TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

T RANSIENTS AND S TEP

R ESPONSES

ELCT222- Lecture Notes

University of S. Carolina

Spring 2012

O UTLINE

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RC transients charging

RC transients discharge

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RC transients Thevenin

P-SPICE

RL transients charging

RL transients discharge

Step responses.

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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.



FIG. 10.27 v

C during the charging phase.



FIG. 10.28 Universal time constant chart.



TABLE 10.3 Selected values of e -x .





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.



FIG. 10.29 Plotting the equation y

C

t/t ) versus time (t).

= E(1 – e -





FIG. 10.32 Revealing the short-circuit equivalent for the capacitor that occurs when the switch is first closed.



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

U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS

FIG. 10.34 Calculator key strokes to determine e -1.2

.


THE CHARGING PHASE


FIG. 10.35 Transient network for

Example 10.6.


THE CHARGING PHASE


FIG. 10.36 v

C versus time for the charging network in Fig. 10.35.


THE CHARGING PHASE


FIG. 10.37 Plotting the waveform in Fig. 10.36 versus time (t).


THE CHARGING PHASE


FIG. 10.38 i

C

and y

R network in Fig. 10.36.

for the charging


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.



PHASE

FIG. 10.39 (a) Charging network; (b) discharging configuration.



PHASE



For the voltage across the capacitor that is decreasing with time, the mathematical expression is:



PHASE

FIG. 10.40 y

C

, i

C

, and y

R contacts in Fig. 10.39(a).

for 5t switching between



PHASE

FIG. 10.41 v

C and i

C for the network in Fig.

10.39(a) with the values in Example 10.6.


THE DISCHARGING PHASE

T HE E FFECT OF ON THE R ESPONSE




FIG. 10.43 Effect of increasing values of C (with R constant) on the

charging curve for v

C

.




FIG. 10.44 Network to be analyzed in

Example 10.8.




FIG. 10.45 v

Fig. 10.44.

C and i

C for the network in




FIG. 10.46 Network to be analyzed in Example 10.9.

FIG. 10.47 The charging phase for the network in Fig.

10.46.




FIG. 10.48 Network in Fig. 10.47 when the switch is moved to position

2 at t = 1t

1

.




FIG. 10.49 v

C

Fig. 10.47.

for the network in




FIG. 10.50 i c

Fig. 10.47.

for the network in

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.

v c



V f



( V i



V f

) e

 t /



INITIAL CONDITIONS

FIG. 10.52 Example

10.10.

INITIAL CONDITIONS

FIG. 10.53 v in Fig. 10.52.

C and i

C for the network

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)(log using the TI-89 calculator.

e

2)

THÉVENIN EQUIVALENT: T =R

TH

C





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.


TH

C


10.11.


TH

C

FIG. 10.57 Applying Thévenin’s theorem to the network in Fig.

10.56.


TH

C

FIG. 10.58 Substituting the Thévenin equivalent for the network in Fig. 10.56.

THÉVENIN EQUIVALENT: T

=R

TH

C

FIG. 10.59 The resulting waveforms for the network in Fig. 10.56.


TH

C

FIG. 10.60

Example 10.12.

FIG. 10.61 Network in

Fig. 10.60 redrawn.


TH

C

FIG. 10.62 yC for the network in

Fig. 10.60.


TH

C


10.13.

THE CURRENT I

C









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: i

R

/R.

= v

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 I

C

FIG. 10.64 v

C

Example 10.14.

for

THE CURRENT I

C

FIG. 10.65 The resulting current i

C voltage in Fig. 10.64.

for the applied

I NDUCTORS

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.


PHASE

FIG. 11.31 Basic R-L transient network.

Remember, for an inductor v

L



L di

L dt


PHASE

FIG. 11.32 i

L

, y

L

, and y

R for the circuit in Fig. 11.31 following the closing of the switch.


PHASE

τ=L/R

If L is large, more flux needed for transient

If R is large, i

L is small, Δi

L small, fast.

FIG. 11.33 Effect of L on the shape of the i storage waveform.

L


PHASE

Fast times, open circuit

High Frequency, open circuit

Long times, short circuit

Low Frequency, short circuit

Because it’s a wound wire.

FIG. 11.34 Circuit in Figure 11.31 the instant the switch is closed.

Current cannot change instantly. Why?


PHASE

FIG. 11.35 Circuit in Fig.

11.31 under steady-state conditions.

FIG. 11.36 Series R-L circuit for Example 11.3.


PHASE

FIG. 11.37 i

L

Fig. 11.36.

and v

L for the network in

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 R l for the practical equivalent) and finding the resulting current through the element.

INITIAL CONDITIONS i

L



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 i

L

Fig. 11.39.

and v

L for the network in

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.


PHASE

FIG. 11.43 Network in Fig. 11.42 the instant the switch is opened.





Current wants to go to 0, but cannot, produces a spark due to large di

L

/dt

Introduce intentional “discharge” path.


PHASE

FIG. 11.42 Initiating the storage phase for an inductor by closing the switch.


PHASE

FIG. 11.43 Network in Fig. 11.42 the instant the switch is opened.

v

L

Apply KVL, and remember that i

L cannot change instantly. Why?

 

(1



R

2

R

1

) E exp(

 t



'

) i

L



E

R

1 exp(

 t



'

)

τ’=L/(R

1

+R

2

)

You can write this down. How?


PHASE


PHASE

FIG. 11.45 The various voltages and the current for the network in Fig. 11.44.

S TEP R ESPONSES

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 t

1 and ends just as abruptly at t

2

.

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 (t r

) and Fall Time (t f

)

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 t r and t f

.

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.


DUTY CYCLE

FIG. 24.11 Periodic pulse trains.

% of time voltage is high


DUTY CYCLE

FIG. 24.12

Example 24.3.


DUTY CYCLE

FIG. 24.13

Example 24.4.


DUTY CYCLE


24.5.

AVERAGE VALUE

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



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

I NSTRUMENTATION





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:


FIG. 24.18 Defining the parameters of

Eq. (24.6).


FIG. 24.19 Example of the use of

Eq. (24.6).



24.8.


FIG. 24.21 y

C and i for the network in

C

Fig. 24.20.


FIG. 24.22

Example 24.9.


FIG. 24.23 v

C

Fig. 24.22.

for the network in

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.


INPUTS

FIG. 24.25 Raising the base-line voltage of a square wave to zero volts.


INPUTS

FIG. 24.26 Applying a periodic square-wave pulse train to an R-C network.

T/2 > 5 T

T/2 = 5 T

T/2 < 5T

T/2 < 5T

FIG. 24.30 v

C

T << 10t.

for T/2 << 5t or


INPUTS


24.10.


INPUTS

FIG. 24.32 v

C in Fig. 24.31.

for the R-C network


INPUTS

FIG. 24.33 i

C in Fig. 24.31.

for the R-C network


INPUTS


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.



FIG. 24.36 X10 attenuator probe.



FIG. 24.37 Capacitive elements present in an attenuator probe arrangement.



FIG. 24.38 Equivalent network in Fig. 24.37.



FIG. 24.39 Thévenin equivalent for C i

Fig. 24.38.

in



FIG. 24.40 The scope pattern for the conditions in Fig. 24.38 with v t

= 200 V peak.



FIG. 24.41 Commercial compensated 10

: 1 attenuator probe. (Courtesy of

Tektronix, Inc.)



Compensating delay pre-delays the input signal

Allows scope electronics time to “catch up”

And remove the RC charging distortion we saw previously.

FIG. 24.42 Compensated attenuator and input impedance to a scope, including the cable capacitance.

TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

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