storey: electrical & electronic systems © pearson education limited 2004 oht 18.1 transient...
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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.1
Transient Behaviour
Introduction Charging Capacitors and Energising Inductors Discharging Capacitors and De-energising Inductors Response of First-Order Systems Second-Order Systems Higher-Order Systems
Chapter 18
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.2
Introduction
So far we have looked at the behaviour of systems in response to:
– fixed DC signals
– constant AC signals
We now turn our attention to the operation of circuits before they reach steady-state conditions
– this is referred to as the transient response
We will begin by looking at simple RC and RL circuits
18.1
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.3
Charging Capacitors and Energising Inductors
Capacitor Charging
Consider the circuit shown here
– Applying Kirchhoff’s voltage law
– Now, in a capacitor
– which substituting gives
18.2
VviR
tv
Cidd
Vvtv
CR dd
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.4
The above is a first-order differential equation with constant coefficients
Assuming VC = 0 at t = 0, this can be solved to give
– see Section 18.2.1 of the course text for this analysis
Since i = Cdv/dt this gives (assuming VC = 0 at t = 0)
– where I = V/R
)e1()e1( t
-CR
t-
VVv
t
-CR
t-
IIi ee
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.5
Thus both the voltage and current have an exponential form
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.6
Inductor energising
A similar analysis of this circuit gives
where I = V/R
– see Section 18.2.2 for this analysis
t
-LRt
-VVv ee
)e1()e1( t
-LRt
-IIi
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.7
Thus, again, both the voltage and current have an exponential form
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.8
Discharging Capacitors and De-energising Inductors
Capacitor discharging
Consider this circuit for discharging a capacitor
– At t = 0, VC = V
– From Kirchhoff’s voltage law
– giving
18.3
0viR
0dd v
tv
CR
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.9
Solving this as before gives
– where I = V/R
– see Section 18.3.1 for this analysis
t
-CR
t-
VVv ee
t
-CR
t-
IIi ee
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.10
In this case, both the voltage and the current take the form of decaying exponentials
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.11
Inductor de-energising
A similar analysis of thiscircuit gives
– where I = V/R
– see Section 18.3.1 for this analysis
t
-LRt
-VVv ee
t
-LRt
-IIi ee
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.12
And once again, both the voltage and the current take the form of decaying exponentials
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.13
A comparison of the four circuits
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.14
Response of First-Order Systems
Initial and final value formulae
– increasing or decreasing exponential waveforms (for either voltage or current) are given by:
– where Vi and Ii are the initial values of the voltage and current
– where Vf and If are the final values of the voltage and current
– the first term in each case is the steady-state response– the second term represents the transient response– the combination gives the total response of the arrangement
18.4
/e)( tfif VVVv
/e)( tfif IIIi
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.15
Example – see Example 18.3 from course text
The input voltage to the following CR network undergoes a step change from 5 V to 10 V at time t = 0. Derive an expression for the resulting output voltage.
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.16
Here the initial value is 5 V and the final value is 10 V. The time constant of the circuit equals CR = 10 103 20 10-6 = 0.2s. Therefore, from above, for t 0
volts e510
e)105(10
e)(
2.0/
2.0/
/
t
t
tfif VVVv
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.17
The nature of exponential curves
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.18
Response of first-ordersystems to a squarewaveform– see Section 18.4.3
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.19
Response of first-ordersystems to a squarewaveform of differentfrequencies– see Section 18.4.3
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.20
Second-Order Systems
Circuits containing both capacitance and inductance are normally described by second-order differential equations. These are termed second-order systems
– for example, this circuit is described by the equation
18.5
Vvt
vRC
t
vLC C
CC d
d
d
d2
2
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.21
When a step input is applied to a second-order system, the form of the resultant transient depends on the relative magnitudes of the coefficients of its differential equation. The general form of the response is
– where n is the undamped natural frequency in rad/s and (Greek Zeta) is the damping factor
xyty
t
y
nn
dd2
d
d12
2
2
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.22
Response of second-order systems
=0 undamped
<1 under damped
=1 critically damped
>1 over damped
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.23
Higher-Order Systems
Higher-order systems are those that are described by third-order or higher-order equations
These often have a transient response similar to that of the second-order systems described earlier
Because of the complexity of the mathematics involved, they will not be discussed further here
18.6
Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.24
Key Points
The charging or discharging of a capacitor, and the energising and de-energising of an inductor, are each associated with exponential voltage and current waveforms
Circuits that contain resistance, and either capacitance or inductance, are termed first-order systems
The increasing or decreasing exponential waveforms of first-order systems can be described by the initial and final value formulae
Circuits that contain both capacitance and inductance are usually second-order systems. These are characterised by their undamped natural frequency and their damping factor