). To determine the initial response, f(0+), we can first
apply continuity condition obtaining vC(0+)=vC(0-) or iL(0+)=iL(0-). Then
substitute a voltage source vC(0+) for the capacitor or a current source iL(0+) for
the inductor in the circuit.
In the equivalent circuits for t =0+,
which are resistive networks, we
can determine the initial instant
responses v(0+) and i(0+).

Example 1
The following circuit has been in stable state before the switch is closed at
t =0. Determine the voltage cross the resistor R1 after the switch is closed.
Solution:
Before the switch is closed, the inductor
is connected with a resistor and without
dependent sources connected to it.
Therefore, the stable response for t<0 is
iL(0-)=0 (no energy stored).
After the switch action (t >0), we calculate
the three factors.
(1) time constant τ = L/R
Thévenin equivalent resistance R = RS||R1
(2) Final stable response v1(∞) = 0 (because the inductor is equivalent to a short-
circuit in final DC stable state).

(3) Initial instant response v1(0+)
Since iL(0)= iL(0)=0 (continuity condition), the inductor is equivalent to a
zero current source (open-circuit) in the instant equivalent circuit at t=0+.
In this equivalent circuit, we have
  1
1 1 0 0
1
(0 ) || S
S
S
R R
v R R I I
R R
 
   

Finally, we write the response as
 
1
1
1 1 1 1 1
1
0 1
1
( ) ( ) (0 ) ( ) ( )
( )
S
S
t
R R t
R R L
S
S
v t v v v e u t
R R
I e u t
R R





 


 
      
 
 

   


6.4 Circuit Applications
6.4.1 Analog Integration and Differentiation模拟积分和微分电路
Having now introduced the inductor and capacitor and examined how to
treat step responses in simple networks, we can inquire into some new
circuits that include energy storage elements.
Analog integration 模拟积分电路
(1) Ideal integration
Because of the feedback to the negative
terminal, we can presume the op-amp
is working in linear region
0
0
S S
S
S
F S
v v v
v v i
R R
v
i i i i
R

 
 

    
    

Therefore, the relation between input and output voltage is
0
1 1
C C F S
v v v v i dt v dt
C RC

       
 
In order the output to be set to zero prior to starting the calculation of an
integral, we often add a switch in the circuit.

(2) Non-ideal integration
Now we consider the non-ideality of an op-amp. All op-amps require small DC
bias currents at both inputs. In the above circuit the bias current can enter the
+ terminal from ground, but the - terminal gets this bias current from the
capacitor. This steady current charges the capacitor gradually up until vC
reaches +VCC, at which point the op-amp saturates making the integrator
inoperable.
There are many ways of compensating for the bias current, but each method
introducing its own problems. Following two of them.
(a) Add a resistance negative feedback (b) With bias adjust potentiometer

2. Analog differentiation 模拟微分电路
Because of the feedback to the negative
terminal, the resistance, we can also
presume the op-amp is working in
linear region.
( )
0
0
S S
S
S
F S
d v v dv
v v i C C
dt dt
dv
i i i i C
dt

 
 

    
    
Therefore, the relation between input and output voltage is
0
S
R R F
dv
v v v v R i RC
dt

        

6.4.2 The Analog Computer模拟计算器
An analog computer needs four types of basic operations
1. Addition/Subtraction 2. Multiply/Divide
3. Integration 4. Differentiation
By means of these basic operation circuit, we can implement various
analog operations.
For an example following circuit implements the operation
2
2
d x
K x
dt
  
 
2
Let
Then (implemented by inverting amplifier)
1
(implemented by integration)
1
(implemented by integration)
dx dy
y RC z RC
dt dt
K
z x
RC
y z dt
RC
x y dt
RC
   
  
  
  



 
1
0 2
2
K R
k
R
RC
 
 

6.4.3 A Square-Wave Oscillator 方波振荡器
When the input voltage in the Schmitt trigger comes from the output through
a RC integration network, the circuit can generates square waveform output.
Recall the transfer characteristic of
Schmitt trigger
• When v1 increases and reaches
v1≥VTd, the output v2 from +VCC
down to VCC
• When v1 decreases and reaches
v1≤VTu, the output v2 from VCC
up to VCC
Recall the operation of RC single-time-constant circuit
• When v2 is positive DC voltage, the capacitor is charged and v3 increases
gradually.
• When v2 is negative DC voltage, the capacitor is discharged and v3
decreases gradually.

Therefore, the operation principle of the above network can be concluded

The circuit and waveform are shown as follows.
v1=v3
t
0
v2
t
0
Where,
2 2
2 3 2 3
;
Td CC Tu CC
R R
V V V V
R R R R

 
  Circuit Demonstration
T1
T2

The time functions of v3 during 0 to T1 and T1 to T2 are
1
1
1
3 1
3 1 2
( ) ( ) 0
( ) ( )
t
R C
CC Tu CC
t T
R C
CC Td CC
v t V V V e t T
v t V V V e T t T



     
      
We can calculate the period of v2
1
1 2
3 1 1 1 1
3
2
( ) ( ) ln ln 1
T
R C CC Tu
Td CC Tu CC
CC Td
V V R
v T V V V V e T R C R C
V V R
  

       
 
  
2 1
1 2
3 2 2 1 1 1
3
2
( ) ( ) ln ln 1
T T
R C CC Td
Tu CC Td CC
CC Tu
V V R
v T V V V V e T T R C R C
V V R

  

         
 
  
When R2=R3, we have
1 2 1 1 1
ln3 1.1
T T T RC RC
   
The output of the circuit is a square-wave.

The exercises of Chapter six:
E6.1, E6.2, E6.3, E6.4, E6.5, E6.6, E6.8

Chapter 06.ppt Capacitance and inductance

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