Download ECE 2006 - Lecture 2

MT-144
NETWORK ANALYSIS
Mechatronics Engineering
(07)
1
TRANSIENT RESPONSE OF FIRST
ORDER CIRCUITS: (Chapter 8)
8.1
8.2
8.3
8.4
8.5
Basic RC and RL Circuits
Transients in First-order Networks
Step, Pulse and Pulse-Train Responses
First-order Op-Amp Circuits
Transient Analysis Using SPICE
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
Introduction.
In Section 7.4 we found that the response to a dc forcing function
consists of a transient component and a dc steady-state component.
The transient component has the same functional form as the natural
response, which is an exponentially decaying function. The dc steadystate component has the same functional form as the forcing function,
which is a constant function.
We now wish to apply the mathematical concepts of Sections 7.3 and
7.4 to a variety of circuit configurations, but with greater attention to
physical behavior. An engineer must always use physical insight to
interpret mathematical results.
In the present chapter we concentrate on first-order circuits, that is,
circuits that contain only one energy-storage element or that contain
multiple elements, but in such a way that they can be reduced to a single
equivalent element via series / parallel combinations.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
Introduction…
In Chapter 9 we turn our attention to second-order circuits, with special
emphasis on circuits that contain one capacitance or one inductance.
We begin by examining in great physical detail how energy builds up
and decays in simple capacitive and inductive circuits. In so doing, we
develop a set of useful rules to facilitate the transient analysis of more
complex networks.
Next, we study the step and pulse responses of the basic R-C, L-R, C-R,
and R-L circuits, and observe how these circuits behave in terms of
signal distortion as well as passing or blocking the dc component of an
input pulse train.
We then turn to capacitive circuits using op amps.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
Introduction…
After examining the classical integrator and differentiator configurations,
we use the op amp to illustrate another important systems concept, this
time the concept of root location control.
In particular, we illustrate the use of an op amp to create a root in the
right half of the s plane and thus obtain a diverging response, a situation
that cannot be achieved with a purely passive circuit.
As usual, we conclude the chapter by illustrating the use of SPICE to
perform the transient analysis of circuits. The SPICE facilities introduced
in this chapter are the . TRAN statement and the PULSE function.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS
In this section we take a closer look at the physical aspects of the
transient behavior of basic RC and RL pairs. However, we must first
state two important rules that will greatly facilitate our transient analysis
of these circuits.
Instantaneous Behavior: The Continuity Conditions
Capacitance current and voltage are related as ic = C dvc/dt. The faster
the change in vc, the greater ic. If vc were to change instantaneously,
dvc/dt would become infinite, and so would ic. An infinite current,
however, would require the existence of an infinite amount of power at
the capacitor terminals. Since this is physically impossible, we have
the following rule, known as the voltage continuity rule for capacitance:
Capacitance Rule 1: The voltage across a capacitance cannot change
instantaneously, that is, at any instant to we must have vC(t0+) = vc(t0-),
where t0+ is the instant right after t0 , and t0- is the instant right before to.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
Instantaneous Behavior: The Continuity Conditions …
Capacitance Rule 1: The voltage across a capacitance cannot change
instantaneously, that is, at any instant to we must have vc(to+) = vc(to-),
where to+ is the instant right after to, and to- is the instant right before to.
Note that the continuity rule applies to vc, not to ic ; ic can indeed
change instantaneously without violating Rule 1.
We can readily visualize this rule using the water tank analogy, where
water level is likened to voltage. and water flow to current. To bring
about an instantaneous level change would require adding or removing
water from the tank in zero time, a physically impossible task. Current
flow can undergo brusque changes, however, as when we suddenly
open a spigot to let water out of the tank.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
Instantaneous Behavior: The Continuity Conditions …
Turning now to inductance, whose voltage and current are related as
vL = L diL/dt, we observe that if iL were to change instantaneously, diL/dt
would become infinite and so would vL. This is again physically
impossible, as it would require the existence of an infinite amount of
power at the inductor terminals. Hence, we have the following current
continuity rule for inductance:
Inductance Rule 1: The current through an inductance cannot change
instantaneously, that is, at any instant to we must have iL(t0+) = iL(t0-).
Inductors are also called chokes to reflect their tendency to oppose
sudden changes in current. Note that the continuity rule applies to
inductive current, not to inductive voltage. The latter i.e. (vL) can indeed
undergo instantaneous changes without violating Rule 1.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior
If the voltage vC across a capacitance is known to have stabilized as
some constant value Vc, not necessarily zero, the capacitance is said to
be in dc steady state. Once in this state, it draws no current from the rest
of the circuit because iC = C dvC / dt= C x 0 = 0. This behavior is
summarized as follows:
Capacitance Rule 2: In dc steady state, a capacitance behaves as an
open circuit, that is, ic = 0.
Likewise, if the current iL through an inductance is known to have
stabilized as some constant value IL , not necessarily zero, the
inductance is said to be in dc steady state, and the voltage across its
terminals is vL = L diL /dt = L x 0 = 0. Consequently,
Inductance Rule 2: In dc steady state, an inductance behaves as a
short circuit, that is, vL = 0.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…
Inductance Rule 2: In dc steady state, an inductance behaves as a
short circuit, that is, vL = 0.
These rules indicate that if an energy-storage element in a circuit is
known to be in dc steady state, we can replace it with an open circuit
if it is a capacitance, or a short circuit if it is an inductance, and thus
simplify our analysis considerably (assuming ideal capacitors and
inductors).
Table 8.1 compares capacitance and inductance behavior. Note again
the dual behavior of the two elements. (next slide)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…
Table 8.1 compares capacitance and inductance behavior. Note again
the dual behavior of the two elements. (next slide)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…
The RC Circuit
In the circuit of Figure 8.1(a) assume the capacitance is initially
discharged so that q= 0. Then, we know q= cv , or we have
v = q/C = 0/C = 0. Moreover, i = (0 – v )/ R = (0 – 0) /R = 0.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
In the circuit of Figure 8.1(a) assume the capacitance is initially
discharged so that q= 0. Then, we know q= cv , or we have
v = q/C = 0/C = 0. Moreover, i = (0 – v )/ R = (0 – 0) /R = 0.
At an instant that we arbitrarily choose as t= 0 we flip the switch to
connect R to VS , ending up with the situation of Figure 8.1(b). The
resulting current i = (VS - v )/ R will charge C toward VS according to
Equation (7.23), which now becomes: RC dv/ dt + v = VS
…(8.1)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
A word of caution is necessary at this point. The instant t= 0 is
exceptional because the switch voltage is in the process of changing
from 0 to VS and, as such, is not a single-valued function of time.
RC dv/ dt + v = VS
…(8.1)
To avoid any ambiguities, we assume Equation (8.1) to begin at t= 0+,
where 0+ denotes the instant just after t = 0, when the switch voltage has
fully attained the value VS.
Likewise, t = 0- shall denote the instant just before t = 0, when the switch
voltage is still zero. With this in mind, the solution to Equation (8.1) shall
also be assumed to begin at t= 0+. Such a solution is provided by
Equation (7.41), but with t = 0+ instead of t = 0.
y(t)= y(0) e-t/Ƭ + XS (1- e-t/Ƭ) … (7.41)
Because of the voltage continuity rule, we have v(0+) = v(0-) = 0,
indicating that the natural component in Equation (7.41) is zero.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
-t/Ƭ
-t/Ƭ
8.1 BASIC RC AND RL CIRCUITS… y(t)= y(0) e + XS (1- e ) … (7.41)
DC Steady-State Behavior…The RC Circuit…
The solution is thus the forced component. To summarize, just prior to
switch activation we have
The responses are shown in Figure 8.2. It is interesting to note that v is
continuous but i is discontinuous at the origin.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
The responses are shown in Figure 8.2. It is interesting to note that v is
continuous but i is discontinuous at the origin. This does not violate
Rule 1, however, as continuity applies to voltage, not to current.
Because of its abrupt jump in magnitude, the current waveform is
referred to as a spike.
v(t)= vS (1- e-t/RC)
i(t)= (vS/R) e-t/RC)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
Both transients of Figure 8.2 are governed by the time constant Ƭ = RC.
Increasing R reduces the charging current, and increasing C increases
the amount of charge that needs to be transferred. In either case the
transients will be slowed down. Conversely, decreasing R results in
faster transients because the charging current is increased.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
In the limit R  0, C would try to charge instantaneously. But to do so
would require an infinitely large current spike from VS, which we know to
be physically impossible. In a practical situation current is limited by the
internal resistance of the source and the connecting wires, indicating
that charge buildup, however rapid, cannot be instantaneous. The spark
that we observe in the laboratory when we connect a capacitor directly
across a battery attests to the large current spike involved.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
Example 8.1
v(t ≥ 0+)= vS (1- e-t/RC)
…(8.2b)
Home Work: Do Exercise 8.1 (page 330 of text book)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS…
DC Steady-State Behavior…The RC Circuit…
Suppose that after the capacitance of Figure 8.1(a) has fully charged to
VS, the switch is flipped back down at an instant that we again choose as
t = 0 for convenience. This is shown in Figure 8.3(a). With the source out
of the way, we end up with the source-free circuit of Figure 8.3(b). By the
voltage continuity rule we must have v(0+) = v(0-) = VS. Consequently, at
t = 0+, the top plate holds the charge +CVS and the bottom
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… Consequently, at t = 0+, the top plate holds the charge
+CVS and the bottom plate the charge -CVS. Moreover, the initial stored
energy in the capacitance is w(0+) = (1/2)CV2S. Thanks to this energy,
C will be able to sustain a nonzero voltage even though VS has been
switched out of the circuit. This voltage, in turn, will cause R to draw
current and gradually discharge C. Though the reference direction is still
shown clockwise for consistency, the discharge current actually flows
counterclockwise, out of the top plate, through the resistance, and back
into the bottom plate.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
The discharge process is governed by Equation (7.23), which now
becomes:
RC dv/ dt + v = 0
…(8.4)
Rewriting as dv/ dt = -v/ (RC) indicates that v decreases at a rate
proportional to v itself. Initially, when v is large, we have a proportionally
rapid discharge rate. As the discharge progresses, the rate decreases in
proportion, making the discharge slower and slower.
This process, similar to emptying a water tank by opening a spigot at the
bottom, is an exponential decay.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
In fact, it is the natural response predicted by Equation (7.33). We thus
have:
y (t) = y(0)e-t/Ƭ ... (7.33),
(Fig 8.4) on the next slide
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
Fig 8.3
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
The rate of decay is governed by the time constant RC. The larger R or
C, the slower the decay. In the limit R  ∞ the decay would become
infinitely slow because there would be no current path for C to
discharge. Hence, C would retain its initial voltage VS indefinitely, thus
providing a memory function.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
Fig 8.3
Example 8.2
tƐ = - Ƭ lnƐ
…(7.34)
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… Example 8.3
y (t) = y(0)e-t/Ƭ ... (7.33)
or v = Vs e-t/RC
v2 = V2S e-2t/RC
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit.
Interchanging the roles of the capacitance and resistance in the R-C
circuit of Figure 8.1(a) turns it into the C-R circuit of Figure 8.5 (a).
Though the current response is not affected by this interchange, the
voltage response will be different because we are now observing it
across R rather than C.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit…
The CR Circuit…
Assume prior to switch activation the circuit is in steady state so that C,
by Rule 2, acts as an open circuit. Then, by Ohm's Law,
v(0-) = Ri(0-) = R x 0 = 0, indicating that the voltage across C must also
be zero.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit…
As the switch is flipped up, the voltage of the left plate jumps from 0 V to
Vs. By Rule 1, the voltage across C right after switch activation must still
be 0 V. For this to be possible, the voltage of the right plate must also
jump from 0 V to VS, so that v(0+) = VS. With a nonzero voltage drop, R
will draw current and cause C to charge exponentially.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit…
In summary,
This response is shown in Figure 8.5(b). We observe that v now exhibits
a discontinuity at t = 0. This does not violate Rule 1, however, because
this rule applies to the voltage across the capacitance, not to the
individual plate voltages.
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TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit…
In summary,
If the switch is left in the up position long enough to allow v to fully decay
to 0 V, the capacitance will achieve its dc steady state, in which the
voltage at the left plate will be VS and that at the right plate will be 0 V.
Consequently, the final stored energy will be wc = (1/2) CVS2.
32
TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit…
Suppose now the switch is flipped back down, at a time that we again
choose as t=0, as shown in Figure 8.6(a). This causes the voltage of the
left plate to jump from VS to 0 V, and that of the right plate to jump from
0 V to -Vs, by the continuity rule. Consequently, v(0+) = -Vs, causing R
to draw current and discharge C,
Note the current will
flow from R to C
through the closed
switch to the lower
node.
33
TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
RC Circuit… The CR Circuit…
This response is shown in Figure 8.6(b). Voltage is again discontinuous
at t = 0, but without violating Rule 1. We also note the creation of a
negative voltage transient using a positive voltage source. This feature is
exploited in the design of certain types of voltage inverters.
Home Work: Do Exercise 8.2 (page 336 of text book)
34
TRANSIENT RESPONSE OF FIRST ORDER
CIRCUITS: (Chapter 8)
8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The
The RL Circuit
Figure 8.7 Flipping a switch to
investigate the forced response
of the RL Circuit
In the circuit of Figure 8.7 assume the inductance has zero initial stored
energy so that, by Equation (7.19), i(0-) = 0. Flipping up the switch
connects the source IS to the RL pair, causing current and, hence,
energy to build up in the inductance. For t ≥ 0+, this process is governed
by Equation (7.24)
By the current continuity rule we must have i(0+)= i(0-)= 0.
Consequently, the solution to Equation (8.9) is the forced response
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