TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

Report
TRANSIENTS AND STEP
RESPONSES
ELCT222- Lecture Notes
University of S. Carolina
Spring 2012
OUTLINE
RC transients charging
 RC transients discharge
 RC transients Thevenin
 P-SPICE
 RL transients charging
 RL transients discharge
 Step responses.
 P-SPICE simulations
 Applications

Reading:
Boylestad Sections
10.5, 10.6, 10.7, 10.9,10.10
24.1-24.7
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
 The
placement of
charge on the
plates of a
capacitor does not
occur
instantaneously.
 Instead, it occurs
over a period of
time determined
by the components
of the network.
FIG. 10.26 Basic R-C
charging network.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
FIG. 10.27 vC during the
charging phase.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
FIG. 10.28 Universal time
constant chart.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
TABLE 10.3 Selected
values of e-x.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
 The
factor t, called the time constant of
the network, has the units of time, as
shown below using some of the basic
equations introduced earlier in this text:
The larger R is, the lower the charging current, longer time to charge
The larger C is, the more charge required for a given V, longer time.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
FIG. 10.29 Plotting the equation yC = E(1 – et/t) versus time (t).
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
FIG. 10.32 Revealing the short-circuit
equivalent for the capacitor that occurs when
the switch is first closed.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
FIG. 10.31 Demonstrating that a capacitor
has the characteristics of an open circuit after
the charging phase has passed.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE CHARGING PHASE
TRANSIENTS IN CAPACITIVE NETWORKS:
THE CHARGING PHASE
USING THE CALCULATOR TO SOLVE EXPONENTIAL FUNCTIONS
FIG. 10.34 Calculator key strokes to
determine e-1.2.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE CHARGING PHASE
USING THE CALCULATOR TO SOLVE EXPONENTIAL FUNCTIONS
Equation vC = E(1 – e-t/τ)
FIG. 10.35 Transient network for
Example 10.6.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE CHARGING PHASE
USING THE CALCULATOR TO SOLVE EXPONENTIAL FUNCTIONS
FIG. 10.36 vC versus time for the charging
network in Fig. 10.35.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE CHARGING PHASE
USING THE CALCULATOR TO SOLVE EXPONENTIAL FUNCTIONS
FIG. 10.37 Plotting the waveform in Fig. 10.36
versus time (t).
TRANSIENTS IN CAPACITIVE NETWORKS:
THE CHARGING PHASE
USING THE CALCULATOR TO SOLVE EXPONENTIAL FUNCTIONS
FIG. 10.38 iC and yR for the charging
network in Fig. 10.36.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE DISCHARGING
PHASE
 We
now investigate how to discharge a
capacitor while exerting some control on
how long the discharge time will be.
 You can, of course, place a lead directly
across a capacitor to discharge it very
quickly—and possibly cause a visible
spark.
 For larger capacitors such those in TV
sets, this procedure should not be
attempted because of the high voltages
involved—unless, of course, you are
trained in the maneuver.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE DISCHARGING
PHASE
FIG. 10.39 (a) Charging
network; (b) discharging
configuration.
TRANSIENTS IN CAPACITIVE
NETWORKS: THE DISCHARGING
PHASE
 For
the voltage across the capacitor that
is decreasing with time, the
mathematical expression is:
TRANSIENTS IN CAPACITIVE
NETWORKS: THE DISCHARGING
PHASE
FIG. 10.40 yC, iC, and yR for 5t switching between
contacts in Fig. 10.39(a).
TRANSIENTS IN CAPACITIVE
NETWORKS: THE DISCHARGING
PHASE
FIG. 10.41 vC and iC for the network in Fig.
10.39(a) with the values in Example 10.6.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.43 Effect of increasing
values of C (with R constant) on the
charging curve for vC.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.44 Network to be analyzed in
Example 10.8.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.45 vC and iC for the network in
Fig. 10.44.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.46 Network to be
analyzed in Example 10.9.
FIG. 10.47 The charging
phase for the network in Fig.
10.46.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.48 Network in Fig. 10.47
when the switch is moved to position
2 at t = 1t1.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.49 vC for the network in
Fig. 10.47.
TRANSIENTS IN CAPACITIVE NETWORKS:
THE DISCHARGING PHASE
THE EFFECT OF ON THE RESPONSE
FIG. 10.50 ic for the network in
Fig. 10.47.
INITIAL CONDITIONS
 The
voltage across the capacitor at this
instant is called the initial value, as
shown for the general waveform in Fig.
10.51.
FIG. 10.51 Defining the regions
associated with a transient response.
-t /t
c
f
i
f
v = V + (V - V )e
INITIAL CONDITIONS
FIG. 10.52 Example
10.10.
INITIAL CONDITIONS
FIG. 10.53 vC and iC for the network
in Fig. 10.52.
INITIAL CONDITIONS
FIG. 10.54 Defining the parameters
in Eq. (10.21) for the discharge phase.
INSTANTANEOUS VALUES
 Occasionally,
you may need to determine
the voltage or current at a particular
instant of time that is not an integral
multiple of t.
FIG. 10.55 Key strokes to determine (2 ms)(loge2)
using the TI-89 calculator.
THÉVENIN EQUIVALENT:
T
=RTHC
You may encounter instances in which the
network does not have the simple series form in
Fig. 10.26.
 You then need to find the Thévenin equivalent
circuit for the network external to the capacitive
element.

THÉVENIN EQUIVALENT:
FIG. 10.56 Example
10.11.
T
=RTHC
THÉVENIN EQUIVALENT:
FIG. 10.57 Applying Thévenin’s
theorem to the network in Fig.
10.56.
T
=RTHC
THÉVENIN EQUIVALENT:
FIG. 10.58 Substituting the Thévenin
equivalent for the network in Fig. 10.56.
T
=RTHC
THÉVENIN EQUIVALENT:
=RTHC
T
FIG. 10.59 The resulting
waveforms for the
network in Fig. 10.56.
THÉVENIN EQUIVALENT:
FIG. 10.60
Example 10.12.
T
=RTHC
FIG. 10.61 Network in
Fig. 10.60 redrawn.
THÉVENIN EQUIVALENT:
FIG. 10.62 yC for the network in
Fig. 10.60.
T
=RTHC
THÉVENIN EQUIVALENT:
FIG. 10.63 Example
10.13.
T
=RTHC
THE CURRENT IC
There is a very special relationship between the
current of a capacitor and the voltage across it.
 For the resistor, it is defined by Ohm’s law: iR =
vR/R.
 The current through and the voltage across the
resistor are related by a constant R—a very
simple direct linear relationship.
 For the capacitor, it is the more complex
relationship defined by:

THE CURRENT IC
FIG. 10.64 vC for
Example 10.14.
THE CURRENT IC
FIG. 10.65 The resulting current iC for the applied
voltage in Fig. 10.64.
INDUCTORS
R-L TRANSIENTS: THE STORAGE
PHASE
 The
storage waveforms have the same
shape, and time constants are defined for
each configuration.
 Because these concepts are so similar
(refer to Section 10.5 on the charging of a
capacitor), you have an opportunity to
reinforce concepts introduced earlier and
still learn more about the behavior of
inductive elements.
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.31 Basic R-L
transient network.
Remember, for an inductor
diL
vL = L
dt
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.32 iL, yL, and yR
for the circuit in Fig. 11.31
following the closing of the
switch.
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.33 Effect of L on the shape of the iL
storage waveform.
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.34 Circuit in Figure 11.31 the instant the
switch is closed.
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.35 Circuit in Fig.
11.31 under steady-state
conditions.
FIG. 11.36 Series R-L circuit
for Example 11.3.
R-L TRANSIENTS: THE STORAGE
PHASE
FIG. 11.37 iL and vL for the network in
Fig. 11.36.
INITIAL CONDITIONS
Since the current through a coil cannot change
instantaneously, the current through a coil
begins the transient phase at the initial value
established by the network (note Fig. 11.38)
before the switch was closed.
 It then passes through the transient phase until
it reaches the steady-state (or final) level after
about five time constants.
 The steadystate level of the inductor current can
be found by substituting its shortcircuit
equivalent (or Rl for the practical equivalent) and
finding the resulting current through the
element.

INITIAL CONDITIONS
iL = I f + (I i - I f )
FIG. 11.38 Defining the three phases of a
transient waveform.
INITIAL CONDITIONS
FIG. 11.39
Example 11.4.
INITIAL CONDITIONS
FIG. 11.40 iL and vL for the network in
Fig. 11.39.
R-L TRANSIENTS: THE RELEASE
PHASE
FIG. 11.41 Demonstrating the effect of
opening a switch in series with an
inductor with a steady-state current.
R-L TRANSIENTS: THE RELEASE
PHASE
FIG. 11.42 Initiating the storage phase for an inductor by
closing the switch.
R-L TRANSIENTS: THE RELEASE
PHASE
FIG. 11.43 Network in Fig. 11.42 the instant
the switch is opened.
R-L TRANSIENTS: THE RELEASE
PHASE
R-L TRANSIENTS: THE RELEASE
PHASE
FIG. 11.45 The various
voltages and the current for
the network in Fig. 11.44.
STEP RESPONSES
OBJECTIVES
Become familiar with the specific terms that
define a pulse waveform and how to calculate
various parameters such as the pulse width, rise
and fall times, and tilt.
 Be able to calculate the pulse repetition rate and
the duty cycle of any pulse waveform.
 Become aware of the parameters that define the
response of an R-C network to a square-wave
input.
 Understand how a compensator probe of an
oscilloscope is used to improve the appearance of
an output pulse waveform.

IDEAL VERSUS ACTUAL
 The
ideal pulse in Fig. 24.1 has vertical
sides, sharp corners, and a flat peak
characteristic; it starts instantaneously
at t1 and ends just as abruptly at t2.
FIG. 24.1 Ideal pulse
waveform.
IDEAL VERSUS ACTUAL
FIG. 24.2 Actual pulse
waveform.
IDEAL VERSUS ACTUAL
Amplitude
 Pulse Width
 Base-Line Voltage
 Positive-Going and Negative-Going Pulses
 Rise Time (tr) and Fall Time (tf)
 Tilt

IDEAL VERSUS ACTUAL
FIG. 24.3 Defining the baseline voltage.
IDEAL VERSUS ACTUAL
FIG. 24.4 Positivegoing pulse.
IDEAL VERSUS ACTUAL
FIG. 24.5 Defining
tr and tf.
IDEAL VERSUS ACTUAL
FIG. 24.6
Defining tilt.
IDEAL VERSUS ACTUAL
FIG. 24.7 Defining preshoot, overshoot,
and ringing.
IDEAL VERSUS ACTUAL
FIG. 24.8
Example 24.1.
IDEAL VERSUS ACTUAL
FIG. 24.9 Example
24.2.
PULSE REPETITION RATE AND
DUTY CYCLE
A
series of pulses such as those appearing
in Fig. 24.10 is called a pulse train.
 The varying widths and heights may
contain information that can be decoded
at the receiving end.
 If the pattern repeats itself in a periodic
manner as shown in Fig. 24.11(a) and (b),
the result is called a periodic pulse
train.
PULSE REPETITION RATE AND
DUTY CYCLE
FIG. 24.11 Periodic
pulse trains.
PULSE REPETITION RATE AND
DUTY CYCLE
FIG. 24.12
Example 24.3.
PULSE REPETITION RATE AND
DUTY CYCLE
FIG. 24.13
Example 24.4.
PULSE REPETITION RATE AND
DUTY CYCLE
FIG. 24.14 Example
24.5.
AVERAGE VALUE



The average value of a pulse waveform can be
determined using one of two methods.
The first is the procedure outlined in Section 13.7,
which can be applied to any alternating waveform.
The second can be applied only to pulse waveforms
since it utilizes terms specifically related to pulse
waveforms; that is,
AVERAGE VALUE
FIG. 24.15
Example 24.6.
AVERAGE VALUE
FIG. 24.16 Solution to part (b) of
Example 24.7.
AVERAGE VALUE
INSTRUMENTATION
The average value (dc
value) of any
waveform can be
easily determined
using the oscilloscope.
 If the mode switch of
the scope is set in the
ac position, the
average or dc
component of the
applied waveform is
blocked by an internal
capacitor from
reaching the screen.

FIG. 24.17 Determining the
average value of a pulse
waveform using an
oscilloscope.
TRANSIENT R-C NETWORKS
In Chapter 10, the general solution for the
transient behavior of an R-C network with or
without initial values was developed.
 The resulting equation for the voltage across a
capacitor is repeated here for convenience:

TRANSIENT R-C NETWORKS
FIG. 24.18 Defining the parameters of
Eq. (24.6).
TRANSIENT R-C NETWORKS
FIG. 24.19 Example of the use of
Eq. (24.6).
TRANSIENT R-C NETWORKS
FIG. 24.20 Example
24.8.
TRANSIENT R-C NETWORKS
FIG. 24.21 yC and iC
for the network in
Fig. 24.20.
TRANSIENT R-C NETWORKS
FIG. 24.22
Example 24.9.
TRANSIENT R-C NETWORKS
FIG. 24.23 vC for the network in
Fig. 24.22.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
 The
square wave in Fig. 24.24 is a
particular form of pulse waveform.
 It has a duty cycle of 50% and an average
value of zero volts, as calculated as
follows:
FIG. 24.24 Periodic
square wave.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
FIG. 24.25 Raising the base-line voltage of a square
wave to zero volts.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
FIG. 24.26 Applying a periodic square-wave pulse train
to an R-C network.
T/2 > 5T
T/2 = 5T
T/2 < 5T
T/2 < 5T
FIG. 24.30 vC for T/2 << 5t or
T << 10t.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
FIG. 24.31 Example
24.10.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
FIG. 24.32 vC for the R-C network
in Fig. 24.31.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
FIG. 24.33 iC for the R-C network
in Fig. 24.31.
R-C RESPONSE TO SQUARE-WAVE
INPUTS
R-C RESPONSE TO SQUARE-WAVE
INPUTS
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
The X10 attenuator probe used with
oscilloscopes is designed to reduce the magnitude
of the input voltage by a factor of 10.
 If the input impedance to a scope is 1 MΩ, the
X10 attenuator probe will have an internal
resistance of 9 MΩ, as shown in Fig. 24.36.

OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.36 X10
attenuator probe.
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.37 Capacitive elements present in an
attenuator probe arrangement.
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.38 Equivalent network
in Fig. 24.37.
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.39 Thévenin equivalent for Ci in
Fig. 24.38.
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.40 The scope pattern for the
conditions in Fig. 24.38 with vt = 200 V peak.
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.41 Commercial compensated 10
: 1 attenuator probe. (Courtesy of
Tektronix, Inc.)
OSCILLOSCOPE ATTENUATOR
AND COMPENSATING PROBE
FIG. 24.42 Compensated attenuator
and input impedance to a scope,
including the cable capacitance.

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