CIRCUIT WORKS - Wiley · V-Lab #20 Active Bandpass Filter Design. ... Display both circuit’s pole-zero patterns and compare the measured values of ... Circuit Works Virtual Lab

Circuit Works Virtual Lab Exercise - Copyright 2001, All rights Reserved.CircuitWorks, LLC.

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C I R C U I T W O R K S

Virtual Lab Exercises

Copyright 2001, CircuitWorks, LLC.www.cktworks.com All rights reserved.

Taking the work out of circuits!Circuit Works

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Taking the work out of circuits!

Circuit Works Virtual Lab Exercise - Copyright 2001, All rights Reserved.CircuitWorks, LLC

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Circuit Works Virtual Lab Exercises

Index

V-Lab #1 Zero-Input Response of a Series RC CircuitV-Lab #2 Zero-Input Response of a Series RL CircuitV-Lab #3 Zero-State Response of a Series RC Circuit (Step Input)V-Lab #4 Zero-Input Response of a Series RL Circuit (Step Input)V-Lab #5 Initial-State and Superposition (Series RC Circuit)V-Lab #6 Initial-State and Superposition (Series RL Circuit)V-Lab #7 Switched RC and RL CircuitsV-Lab #8 Pulse and Impulse ResponseV-Lab #9 RC Circuit - Sinusoidal Input SignalV-Lab #10 RL Circuit - Sinusoidal Input SignalV-Lab #11 RLC Circuit - Zero-Input Response (Overdamped)V-Lab #12 RLC Circuit - Zero-Input Response (Critically Damped and Underdamped)V-Lab #13 RLC Circuit - Zero-State Response (Step Input)V-Lab #14 RLC Circuit - Initial-State Response (Step Input)V-Lab #15 RLC Circuit - Pulse and Impulse ResponseV-Lab #16 RLC Circuit - Sinusoidal ResponseV-Lab #17 RC Circuit - Frequency Response and FiltersV-Lab #18 Periodic and Aperiodic Signal SpectraV-Lab #19 Filters and Rectified SignalsV-Lab #20 Active Bandpass Filter Design


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CircuitWorks Circuit Works Virtual Lab - Exercise 1Zero-Input Response of a Series RC Circuit

Objectives: The zero-input response (ZIR) of a linear circuit is its response to initial capacitorvoltages and inductor currents when no source is applied (zero-input conditions). The ZIR isaffected by the value of the circuit’s components, and its qualitative features can be predictedfrom the circuit’s pole-zero pattern (PZP). This exercise will use Circuit Works to examine the ZIRof the series RC circuit to explore the relationship between a circuit's component values, its pole-zero pattern, initial stored energy and its zero-input time domain response.

1. Select and configure the first-order series RC circuit shown below.Component values:

Primary SecondaryR: 5 kΩ 5 kΩC: 1 µF 1.15 µF

Initial conditions: v1: 5 VOutput: v1

2. Compare the pole-zero pattern of the capacitor voltage in the secondary circuit to that in the primarycircuit. How did changing C affect the pole-zero pattern?___ a. Increasing C caused the pole to move away from the origin.___ b. Increasing C caused the pole to move toward the origin.___ c. Increasing C caused the pole to move toward the origin and caused the zero to move up the j-axis.___d. Increasing C has no effect on the PZP.

3. For the component values assigned in Part 1 create a hardcopy of the screen image of the zero inputresponse and measure each circuit's time constant, τ. To do this, you will need to measure the time intervalover which the response decays by a certain amount (37%). Show your measurements on the screenimage. Display both circuit’s pole-zero patterns and compare the measured values of the circuit’s timeconstant to the value implied by the pole-zero pattern (i.e. calculate the time constant of the series RCcircuit from τ = 1/|s|, where s is the location of the transfer function's pole in the s-plane). For this circuit, s= -1/RC. Enter the measured and calculated values of each circuit's time constant below.

Calculated: τPrimary = __________ τSecondary = __________

Measured: τPrimary = __________ τSecondary = __________

4. Change the initial capacitor voltage to 20 Volts, select the unit step input signal (signal #21), and displaythe zero-input response of the capacitor voltage for the primary and secondary component values definedabove in Part 1. How did increasing C affect the zero-input response?___ a. Increasing C caused the circuit's zero-input response to decay to 0 in less time.___ b. Increasing C caused the circuit's zero-input response to decay to 0 more slowly.___ c. Increasing C does not affect the ZIR.

5. The circuit behavior observed in #4 occurs because:___ a. A smaller capacitor takes more time to charge.___ b. A larger resistor dissipates less energy.___ c. A larger capacitor discharges faster.


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6. The ZIR of the capacitor voltage is linear with respect to the initial capacitor voltage. How does doublingthe initial capacitor voltage affect the ZIR?___ a. Doubling the initial capacitor voltage scales the ZIR waveform by a factor of 0.5.___ b. Doubling the initial capacitor voltage scales the ZIR waveform by a factor of 2.___ c. The ZIR did not change.

7. To examine the relationship between the time constant and the initial slope of the zero-input response,make a hardcopy of the screen image of the ZIR of the capacitor voltage for the primary and secondarycomponents and annotate the image to show the initial slope of the response in each case. Compare themeasured value of the slope to the value of v1(0+)/τtc, with v1(0-) = 20 Volts.

8. How are the initial slope of the ZIR and the circuit’s time constant τtc related?:___ a. The initial slope does not depend on v1(0-).___ b. The magnitude of the initial slope increases with τtc.___ c. The magnitude of the initial sloped decreases with τtc.___ d. The initial slope of the ZIR equals τtc.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 2Zero-Input Response of a Series RL Circuit

Objectives: The zero-input response (ZIR) of a linear circuit is its response to initial capacitorvoltages and inductor currents when no source is applied (zero-input conditions). The ZIR isaffected by the value of the circuit’s components, and its qualitative features can be predictedfrom the circuit’s pole-zero pattern. This exercise will use Circuit Works to examine the ZIR of theseries RL circuit to explore the relationship between a circuit's component values, its pole-zeropattern, initial stored energy and its zero-input time domain response.

1. Select and configure the first-order series RL circuit as shown below.Component values:Primary SecondaryR: 5 kΩ 5 kΩL: 1 H 1.15 H

Initial conditions: i1: 5 mAOutput: i1

Compare the pole-zero pattern of the inductor current in the secondary circuit to that obtained for theprimary component values (display both simultaneously).

2. How did the changing L affect the pole-zero pattern?___ a. Increasing L caused the pole to move away from the origin.___ b. Increasing L caused the pole to move toward the origin.___ c. Increasing L caused the pole to move toward the origin and caused the zero to move up the j-axis.___ d. Increasing L has no effect on the PZP.

3. Create a hardcopy of the screen image of the ZIR and measure each circuit's time constant, τtc, bymeasuring the time interval over which the response decays by 37%. Show your measurements on thescreen image. Display both circuit’s pole-zero patterns and compare the measured values of the circuit’stime constant to the value implied by the pole-zero pattern (i.e. calculate the time constant of the series RLcircuit from τ = 1/|s|, where s is the location of the transfer function's pole in the s-plane). For this circuit, s= -R/L. Enter the measured and calculated values of the time constant below.

Calculated: τPrimary = __________ τSecondary = __________

Measured: τPrimary = __________ τSecondary = __________

4 . Change the initial inductor current to 20 mA, select the unit step input signal (signal 21), and display thezero-input response of the inductor current for the primary and secondary component values defined above inPart 1. How did increasing L affect the zero-input response?___ a. Increasing L caused the circuit's zero-input response to decay to 0 in less time.___ b. Increasing L caused the circuit's zero-input response to decay to 0 more slowly.___ c. Increasing L does not affect the ZIR.

5. The circuit behavior observed in #4 occurs because:___ a. A smaller inductor takes more time to energize.___ b. A larger resistor dissipates less energy.___ c. A larger inductor de-energizes faster.


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6. The ZIR of the inductor current is linear with respect to the initial inductor current. How does doubling theinitial inductor current affect the ZIR of the primary circuit.___ a. Doubling the initial inductor current scales the ZIR waveform by a factor of 0.5.___ b. Doubling the initial inductor current scales the ZIR waveform by a factor of 2.___ c. The ZIR did not change.

7. To examine the relationship between the time constant and the initial slope of the zero-input response,make a hardcopy of the screen image of the ZIR of the inductor current for the primary and secondarycomponents and annotate the image to show the initial slope of the response in each case. Compare themeasured value of the slope to the value of i1(0+)/τtc, with i1(0-) = 20 mA.

8. Describe the relationship between the initial slope of the ZIR and the circuit’s time constant:___ a. The initial slope does not depend on i1(0-).___ b. The magnitude of the initial slope increases with τtc.___ c. The magnitude of the initial sloped decreases with τtc.___ d. The initial slope of the ZIR equals τtc.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 3Zero-State Response of a Series RC Circuit (Step Input)

Objectives: The zero-state response (ZSR) of a circuit describes its response to an applied sourcewhen the circuit is initially at rest, i.e. the capacitors are discharged and the inductors are de-energized. In this condition the circuit has no internal stored energy, and the circuit is said to bein the “zero-state.” The zero-state response of a circuit can be modeled analytically to describethe relationships between the circuit's response and its component values and applied source.This exercise will display how the values of a circuit's components simultaneously affect its pole-zero pattern, natural frequency, and time-domain response waveforms.

1. Select and configure the first-order series RC circuit as shown below:

Component values:Primary Secondary

R: 100 kΩ 200 kΩC: 15 µF 15 µF


Input signal: Step (#21), A = 5

For these choices the circuit will be driven by a constant 5 volt source for t > 0.

Pole-Zero Patterns: Observe the pole-zero pattern of the transfer function relating the capacitor voltage(circuit output) to the applied voltage source (circuit input). What is the natural frequency (pole) of theresponse of the capacitor voltage?

s = ______ (Primary) s = ______ (Secondary)

2. Which circuit's response can be expected to take the shortest time to reach steady-state?___ Primary circuit ___ Secondary circuit

Capacitor Voltage – ZSR: Display the zero-state response of the capacitor voltage for the primary andsecondary components given in Part 1. Observe the waveform of the ZSR in each case. (Remember – thedisplay of the output is scaled by the factor Ys).

3. What are the initial values of the capacitor voltage response for the primary and secondary componentvalues?

v1(0+) = ______ (Primary) v1(0+) = ______ (Secondary)

4. What are the final values of the capacitor voltage response for the primary and secondary componentvalues?

v1(∞) = ______ (Primary) v1(∞) = ______ (Secondary)

Capacitor Voltage Response – Steady-State Values: The steady-state value of a circuit's response isthe value that remains after the circuit has been driven by a source for an arbitrarily long time. As anengineering “rule-of-thumb,” a circuit can be considered to reach steady-state after the source has been


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applied for a period of four time constants (provided that all of the circuit's poles are in the left half-plane). Ifthe input is a step function the response is within 2% of its final value when t > 4τmax where τmax is thecircuit's longest time constant.

5. Measure the steady-state value of the ZSR of the capacitor voltage for each set of components specifiedin Part 1. Indicate the location of your measurement on a hard copy of the screen image. What is thesteady-state capacitor voltage?

vss = ______ (Primary) vss = ______ (Secondary)

Response Rise Time: If the ZSR to a step input has a steady-state value that exceeds the initial value ofthe response, the “rise time” (tr) is used as a measure of how fast the circuit responds to the input. In thiscase the rise time is the time taken for the zero-state response to make the transition from 10% to 90% ofits steady-state value. Using the zero-state responses generated in Part 2, measure the rise time of thecapacitor voltage.

6. What are the measured values of the rise time?

tr = ______ (Primary) tr = ______ (Secondary)

Change the source signal delay time, τ, to have a value of 2 secs. Display the ZSR, and measure the risetime for the primary and secondary component values.

7. What effect does the change in τ have on the zero-state response?___ a. The steady-state value of the capacitor voltage increases.___ b. The steady-state value of the capacitor voltage decreases.___ c. The zero-state response curve is just translated on the time axis to create a delay of τ secs.___ d. The zero-state response is advanced by τ seconds.



Capacitor Current Response – ZSR: Observe the capacitor current, and display the pole-zero pattern ofthe transfer function relating its response to the applied voltage source, for the primary and secondarycomponent values given in Part 1.

9. What are the natural frequencies of the response of the capacitor current?


10. What is the location of the zero in the pole-zero pattern?


11. Explain the significance of the zero in the pole-zero pattern.___ a. The zero is located at the origin because the circuit cannot create a current when a step is applied.___ b. The zero is located at the origin because the circuit's transient response is zero.___ c. The zero located at the origin indicates that the "forced" component of the current to a step input will be zero.


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12. Which circuit's response can be expected to take the shortest time to reach steady- state?___ Primary circuit ___ Secondary circuit

Response Fall Time: If a circuit's ZSR has a constant steady-state value that is less than the initial value,then the fall time, tf, of the ZSR is the time required for the response, y(t), to make the transition from thevalue yi - 0.1(yi - yss) to the value yss + 0.1(yi -yss), i.e. the transition from within 10% of its initial value towithin 10% of its final value. Like the rise time, the fall time depends on the time constants of the circuit.

Examine the zero-state response of the capacitor current for both sets of component values and calculatethe fall time from the screen image data.

13. What are the measured values of the fall time?

tf = ______ (Primary) tf = ______ (Secondary)

Initial Slope of the Response : Now re-examine the ZSR generated in Part 1 for the capacitor voltage andnote the initial slope of the ZSR waveforms.

14. If τtc is the time constant of the circuit, how does the initial slope of the ZSR beginning at t = τ compareto the value of A/τtc?___ a. The initial slope equals A/τtc.___ b. The initial slope does not depend on A.___ c. The initial slope increases with τtc.___ d. The initial slope equals τtc /A.

15. Describe the relationship between the zero-state response of the capacitor voltage, the circuit's pole-zero pattern, and its component values.___ a. Increasing the value of R causes the circuit's pole to move farther from the

origin, and its time response to have a shorter time constant.___ b. Increasing the value of C causes the circuit's pole to move closer to the origin, and its time response to take less time to reach steady-state.___ c. Increasing R or C will cause the circuit's pole to move closer to the origin, and will reduce the value of the time constant of the response.___ d. Decreasing R or C will cause the circuit's pole to move farther from the origin, and will reduce the value of the time constant of the response.

Scaling the Amplitude of the Source. Increase A to 10 Volts, and generate the ZSR for τ = 0. Comparethe ZSR to that found in Part I (A = 5).

16. How does changing the value of A affect the ZSR?___ a. Decreasing A will cause the ZSR curve to increase by the same factor.___ b. The ZSR does not depend on A.___ c. Increasing A by a factor of 2 scales the response by a factor of 2.___ d. Increasing A by a factor of 2 scales the response by a factor of 4.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 4Zero-Input Response of a Series RL Circuit (Step Input)

Objectives: The zero-state response (ZSR) of a circuit describes its response to an applied sourcewhen the circuit is initially at rest, i.e. the capacitors are discharged and the inductors are de-energized. In this condition the circuit has no internal stored energy, and the circuit is said to bein the “zero-state.” The zero-state response of a circuit can be modeled analytically to describethe relationships between the circuit's response and its component values and applied source.This exercise will display how the values of a circuit's components simultaneously affect its pole-zero pattern, natural frequency, and time-domain response waveforms.

1. Select and configure the first-order series RL circuit (Circuit 25) as shown below:


R: 100 kΩ 200 kΩL: 20 H 20 H

Initial conditions: i1: 0 AOutput: i1Input signal: Step (#21), A = 5


Pole-Zero Patterns: Observe the pole-zero pattern of the transfer function relating the inductor current(circuit output) to the applied voltage source (circuit input). What is the natural frequency of the response ofthe inductor current?


2. Which circuit's response can be expected to take the shortest time to reach steady-state?___ Primary circuit ___ Secondary circuit

Inductor Current – ZSR: Display the zero-state response of the inductor current for the primary andsecondary components given in Part 1. Observe the waveform of the ZSR in each case. (Remember – thedisplay of the output is scaled by the factor Ys).

3. What are the initial values of the inductor current response for the primary and secondary componentvalues?

i1(0+) = ______ (Primary)i1(0+) = ______ (Secondary)

4. What are the final values of the inductor current response for the primary and secondary componentvalues?

i1(∞) = ______ (Primary) i1(∞) = ______ (Secondary)

Inductor Current Response – Steady-State Values: The steady-state value of a circuit's response is thevalue that remains after the circuit has been driven by a source for an arbitrarily long time. As an engineering“rule-of-thumb,” a circuit can be considered to reach steady-state after the source has been applied for a


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period of four time constants (provided that all of the circuit's poles are in the left half-plane). If the input is astep function the response is within 2% of its final value when t > 4τmax where τmax is the circuit's longesttime constant.

5. Measure the steady-state value of the ZSR of the inductor current for each set of components specified inPart 1. Indicate the location of your measurement on a hard copy of the screen image. What is thesteady-state inductor current?

iss = ______ (Primary) iss = ______ (Secondary)

Response Rise Time: If the ZSR to a step input has a steady-state value that exceeds the initial value ofthe response, the “rise time” (tr) is used as a measure of how fast the circuit responds to the input. In thiscase the rise time is the time taken for the zero-state response to make the transition from 10% to 90% ofits steady-state value. Using the zero-state responses generated in Part 2, measure the rise time of theinductor current.



Change the source signal delay time, τ, to have a value of 2 secs. Display the ZSR, and measure the risetime for the primary and secondary component values.

7. What effect does the change in τ have on the zero-state response?___ a. The steady-state value of the inductor current increases.___ b. The steady-state value of the inductor current decreases.___ c. The zero-state response curve is just translated on the time axis to create a delay of τ secs.___ d. The zero-state response is advanced by τ seconds.



Inductor Current Response– ZSR: Observe the inductor voltage, and display the pole-zero pattern of thetransfer function relating its response to the applied current source.

9. What is the natural frequency of the response of the inductor voltage?


10. What is the location of the zero in the pole-zero pattern?


11. Explain the significance of the zero in the pole-zero pattern.

___ a. The zero is located at the origin because the circuit cannot create an inductor voltage when a step is applied.___ b. The zero is located at the origin because the circuit's transient response is zero.___ c. The zero located at the origin indicates that the "forced" component of the inductor voltage to a step input will be zero.

12. Which circuit's response can be expected to take the shortest time to reach steady-state?


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___ Primary circuit ___ Secondary circuit

Response Fall Time: If a circuit's ZSR has a constant steady-state value that is less than the initial value,then the fall time, tf, of the ZSR is the time required for the response, t(t), to make the transition from thevalue yi - 0.1(yi - yss) to the value yss + 0.1(yi -yss), i.e. the transition from within 10% of its initial value towithin 10% of its final value. Like the rise time, the fall time depends on the time constants of the circuit.

Examine the zero-state response of the inductor current for both sets of component values and calculate thefall time from the screen image data.

13. What are the measured values of the fall time?

tf = ______ (Primary) tf = ______ (Secondary)

Initial Slope of the Response : Now re-examine the ZSR generated in Part 1 for the inductor current andnote the initial slope of the ZSR waveforms.

14. If τ is the time constant of the circuit, how does the initial slope of the ZSR beginning at t = τ compare tothe value of (A/R)/τtc?___ a. The initial slope equals (A/R)/τtc.___ b. The initial slope does not depend on A/R.___ c. The initial slope increases with τtc.___ d. The initial slope equals (τtc /A)/R.

15. Describe the relationship between the zero-state response of the inductor current, the circuit's pole-zeropattern, and its component values.___ a. Increasing the value of R causes the circuit's pole to move farther from the origin, and its time response to have a shorter time constant.___ b. Increasing the value of L causes the circuit's pole to move closer to the origin, and its time response to take less time to reach steady-state.___ c. Increasing R or L will cause the circuit's pole to move closer to the origin, and will reduce the value of the time constant of the response.___ d. Decreasing R or L will cause the circuit's pole to move farther from the origin, and will reduce the value of the time constant of the response.

Source Amplitude Scaling: For the primary component values, increase A to 10 Volts, and generate theZSR for τ = 0. Compare the ZSR to that found in Part I (A = 5).

16. Describe the effect of changing the value of A.___ a. Decreasing A will cause the ZSR curve to increase by the same factor.___ b. The ZSR does not depend on A.___ c. Increasing A by a factor of 2 scales the response by a factor of 2.___ d. Increasing A by a factor of 2 scales the response by a factor of 4.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 5Initial-State Response and Superposition

( Series RC Circuit)

Objectives: The initial-state response (ISR) of a circuit depends on the circuit's initial storedenergy and on its applied source. It is also related to the circuit's zero-input response (ZIR) and toits zero-state response (ZSR). This exercise will examine the relationship between the RC circuit'scomponent values, its pole-zero pattern, initial-state response, its zero-input response, and itszero-state response.

1. Select and configure the first-order series RC circuit (Circuit 1) as shown below:


R: 250 kΩ noneC: 2 µF none


Input signal: Step (#21), A = 10


Pole-Zero Patterns: Observe the pole-zero pattern of the transfer function relating the capacitor voltage(circuit output) to the applied voltage source (circuit input).

2. What is the circuit's natural frequency (pole)? s = ______ (Primary)

Superposition of ZIR and ZSR: Display the superimposed plots of the capacitor voltage's zero-inputresponse, zero-state response, and initial-state response.

3. Describe the relationship between the ISR, the ZIR and the ZSR:___ a. The initial-state response has a longer time constant.___ b. The initial-state response is formed by adding the zero-input response and the transient response.___ c. The initial-state response is the sum of the zero input response and the zero-state response.___ d. The initial-state response always has the same transient as the zero-state response.

Scaling the Input Signal: Generate a hard copy of the screen image of the initial-state response of thecircuit configured in Part 1. Then generate the screen image hardcopy of the ISR again, but with A = -10Volts.

4. Describe the effect of scaling the height of the input signal, A, by the factor -1:___ a. Scaling the height the input signal by a factor of -1 will cause the ISR to be scaled by the same factor.___ b. Scaling the height of the input signal by a factor of -1 will scale the ZSR component of the initial-state response by 1.___ c. Scaling the height of the step input signal by a factor of -1 will scale the ZSR component of the initial-state response by -1.


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___ d. Scaling the height of the step input signal by a factor of -1 will cause the transient response to decay faster.

Now reconfigure the circuit to have:

v1 = 5 V (Initial capacitor voltage)τ = 2 secs. (Source delay time)A = 5

In this configuration, the capacitor voltage will decay from t = 0 until the source is turned on at time t = τ = 2secs.

Generate a hardcopy of the screen image of the ISR of the capacitor voltage. Change the height of the inputstep signal to be A = 10 and again create a hardcopy of the screen image of the ISR.

5. What is the initial value of the response of the capacitor voltage when A = 5?

v1(0+) = _____

6. What is the initial value of the response of the capacitor voltage when A = 10?

v1(0+) = _____

7. What is the steady-state value of the capacitor voltage when A = 5?

v1(∞) = _____

8. What is the steady-state value of the capacitor voltage when A = 10?

v1(∞) = _____

9. How did changing in the value of the height of the source step affect the response?___ a. The response for t < τ was doubled in value.___ b. The ZSR component of the response for t > τ was doubled in value.___ c. The source was doubled, so the response was doubled.

Scaling the Initial Conditions: With A = 5, and τ = 2 secs, display the screen image copy of the ISR ofthe capacitor voltage when the initial capacitor voltage is 3 V, and again with an initial voltage of 6 V.

10. How did changing the initial capacitor voltage affect the ISR?___ a. The initial-state response does not depend on v1(0-) . ___ b. Doubling the value of v1(0-) scales the initial-state response by a factor of 2.___ c. Doubling the value of v1(0-) scales the zero-state response component of the initial-state response by a factor of 2.___ d. Doubling the value of v1(0-) scales the zero-input response component of the initial-state response by a factor of 2.

Scaling Initial Conditions and Source : With A = 5, τ = 2, v1(0-) = 3, generate the ISR of the capacitorvoltage. Then generate the ISR with A = 10, τ = 2, and v1(0-) = 6.

11. Describe the effect of doubling both the initial voltage and the source amplitude.___ a. Doubling the initial voltage and the source amplitude scales the initial-state response by a factor of 2.


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___ b. The zero-input response component is scales by a factor of 2, but the zero-state response component is scaled by a factor of 4.___ c. The zero-state response component is scaled by a factor of 2, but the zero-input response component is scaled by a factor of 4.___ d. Doubling the initial voltage and the source amplitude scales the zero-input response by a factor of 2, scales the zero-state response by a factor of 2, and scales the initial-state response by a factor of 4.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 6Initial-State Response and Superposition

(Series RL Circuit)

Objectives: The initial-state response (ISR) of a circuit depends on the circuit's initial storedenergy and on its applied source. It is also related to the circuit's zero-input response (ZIR) and toits zero-state response (ZSR). This exercise will examine the relationship between the RL circuit'scomponent values, its pole-zero pattern, initial-state response, its zero-input response, and itszero-state response.

1. Select and configure the first-order series RL circuit as shown below:


R: 2.5 kΩ noneL: 5 H none

Initial conditions: i1: 5 mAOutput: i1Input signal: Step (#21), A = 10


Pole-Zero Patterns: Observe the pole-zero pattern of the transfer function relating the inductor current(circuit output) to the applied voltage source (circuit input).

2. What is the circuit's natural frequency (pole)? s = ______ (Primary)

Superposition of ZIR and ZSR: Display the superimposed plots of the inductor current's zero-inputresponse, zero-state response, and initial-state response.

3. Describe the relationship between the ISR, the ZIR and the ZSR:___ a. The initial-state response has a longer time constant.___ b. The initial-state response is formed by adding the zero-input response and the transient response.___ c. The initial-state response is the sum of the zero input response and the zero-state response.___ d. The ISR always has the same transient as the ZSR.

Scaling the Input Signal: Generate a hard copy of the screen image of the initial-state response of thecircuit configured in Part 1. Then generate the screen image hardcopy of the ISR again, but with A = -10Volts.

4. Describe the effect of scaling the height of the input signal, A, by the factor -1:___ a. Scaling the height the input signal by a factor of -1 will cause the ISR to be scaled by the same factor.___ b. Scaling the height of the input signal by a factor of -1 will scale the ZSR component of the initial-state response by 1.___ c. Scaling the height of the step input signal by a factor of -1 will scale the ZSR component of the initial-state response by -1.___ d. Scaling the height of the step input signal by a factor of -1 will cause the transient


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response to decay faster.

Now reconfigure the circuit to have:

i1 = 5 mA (Initial inductor current)τ = 2 secs. (Source delay time)A = 5

In this configuration, the inductor current will decay from t = 0 until the source is turned on at time t = τ = 5secs.

Generate a hardcopy of the screen image of the ISR of the inductor current. Change the height of the inputstep signal to be A = 10 and again create a hardcopy of the screen image of the ISR.

5. What is the initial value of the response of the inductor current when A = 5?

v1(0+) = _____

6. What is the initial value of the response of the inductor current when A = 10?

v1(0+) = _____

7. What is the steady-state value of the inductor current when A = 5?

v1(∞) = _____

8. What is the steady-state value of the inductor current when A = 10?

v1(∞) = _____

9. How did the change in the value of the height of the source step affect the response?___ a. The response for t < τ was scaled by a factor of 2.___ b. The ZSR component of the response for t > τ was scaled by a factor of 2.___ c. The source was doubled, so the response was scaled by a factor of 2.

Scaling the Initial Conditions: With A = 5, and τ = 2 secs, display the screen image copy of the ISR ofthe inductor current when the initial inductor current is 3 mA, and again with an initial voltage of 6 mA.

10. How did the change in the initial inductor current affect the ISR?___ a. The initial-state response does not depend on i1(0-) . ___ b. Doubling the value of i1(0-) scales the ISR by a factor of 2.___ c. Doubling the value of i1(0-) scales the ZSR component of the ISR by a factor of 2.___ d. Doubling the value of i1(0-) scales the ZSR component of the ISR by a factor of 2.

Scaling Initial Conditions and Source : With A = 5, τ = 2, i1(0-) = 3 mA, generate the ISR of the inductorcurrent. Then generate the ISR with A = 10, τ = 2, and i1(0-) = 6 mA.

11. Describe the effect of doubling both the initial inductor current and the source amplitude.___ a. Doubling the initial current and the source amplitude scales the initial-state response by a factor of 2.___ b. The zero-input response component is scaled by a factor of 2, but the zero-state response component is scaled by a factor of 4.___ c. The zero-state response component is scaled by a factor of 2, but the zero-input


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response component is scaled by a factor of 4.___ d. Doubling the initial current and the source amplitude scales the ZIR by a factor of 2, scales the ZSR by a factor of 2, and scales the ISR by a factor of 4.


19

CircuitWorks

Circuit Works Virtual Lab -Exercise 7

Switched RC and RL Circuits

Objectives: Switches can be used to form the initial capacitor voltages and inductor currents in acircuit. In a typical situation, the switches will be in one configuration for t < 0 and in another for t> 0. The activity of the circuit before t = 0 creates the values of the circuit variables at t = 0-, andcontinuity conditions govern the capacitor voltages and inductor currents at t = 0+. These valuesand the source waveform for t > 0 account for the circuit's initial-state response.

Switched RC Circuit: In the switched RC circuit shown below the switches are assumed to have beenclosed long enough to allow the circuit to be in steady-state at t = 0-. Let the circuit have primarycomponent values of R1 = 4Ω, R2 = 4Ω, C = 2 F and K = 2.

Component ValuesPrimary SecondaryR1 4 Ω noneR2 4 Ω noneC 2 F noneK 2 none

1. What are the values of i1, v1, and v2 immediately before the switches change their configuration?

i1(0-) = ______ v1(0-) = ______ v2(0-) = ______

2. What are the values of i1, v1 and v2 immediately after the switches change their configuration?

i1(0+) = ______ v1(0+) = ______ v2(0+) = ______

3. For what value of K will the circuit have a pole in the left half-plane?

____ K = -5 ____ K < -1 ____ K > -1 ____ K = -2

Choose an initial value of v1 to correspond to the boundary conditions determined above and generate thescreen image copy of initial-state responses of i1, v1 and v2 to a unit step. Note their values at t=0+.

Switched RL Circuit: In the switched RL circuit (#83) the switch is assumed to have been closed longenough to allow the circuit to be in steady-state at t = 0-. Let the circuit have primary component values ofR1 = 2 Ω, R2 = 6 Ω, R3 = 1 Ω, and L = 2 H.

Component ValuesPrimary SecondaryR1 2 Ω noneR2 6 Ω noneR3 1 Ω noneL 2 H none


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4. What are the values of i1, i2, v1 and v2 immediately before the switch opens?

i1(0-) = ______ i2(0-) = ______ v1(0-) = ______ v2(0-) = ______

5. What are the values of i1, i2, v1, and v2 immediately after the switch opens?

i1(0+) = ______ i2(0+) = ______ v1(0+) = ______ v2(0+) = ______

6. Choose an initial values of i1 to correspond to the boundary conditions determined above and generatethe zero-input responses of i1, i2, v1 and v2. Note their values at t=0+.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 8Pulse and Impulse Response

Objectives: A circuit whose time constants are short compared to the width of an input pulse willpass the pulse. It will reject (attenuate) a pulse whose width is relatively short compared to itstime constants. In effect, a circuit having relatively long time constants cannot respond to rapidlychanging signals. This exercise will examine the relationship between the time constant of theseries RC circuit and its zero-state response to various pulse and impulse signals.

Series RC Circuit – Pulse and Impulse Response


R: 10 KΩ 10 kΩC: 1 µF 5 µF


Input signal: Pulse (#28), A = 10 ∆ = 0.05 sec

Examine the zero-state response of the capacitor voltage to a rectangular pulse (Signal #28) with amplitudeA = 10 and width ∆ = 0.05 sec. Repeat for a rectangular pulse having A = 10 and ∆ = 0.1 sec (Use thezoom feature if necessary).

1. What are the time constants of the circuits?

τ1 = ______ (Primary) τ2 = ______ (Secondary)

2. Which circuit has the shorter time constant?

___ Primary circuit ___ Secondary circuit

Compare the zero-state response of each circuit to the two pulses.

3. What can be said about how the two circuits respond to the pulses?___ a. The circuit with the longer time constant takes too long to respond to a long pulse, but can pass the short pulse.___ b. The circuit with the shorter time constant cannot respond quickly enough to pass the narrower pulse.___ c. The circuit with the shorter time constant passes the pulse with ∆ = 60 µs, and the circuit with the longer time constant passes both pulses.___ d. The circuit with the longer time constant passes the pulse with ∆ = 60 µs, and the circuit with the shorter time constant passes both pulses.

For the circuit components used in Part 1, generate the hardcopy images of the zero-state response to atriangular pulse (signal #30), and again to a sine pulse(signal #31), both having amplitude A = 10, and width∆ = 0.05 sec, and again with ∆ = 0.1 sec. The sine pulse is to have the same width.

4. How does the shape of the pulse affect whether it is passed by the circuit?___ a. The relationship between the pulse width and the circuit's time constant


22

determines whether a pulse will pass, not its shape.___ b. Pulses having a smoother shape will be passed by the circuit.___ c. Pulses with a smooth shape are passed by circuits that have a long time constant.

Apply an impulse of height A = 10 to the series RC circuit having the primary component values given inPart 1. Generate the screen image hardcopy of the zero-state response of the capacitor voltage with τ = 0.

5. For what value of the initial capacitor voltage will the circuit's zero-input response equal its impulseresponse?

v1(0-) = ______.

Verify your answer by generating the screen image copy of zero-input response.

6. Explain why the discontinuity in the capacitor voltage occurs. (Compare v1(0-) and v1(0+)).___ a. The resistor voltage may be discontinuous.___ b. The capacitor current impulse is integrated.___ c. The capacitor current impulse is differentiated.


23

Series RL Circuit – Pulse and Impulse Response


R: 5 KΩ 5 kΩL: 10 H 1 H

Output: i1

Examine the zero-state response of the inductor current to a rectangular pulse (signal #28) with amplitude A= 10 and width ∆ = 1 ms. Repeat for a rectangular pulse having A = 10 and ∆ = 15 ms (Use the zoomfeature if necessary).

7. What are the time constants of the circuits?

τ1 = ______ (Primary) τ2 = ______ (Secondary)

8. Which circuit has the shorter time constant?___ a. The circuit with the primary component values.___ b. The one with the shorter resistor.___ c. The circuit with the secondary component values.

Compare the zero-state response of each circuit to the two pulses.

9. What can be said about how the two circuits respond to the pulses?___ a. The circuit with the longer time constant takes to long to respond to a long pulse, but can pass the short pulse.___ b. The circuit with the shorter time constant cannot respond quickly enough to pass the narrower pulse.___ c. The circuit with the shorter time constant passes the pulse with ∆ = 15 ms, and the circuit with the longer time constant passes both pulses.___ d. The circuit with the longer time constant passes the pulse with ∆ = 15 ms, and the circuit with the shorter time constant passes both pulses.

For the circuit components used in Part 7, generate the hardcopy images of the zero-state response to atriangular pulse (signal #30), and a sine pulse (signal #31) both having amplitude A = 10, and width ∆

= 1 ms, and again with ∆ = 15 ms. The sine pulse is to have the same width as the other pulses.

10. How does the shape of the pulse affect whether it is passed by the circuit?___ a. The relationship between the pulse width and the circuit's time constant determines whether a pulse will pass, not its shape.___ b. Pulses having a smoother shape will be passed by the circuit.___ c. Pulses with a smooth shape are passed by circuits that have a long time constant.___ d. Pulses with fewer electrons are easier to filter.

Apply an impulse of height A = 10 to the series RL circuit having the primary component values given in Part1. Generate the screen image hardcopy of the zero-state response of the inductor current with delay time τ= 0.


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11. For what value of the initial inductor current will the circuit's zero-input response equal its impulseresponse?

i1(0-) = ______.

Verify your answer by generating the screen image copy of zero-input response.

12. Explain why the discontinuity in the inductor current occurs. (Compare i1(0-) and i1(0+))___ a. The resistor voltage may be discontinuous.___ b. The inductor voltage impulse is integrated.___ c. The inductor voltage impulse is differentiated.


25

CircuitWorks

Circuit Works Virtual Lab - Exercise 9RC Circuit - Sinusoidal Input

Objectives: This exercise examines the time-domain zero-state and initial-state response of theseries RC circuit to a sinusoidal input signal.


R: 20 KΩ 20 kΩC: 0.1 µF 0.2 µF


Part I. Zero-State Response . For the primary and secondary component values, generate the screenimage hardcopy of the zero-state response of the capacitor voltage to a switched sinusoidal voltage source(signal #26) having A = 5, ωo = 3000 rad/s, φ = 0, and τ = 0. Examine the pole-zero patterns and the Bodeplots of the primary and secondary circuit.

1. What is the "period" of the input signal? Tin = ______

2. How long does it take for the response to reach steady state?

tss = ______ (Primary) tss = ______ (Secondary)

3. How many cycles of the output waveform are displayed before steady-state is reached?

Nss = ______ (Primary) Nss = ______ (Secondary)

4. What is the period of the output signal in steady-state?

Tss = ______ (Primary) Tss = ______ (Secondary)

5. What is the magnitude of the circuit's transfer function between the input voltage and the capacitor voltageat the frequency of the source?

|H(jωo)| = ______ (Primary) |H(jωo)| = ______ (Secondary)

6. What is the phase angle difference between the steady-state output sinusoid and the input sinusoid?

θ = ______ (Primary) θ = ______ (Secondary)

7. Repeat Part 2 using ωo = 5000 rad/s.

8. What is the period of the input signal? Tin = ______

9. How long does it take for the response to reach steady-state?



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11. What is the period of the signal in steady-state?


12. What is the magnitude of the circuit's transfer function between the input voltage and the capacitorvoltage at the frequency of the source?


13. At which frequency does the input signal to the primary circuit undergo the greatest attenuation of itssteady-state amplitude?

___ ωo = 3000 rad/s. ___ ωo = 5000 rad/s.

14. What is the physical basis for your answer in 13?___ a. The lower frequency input sinusoid generates relatively little capacitor voltage compared to the higher frequency sinusoid because its switches polarity too slowly.___ b. The higher frequency input sinusoid generates relatively greater capacitor voltage compared to the higher frequency sinusoid because its switches polarity too quickly.___ c. The lower frequency input sinusoid generates relatively greater capacitor voltage compared to the higher frequency sinusoid because its switches polarity more slowly, thereby allowing the capacitor to charge to a higher value before discharging.___ d. The higher frequency input sinusoid generates relatively greater capacitor voltage compared to the lower frequency sinusoid because its switches polarity more quickly, thereby allowing the capacitor to charge to a higher value before discharging.___ e. The higher frequency input sinusoid has no staying power.



Part II. Initial-State Response . For the primary and secondary component values given in Part I, generatethe screen image hardcopy of the initial-state response of the capacitor voltage to a switched sinusoidalsource (signal # 26) having A = 10, ωo = 3000 rad/s, φ = 0, and τ = 3 ms. Use an initial capacitor voltage ofv1(0-) = 20 Volts.

16. What is the value of the capacitor voltage when the source turns on?

v1(τ) = ______ (Primary) v1(τ) = ______ (Secondary)

Part III. Repeat Part II with initial capacitor voltage v1(0-) = - 20 Volts.


27



Part IV. Repeat Part III with the source delay time of τ = 6ms.




28

CircuitWorks

Circuit Works Virtual Lab - Exercise 10RL Circuit - Sinusoidal Input Signal

Objectives: This exercise examines the time-domain zero-state and initial-state response of theinductor current in a series RL circuit to a sinusoidal input signal.


R: 20 KΩ 20 kΩL: 40 H 80 H

Output: i1Initial conditions: v1: 5 V

i1 = 0 A

Part I. Zero-State Response. Examine the pole-zero patterns and generate a screen image hardcopy ofthe zero-state response of the inductor current to a switched sinusoidal voltage source (signal #26) having A= 5, ωo = 400 rad/s, φ = 0, and τ = 0.

1. What is the "period" of the input signal?

Tin = ______

2. How long does it take for the response to reach steady -state?




4. What is the period of the output signal in steady-state?


5. What is the magnitude of the circuit's transfer function between the input voltage and inductor current atthe frequency of the source?




7. Repeat the above question using ωo = 600 rad/s.

8. What is the period of the input signal?

Tin = ______


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9. How long does it take for the response to reach steady-state?






12. What is the magnitude of the circuit's transfer function between the input voltage and inductor current atthe frequency of the source?



a. ___ ωo = 400 rad/s. b. ___ ωo = 600 rad/s.

14. What is the physical basis for your answer in 13?

___ a. The lower frequency input sinusoid generates relatively little inductor current compared to the higher frequency sinusoid because its switches polarity too slowly.___ b. The higher frequency input sinusoid generates relatively greater inductor current compared to the higher frequency sinusoid because its switches polarity too quickly.___ c. The lower frequency input sinusoid generates relatively greater inductor current compared to the higher frequency sinusoid because its switches polarity more slowly, thereby allowing the capacitor to charge to a higher value before discharging.___ d. The higher frequency input sinusoid generates relatively greater inductor current compared to the lower frequency sinusoid because its switches polarity more quickly, thereby allowing the capacitor to charge to a higher value before discharging.___ e. The higher frequency input sinusoid has no staying power.



Part II. Initial-State Response . For the primary and secondary component values given in Part I, generatethe screen image hardcopy of the initial-state response of the inductor current to a switched sinusoidalsource (signal # 26) having A = 10, ωo = 400 rad/s, φ = 0, and τ = 3 ms. Use an initial inductor current ofi1(0-) = 20 mA. (Also examine the superposition waveforms).

16. What is the value of the inductor current when the source turns on?

i1(τ) = ______ (Primary) i1(τ) = ______ (Secondary)

Part III. Repeat Part II with initial inductor current i1(0-) = - 20 mA.


30



Part IV. Repeat Part III with the source delay time of τ = 6ms.




31

CircuitWorks

Circuit Works Virtual Lab - Exercise 11RLC circuit - Zero-Input Response (Overdamped)

Objectives: The component values of a second order circuit determine whether its response willbe overdamped, critically damped or underdamped. This exercise examines the relationshipbetween the component values of the overdamped series RLC circuit and its pole-zero pattern andzero-input response (ZIR). It also examines how the energy initially stored in the inductor andcapacitor affect the zero-input response.

Part I. Pole-Zero Patterns and Time-Domain Response . Select and configure the series RLC circuit asshown below. Examine the screen image of the pole-zero patterns and zero-input responses of the circuitfor both sets of components.


R: 4 kΩ 4 kΩC: 1 F 0.5 FL: 1 H 1 H

Output: v1

1. What are the natural frequencies and time constants of the circuit?

Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___

2. Classify the response of the primary and secondary circuits.

Primary Secondary___ Overdamped ___ Overdamped___ Critically damped ___ Critically damped___ Underdamped ___ Underdamped

3. How did decreasing the value of C (from 1 F to 0.5 F) affect the pole-zero pattern?___ a. The poles became complex.___ b. The poles moved closer together on the real axis.___ c. The poles moved farther apart on the real axis.___ d. The poles became coincident.

4. How did decreasing the value of C affect the zero-input response?___ a. Decreasing C caused the response to oscillate.___ b. Decreasing C caused the response to stop oscillating.___ c. Decreasing C caused the response to reach steady-state sooner.___ d. Decreasing C caused the response to reach steady-state later.

5. Why did the response change in the manner observed?___ a. A smaller C requires less charge to establish a voltage on the capacitor.


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___ b. A smaller C requires more charge to establish a voltage on the capacitor.___ c. A smaller C causes less current to flow.

Part II. Initial and Steady-State Values. Examine the screen image copy of the inductor voltage for bothsets of component values.

6. What are the initial values of the capacitor voltage, v1(0+) and the inductor voltage, v3(0+), responses?

v1(0+) = ______ v3(0+) = ______ (Primary)v1(0+) = ______ v3(0+) = ______ (Secondary)

7. What are the steady-state values of the capacitor and inductor voltages?

v1(∞) = ______ v3(∞) = ______ (Primary)v1(∞) = ______ v3(∞) = ______ (Secondary)

Part III. Effect of Initial Capacitor Voltage . For the same primary and secondary circuit componentvalues as in Part I, and again with an initial inductor current of i1(0-) = 0, increase the initial capacitor voltageto v1(0-) = 10 Volts. Observe the ZIR of the capacitor voltage.

8. What effect did doubling the initial capacitor voltage have on the capacitor voltage's ZIR?___ a. The capacitor voltage waveform was scaled by a factor of 4 .___ b. The capacitor voltage waveform was scaled by a factor of 2.___ c. The capacitor voltage was scaled by a factor of 0.5.

9. What can be said about the effect of changing v1(0-)?___ a. Changing v1(0-) does not affect the ZIR.___ b. Increasing v1(0-) will decrease the ZIR.___ c. Increasing v1(0-) will increase the ZIR by 1/2.___ d. Increasing v1(0-) will increase the ZIR by the same amount if i1(0-) = 0.

10. How does a change in v1(0-) affect the initial inductor voltage response, v3(0+)?___ a. Changing v1(0-) will not affect the initial inductor voltage.___ b. Increasing v1(0-) will scale the initial inductor voltage by the same amount.___ c. Increasing v1(0-) will scale the initial inductor voltage by 1/2.___ d. Increasing v1(0-) will scale the initial inductor voltage by the same amount.

Part IV. Effect of Initial Inductor Current. For the same primary and secondary circuit component valuesas defined in Part I, set the initial capacitor voltage to v1(0-) = 0, and assign an initial inductor current of i1(0-)= 2 Amps. Create the screen image copy of the ZIR of the capacitor voltage for both sets of components.Now change the initial inductor current to v1(0-) = 4 Amps while keeping v1(0-) fixed.

11. How did doubling the initial inductor current affect the ZIR of the capacitor voltage?

___ a. The ZIR of the capacitor voltage was scaled by a factor of 4.___ b. The ZIR of the capacitor voltage was scaled by a factor of 2.___ c. The ZIR of the capacitor voltage was scaled by a factor of 1/2.

12. How does a change in i1(0-) affect the initial inductor voltage, v3(0-)?___ a. Changing i1(0-) will not affect the initial inductor voltage.___ b. Increasing i1(0-) will decrease the initial inductor voltage.___ c. Increasing i1(0-) will increase the initial inductor voltage by 1/2.___ d. Increasing i1(0-) will increase the initial inductor voltage by the same amount.


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13. What can be said about the effect of changing i1(0-)?___ a. Changing i1(0-) does not affect the ZIR.___ b. Increasing i1(0-) will decrease the ZIR.___ c. Increasing i1(0-) will increase the ZIR by 1/2.___ d. Increasing i1(0-) will increase the ZIR by the same amount if v1(0-) = 0.

Part V. Effect of Changing v1(0-) and i1(0

-). For the same primary and secondary circuit componentvalues as in Part I, but with initial capacitor voltage v1(0-) = 5 Volts and initial inductor current i1(0-) = 2Amps, observe the ZIR of the capacitor voltage.

14. How did applying both the initial capacitor voltage and the initial inductor current affect the response?___ a. The capacitor voltage was scaled by a factor of 4.___ b. The ZIR of the capacitor voltage is now the sum of its ZIR to the initial capacitor voltage and its ZIR to the initial inductor current.___ c. The ZIR of the capacitor voltage is now the difference of its ZIR to the initial capacitor voltage and its ZIR to the initial inductor current.___ d. The capacitor voltage is scaled by a factor of 1/2.

Part VI. Effect of Changing v1(0-) and i1(0

-) For the same component values as in Part V, increase v1(0-)to 10 V and increase i1(0-) to 4 A. Observe the ZIR of the capacitor voltage.

15. How did doubling both the initial capacitor voltage and the initial inductor current affect the waveform ofthe zero-input response?___ a. The ZIR of the capacitor voltage was scaled by a factor of 4.___ b. The ZIR of the capacitor voltage was scaled by a factor of 2.___ c. The ZIR of the capacitor voltage was scaled by a factor of 1/2.___ d. The effects cancelled each other.

16. What can be said about simultaneous changes in v1(0-) and i1(0-)?___ a. Changing v1(0-) and i1(0-) causes the ZIR to increase.___ b. Increasing v1(0-) and i1(0-) by the same factor causes the ZIR to decrease by the same factor.___ c. Increasing v1(0-) and i1(0-) by the same factor causes the ZIR to increase by the same factor.___ d. Changing v1(0-) and i1(0-) causes the ZIR to decrease.___ e. Increasing v1(0-) and i1(0-) by the same factor has no effect because the changes cancel each other.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 12RLC Circuit - Zero-Input Response (Critically Damped and

Underdamped)

Objectives: The component values of a second order circuit determine whether its response willbe overdamped, critically damped or underdamped. This exercise examines the relationshipbetween the component values of the critically damped and underdamped series RLC circuits andtheir pole-zero patterns and zero-input responses (ZIR). It also examines how the energy initiallystored in the inductor and capacitor affect the zero-input response.

Part I. Pole-Zero Patterns and Time Constants. Select and configure the series RLC circuit as shownbelow.


R: 4 Ω 4 ΩC: 0.5 F 0.25 FL: 1 H 1 H

Output: v1

Initial conditions: v1: 5 Vi1: 2 A


Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___


Primary: ___ Overdamped ___ Critically Damped ___ UnderdampedSecondary: ___ Overdamped ___ Critically damped ___ Underdamped

3. How did decreasing the value of C (from 0.5 F to 0.25 F) affect the pole-zero pattern?___ a. The poles became complex.___ b. The poles moved closer together on the real axis.___ c. The poles moved farther apart on the real axis.___ d. The poles became coincident.___ e. No effect.

Part II. Zero-Input Response to Initial Conditions. Display the ZIR of the capacitor voltage. Consider theeffect of changing C from 0.5 F to 0.25 F.



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5. Why did the response change in the manner observed?___ a. A smaller C requires less charge to establish a voltage on the capacitor.___ b. A smaller C requires more charge to establish a voltage on the capacitor.___ c. A smaller C causes less current to flow.___ d. A smaller C causes more current to flow.

Part III. Sensitivity to Changes in C. Change the value of the secondary capacitor to have Cprimary = 0.05F


Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___



8. How did decreasing the value of C (from 0.25 F to 0.05 F) affect the pole-zero pattern?___ a. The poles became complex.___ b. The poles moved closer together on the real axis.___ c. The poles moved farther apart on the real axis.___ d. The poles became coincident.___ e. No effect


10. How does decreasing the value of C affect the steady-state capacitor voltage?___ a. Decreasing C causes the steady-state voltage to increase.___ b. Decreasing C causes the steady-state voltage to decrease.___ c. Decreasing C has no effect on the steady-state voltage.

11. Examine the pole-zero pattern; what are the primary circuit’s damping factor and damped frequency ofoscillation?

α = __________ ωd _________

Part IV. Sensitivity to Changes in C. Reconfigure the circuit to have C1 = 0.05 F and C2 = 0.01 F.Examine the pole-zero patterns and the ZIR of both circuits.


Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______


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Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___



14. What are the steady-state values of the capacitor and inductor voltages?


15. What is the steady-state value of the inductor current?

i1(∞) = ______ (Primary)i1(∞) = ______ (Secondary)

16. What are the circuit’s damping factors and damped frequency of oscillation?

α = ______ ωd = _____ (Primary)α = ______ ωd = _____ (Secondary)

17. How did decreasing the value of C (from 0.05 F to 0.01 F) affect the pole-zero pattern?___ a. The poles became complex.___ b. The poles moved closer together on the real axis.___ c. The poles moved farther apart on the real axis.___ d. The poles became coincident.___ e. The poles moved farther from the real axis.___ f. The poles moved closer to the real axis.___ g. No effect

19. What effect did scaling the value of C have on the zero-input response?___ a. Decreasing C caused the damped frequency of oscillation to increase.___ b. Decreasing C caused the damped frequency of oscillation to decrease.___ c. Decreasing C caused the damping factor to increase.___ d. Decreasing C caused the damping factor to decrease.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 13RLC Circuit - Zero-State Response (Step Input)

Objectives: The zero-state response (ZSR) of a circuit is its response to an applied source when theenergy initially stored in the circuit's inductors and capacitors is zero. The ZSR is due entirely tothe applied source. This exercise examines the ZSR of the series RLC circuit when a step inputsignal is applied. It relates changes in the circuit's components to changes in its pole-zero patternand its time domain waveforms. It also examines the response's rise time.

Part I. Pole-Zero Patterns and Time Constants. Select and configure the series RLC circuit as shownbelow.


R: 4 Ω 4 ΩC: 1 F 0.5 FL: 1 H 1 H

Output: v1


1. What are the natural frequencies and time constants of the circuits?

Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Part II. Zero-State Response – Step Input. Select the step input signal (#21) and assign its amplitudeto be A = 10 V. Examine the pole-zero pattern and the zero-state response of the capacitor voltage.



3. How did decreasing the value of C (from 0.5 F to 0.25 F) affect the pole-zero pattern?___ a. The poles became complex.___ b. The poles moved closer together on the real axis.___ c. The poles moved farther apart on the real axis.___ d. The poles became coincident.___ e. No effect.




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5. What are the steady-state values of the capacitor voltage, v1(0+) and the inductor voltage, v3(0+),responses?


6. What are the initial and steady-state values of the zero-state response of the inductor current?

i1(0+) = ______ (Primary) i1(∞) = ______ (Primary)i1(0+) = ______ (Secondary) i1(∞) = ______ (Secondary)

Part III. Effect of Scaling the Amplitude of the Source. Double the source amplitude (Use A = 20 V).Examine the pole-zero pattern and the zero-state response of the capacitor voltage.



8. What are the steady-state values of the capacitor voltage, v1(0+) and the inductor voltage, v3(0+),responses?


9. What is the initial value of the zero-state response of the inductor current?

i1(0+) = ______ (Primary) i1(0+) = ______ (Secondary)

10. What is the steady-state value of the zero-state response of the inductor current?

i1(∞) = ______ (Primary) i1(∞) = ______ (Secondary)

11. How did doubling the value of A affect the initial values of the circuit's voltages and current responses?___ a. The voltages increased, current decreased.___ b. Voltage and current decreased.___ c. Voltage and current were doubled.___ d. Voltage and current were decreased by a factor of 2.___ e. Voltage and current increased by a factor of 4.___ f. Only the capacitor voltage doubled.___ g. Only the inductor voltage doubled.___ h. Only the current doubled.

12. How did doubling the value of A affect the steady-state values of the circuit's voltages and current zero-state response?___ a. The voltages increased, current decreased.___ b. The steady-state voltage and current decreased.___ c. The steady-state voltage and current were doubled.___ d. The steady-state voltage and current were decreased by a factor of 2.___ e. The steady-state voltage and current increased by a factor of 4.

Part IV. Overshoot and Oscillation. Reconfigure the circuit to have the values shown below:

Circuit Works Virtual Lab Exercise - Copyright 2001, All rights Reserved.CircuitWorks, LLC.

39


R: 4 Ω 4 ΩC: 0.25 F 0.05 FL: 1 H 1 H

Examine the pole-zero pattern and the zero-state response of the capacitor voltage to the step input with A= 20 V.



14. Why does the pole-zero pattern of the transfer function of the inductor voltage have a zero at s = 0?___ a. The circuit is underdamped.___ b. The inductor's steady-state voltage is zero for a step input.___ c. Because electrons only move when the pole is positive and real.

15. How long does it take the capacitor voltage to reach steady-state?


16. How many cycles of oscillation occur before the secondary circuit's capacitor voltage response reachessteady-state?

N = ______ (Primary) N = ______ (Secondary)

17. What are the calculated (Tc) and measured periods (Tm) of the damped oscillation of the secondarycircuit's capacitor voltage response?

Tc = ______ Tm = ______

18. What are the measured rise time and % overshoot of the capacitor voltage response in the secondarycircuit?

tr = ______ %os = ______

Part VI. Overshoot and Oscillation. Reconfigure the circuit to have the values shown below:


R: 4 Ω 4 ΩC: 0.25 F 0.01FL: 1 H 1 H

Examine the pole-zero pattern and the zero-state response of the capacitor voltage to the step input with A= 20 V.



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20. Why does the pole-zero pattern of the transfer function of the inductor voltage have a zero at s = 0?

___ a. The circuit is underdamped.___ b. The inductor's steady-state voltage is zero for a step input.___ c. Because electrons only move when the pole is positive and real.






Tc = ______ Tm = ______


tr = ______ %os = ______


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CircuitWorks

Circuit Works Virtual Lab - Exercise 14RLC Circuit - Initial-State Response (Step Input)

Objectives: The initial-state response (ISR) of a circuit is its response to both an applied sourceand the energy initially stored in the circuit's capacitors and inductors. This exercise examinesthe ISR of the series RLC circuit when a step input signal is applied. It relates changes in thecircuit's components to changes in its pole-zero pattern and its time domain waveforms.


R: 4 Ω 4 ΩC: 1 F 0.5FL: 1 H 1 H

Output: v1


Part 1. Zero-State Response – Step Input. Select the step input signal (#21) and assign its amplitude tobe A = 10 V. Examine the pole-zero pattern and the initial-state response of the capacitor voltage.


Primary: s1 = ______ s2 = ______τ1 = ______ τ2 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___



3. What are the initial values of the circuit’s capacitor voltage, v1(0+), inductor voltage, v3(0+), and inductorcurrent, i1(0+), responses?

v1(0+) = ______ v3(0+) = ______ i1(0+) = ______ (Primary)v1(0+) = ______ v3(0+) = ______ i1(0+) = ______ (Secondary)

4. What are the steady-state values of the circuit’s capacitor voltage, v1(∞), inductor voltage, v3(∞), andinductor current, i1(∞), responses?

v1(∞) = ______ v3(∞) = ______ i1(∞) = ______ (Primary)v1(∞) = ______ v3(∞) = ______ i1(∞) = ______ (Secondary)

Part II. Effect of Scaling the Amplitude of the Source. Double the source amplitude (Use A = 20 V).Examine the pole-zero pattern and the initial-state response of the capacitor voltage.


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5. What are the initial values of the circuit’s capacitor voltage, v1(0+), inductor voltage, v3(0+), and inductorcurrent, i1(0+), responses?

v1(0+) = ______ v3(0+) = ______ i1(0+) = ______ (Primary)v1(0+) = ______ v3(0+) = ______ i1(0+) = ______ (Secondary)

6. What are the steady-state values of the circuit’s capacitor voltage, v1(∞), inductor voltage, v3(∞), andinductor current, i1(∞), responses?

v1(∞) = ______ v3(∞) = ______ i1(∞) = ______ (Primary)v1(∞) = ______ v3(∞) = ______ i1(∞) = ______ (Secondary)

7. How did doubling the value of A affect the initial values of the circuit's voltages and current responses?___ a. The initial voltages increased; the current decreased.___ b. The initial voltage and current decreased.___ c. The initial voltage and current were doubled.___ d. The initial voltage and current were decreased by a factor of 2.___ e. The initial voltage and current increased by a factor of 4.___ f. Only the initial capacitor voltage doubled.___ g. Only the initial inductor voltage doubled.___ h. Only the initial inductor current doubled.___ i. Only the zero-state response component of the response was halved.___ j. Only the zero-state response component of the response was doubled.

8. How did doubling the value of A affect the steady-state values of the circuit's voltages and current zero-state response?___ a. The steady-state voltages increased; the current decreased.___ b. The steady-state voltage and current decreased.___ c. The steady-state voltage and current were doubled.___ d. The steady-state voltage and current were decreased by a factor of 2.___ e. The steady-state voltage and current increased by a factor of 4.

Part III. Overshoot and Oscillation. Reconfigure the circuit to have the values shown below:


R: 4 Ω 4 ΩC: 0.25 F 0.05FL: 1 H 1 H

Output: v1, v3, i1Initial conditions: v1: 5 V

i1: 2 A

Examine the pole-zero pattern and the initial-state response of the capacitor voltage, inductor voltage, andinductor current to the step input with A = 20 V.

9. Classify the response of the primary and secondary circuits.Primary: ___ Overdamped ___ Critically Damped ___ UnderdampedSecondary: ___ Overdamped ___ Critically damped ___ Underdamped

10. Why does the pole-zero pattern of the transfer function of the inductor voltage have a zero at s = 0?


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___ a. The capacitor’s steady-state voltage is zero.___ b. The circuit is underdamped.___ c. The inductor’s steady-state voltage is zero for a step input.






Tc = ______ Tm = ______


tr = ______ %os = ______

Part IV. Overshoot and Oscillation. Reconfigure the circuit to have the values shown below:


R: 4 Ω 4 ΩC: 0.25 F 0.01FL: 1 H 1 H


i1: 2 A

Examine the pole-zero pattern and the initial-state response of the capacitor voltage, inductor voltage, andinductor current to the step input with A = 20 V.

15. How long does it take the circuit to reach steady-state?





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Tc = ______ Tm = ______


tr = ______ %os = ______

Part V. Effect of Source Delay Time. Using the parameters and initial conditions shown below (same asPart I), apply the same step input signal, but with τ = 0.5 secs. Examine the screen images of the initial-state response of the capacitor voltage, inductor voltage, and inductor current.


R: 4 Ω 4 ΩC: 1 F 0.5FL: 1 H 1 H


i1: 2 A

19. For the primary circuit components, what are the values of the capacitor voltage, inductor voltage andinductor current immediately before and immediately after the source is turned on?

v1(τ-) = _______ v1(τ+) = _______v3(τ-) = _______ v3(τ+) = _______i1(τ-) = _______ i1(τ+) = _______

20. At what time does the capacitor voltage reach its steady-state value?

tss = _________ (Primary) iss = __________ (Secondary)

Part VI. Effect of Source Delay Time. Using the parameters and initial conditions shown below (same asPart I), apply the same step input signal, but with a delay time of τ = 0.5 secs. Examine the screen imagesof the initial-state response of the capacitor voltage, inductor voltage, and inductor current.


R: 4 Ω 4 ΩC: 0.25 F 0.05FL: 1 H 1 H


i1: 2 A

21. For the primary circuit components, what are the values of the capacitor voltage, inductor voltage andinductor current immediately before and immediately after the source is turned on?


45

v1(τ-) = _______ v1(τ+) = _______v3(τ-) = _______ v3(τ+) = _______i1(τ-) = _______ i1(τ+) = _______

22. At what time does the capacitor voltage reach its steady-state value?

tss = _________ (Primary) tss = __________ (Secondary)

Part VII. Ringing Response . An underdamped second-order circuit is said to "ring" when its oscillation islightly damped. Several cycles of oscillation occur before the circuit reaches steady-state. Reconfigure thecircuit to have the parameters shown below:


R: 4 Ω 4 ΩC: 0.001 F 0.05FL: 1 H 1 H


i1: 2 A

Examine the pole-zero pattern and the initial-state response of the capacitor voltage, inductor voltage, andinductor current of the primary waveform. It will be necessary to “zoom” to adjust the time base to displaythe sinusoidal waveform of the response.

23. What is the time constant of the response of the primary circuit?

τ = __________

24. What is the period of the undamped oscillation of the primary circuit’s response?

T = __________


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CircuitWorks

Circuit Works Virtual Lab - Exercise 15RLC Circuit - Pulse and Impulse Response

Objectives: A circuit whose time constants are short compared to the width of an input pulse willpass the pulse. It will reject (attenuate) a pulse whose width is relatively short compared to itstime constants. In effect, a circuit having relatively long time constants cannot respond to rapidlychanging signals. This exercise will examine how an underdamped series RLC circuit respondsto a rectangular pulse input voltage and to an impulse.

Part I. Pulse Response . Configure the series RLC circuit to have the values shown below:


R: 2 Ω 20 ΩC: 100 µF 100 µFL: 1mH 1 mH

Output: v1


Apply a rectangular pulse (Signal #28) with amplitude A = 10 V, pulsewidth ∆ = 4 ms, and delay time τ = 0.


Primary: α = ______ ω = ______τ1 = ______

Secondary: s1 = ______ s2 = ______τ1 = ______ τ2 = ___

2. What is the period of oscillation of the capacitor voltage's damped oscillation in the primary circuit?

T = ______ (Primary)

3. What can be said about the pole-zero patterns of the capacitor voltage response?___ a. The primary and secondary circuit have the same poles.___ b. The secondary circuit's poles are closer to the imaginary axis.___ c. The secondary circuit's poles are closer to the real axis.___ d. The primary circuit has complex poles.

4. Examine the capacitor voltage response, and indicate which circuit passes the pulse input signal?

___ Neither ___ Primary circuit ___ Secondary circuit ___ Both

Part II. Sensitivity to Pulsewidth. Repeat Part I with pulsewidth ∆ = 0.05 ms.

5. Examine the capacitor voltage response, and indicate which of the circuits passes the pulse input signal?


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Part III. Now change the primary component values to have R = 2 Ω, L = 1 mH, C = 75 µF. Generateand compare screen image plots of the pole zero patterns and waveforms of the zero-state response of theinductor voltage, inductor current and capacitor voltage.

6. What can be said about the pole-zero patterns of the capacitor voltage response?___ a. The primary and secondary circuit have the same poles.___ b. The secondary circuit's poles are closer to the imaginary axis.___ c. The secondary circuit's poles are closer to the real axis.

7. Examine the capacitor voltage response, and indicate which of the circuits passes the pulse input signal?


8. Measure the period of the damped oscillation in the capacitor voltage response.

T = ______ (Primary) T = ______ (Secondary)

Part IV. Impulse Response . Reconfigure the circuit to have the component values below:


R: 2 Ω 20 ΩC: 23 µF 100 µF L: 1 mH 1 mH

Apply an impulse signal (Signal # 22) with A = 0.0005 V and τ = 1 ms. Examine thescreen images of the zero-state response of the inductor voltage, inductor current and capacitor voltage.

9. What are the values of the capacitor voltage, inductor voltage and inductor current in the primary circuitimmediately before and after the impulse is applied.

v1(τ-) = _______ v1(τ+) = _______v3(τ-) = _______ v3(τ+) = _______i1(τ-) = _______ i1(τ+) = _______

10. What explains the discontinuity in the inductor current?___ a. The capacitor voltage must be continuous.___ b. The resistor voltage can be discontinuous.___ c. The voltage impulse is integrated.___ d. The voltage impulse is differentiated.___ e. The impulse doesn't last long enough to be continuous.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 16RLC Circuit - Sinusoidal Response

Objectives: This exercise examines the sinusoidal response of the series RLC circuit. It revealsthe relationship between the Bode magnitude and phase responses of the circuit and the timedomain amplitude and phase of the response.

Part I. Configure the series RLC circuit to have the values shown below:


R: 100 Ω 100 ΩC: 252 µF 126 µFL: 100 m H 50 H

Output: v1


Part I. Time Constants and Oscillations. Select the sinusoidal input signal (#26); assign amplitude A =10, frequency ωo = 2π rad/s, phase angle φ = 0, and source delay time τ = 0. Examine the pole-zeropatterns and the zero-state response of the circuit.







4. What is the magnitude of the circuit's transfer function at the frequency of the source?

|H(j2π)| = ______ (Primary) |H(j2π)| = ______ (Secondary)

5. What is the angle of the circuit's transfer function at the frequency of the source?

/ H(j2π) = ______ (Primary) / H(j2π) = ______ (Secondary)

6. By what amount is the steady-state output signal delayed relative to the input signal?

τd = ______ (Primary) τd = ______ (Secondary)

Part II. Bode Plots. Repeat Part I using the frequency ωo = 4π rad/s.



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10. What is the magnitude of the circuit's transfer function at the frequency of the source?

|H(j4π)| = ______ (Primary) |H(j4π)| = ______ (Secondary)

11. What is the angle of the circuit's transfer function at the frequency of the source?

/ H(j4π) = ______ (Primary) / H(j4π) = ______ (Secondary)

12. By what amount is the steady-state output signal delayed relative to the input signal?

τd = ______ secs. (Primary) τd = ______ secs. (Secondary)


___ ωo = 2π ___ ωo = 4π

14. At which frequency does the output of the secondary circuit undergo the greatest attenuation of itssteady-state amplitude relative to the input signal?

___ ωo = 2π ___ ωo = 4π

15. At which frequency does the primary circuit's steady-state output signal experience the greatest delayrelative to the input signal?

___ ωo = 2π ___ ωo = 4π

16. At which frequency does the secondary circuit's steady-state output signal experience the greatestdelay relative to the input signal?

___ ωo = 2π ___ ωo = 4π

Part III. Response Boundary Conditions. For the primary and secondary component values given in PartI, examine the initial-state response of the capacitor voltage to a switched sinusoidal source (Signal #26)having amplitude A = 10, frequency ωo = 2π rad/s, phase angle φ = 0, and source delay time τ = 0.5 sec.Use an initial capacitor voltage of v1(0-) = 5 V, and an initial inductor current of i1(0-) = - 50 mA.

17. What are the values of the capacitor voltage, inductor voltage, and inductor current immediately beforethe source turns on?

v1(τ-) = _______ (Primary) v1(τ-) = _______ (Secondary)v3(τ-) = _______ (Primary) v3(τ-) = _______ (Secondary) i1(τ-) = _______ (Primary) i1(τ-) = _______ (Secondary)


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18. What are the values of the capacitor voltage, inductor voltage, and inductor current immediately after thesource turns on?

v1(τ+) = _______ (Primary) v1(τ+) = _______ (Secondary)v3(τ+) = _______ (Primary) v3(τ+) = _______ (Secondary) i1(τ+) = _______ (Primary) i1(τ+) = _______ (Secondary)

Part IV. Response Boundary Conditions. Repeat Part III with v1(0-) = - 5 V and i1(0-) = -50 mA.





Part V. Response Boundary Conditions. Repeat Part IV with τ = 2 sec.





23. What is the value of the capacitor voltage immediately before the source turns on?


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CircuitWorks

Circuit Works Virtual Lab - Exercise 17RC Circuit - Frequency Response and Filters

Objectives: A circuit's sinusoidal input signal (switched or otherwise) will create a steady-statesinusoidal output signal whose magnitude and phase are determined by the frequency responsecharacteristics of the circuit (magnitude and phase). A circuit is said to be a filter if it itsmagnitude response characteristic selectively attenuates the magnitude of a sinusoidal inputsignal, or its phase response characteristic selectively shifts its phase angle. A lowpass filter willpass sinusoidal signals whose frequency is low, and will attenuate those whose frequency is high.Likewise, the amount of phase shift between the input signal sinusoid and the steady-state outputsignal sinusoid will depend on the input signal's frequency. This exercise will examine the seriesRC circuit's lowpass and highpass frequency response characteristics, and their dependence onthe circuit's component values. It will also demonstrate the physical significance of a change in asignal's phase angle, and the relationship between the time-domain and frequency-domainresponses of the circuit.

Part I. Lowpass and Highpass Responses. Select and configure the first-order series RC circuit shownbelow. Examine the screen image copies of the pole-zero patterns and Bode plots of the transfer functionsfor the capacitor voltage and resistor voltage responses (zero-state).


R: 50 kΩ 50 kΩC: 0.01 µF 0.005 µF


1. For the primary circuit, which voltage or current has a low pass response?

___ Capacitor voltage ___ Resistor voltage ___ Both ___ Neither

2. Why is the response a lowpass response?___ a. The source frequency is too low.___ b. The capacitor acts like a short circuit at low frequency.___ c. The capacitor accumulates relatively less charge at high frequency, so most of the source voltage appears across the resistor.___ d. The capacitor accumulates relatively more charge at high frequency, so most of the source voltage appears across the capacitor.___ e. None of the above - it's not a lowpass response.

3. Which response is a highpass response?

___ Capacitor voltage ___ Resistor voltage ___ Both ___ Neither


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4. Why is the response a highpass response?___ a. The source frequency is too low.___ b. The capacitor acts like a short circuit at low frequency.___ c. The capacitor accumulates relatively less charge at high frequency, so most of the source voltage appears across the capacitor.___ d. The capacitor accumulates relatively less charge at high frequency, so most of the source voltage appears across the resistor.___ e. None of the above - it's not a highpass response.

5. What is the bandwidth, or cutoff frequency, of the lowpass response? ωc = ______.

6. What is the bandwidth of the highpass response? ωc = ______.

Part II. Filter Bandwidth. Next, we will examine how the value of the circuit's components determine itsbandwidth (cutoff frequency).

7. How did decreasing the value of C from 0.01 µF to 0.005 µF affect the circuit's pole-zero pattern and Boderesponse curves?___ a. The circuit's pole moved closer to the origin.___ b. The circuit's bandwidth decreased.___ c. The high-frequency slope of the Bode response curves decreased.___ d. The pole of the capacitor voltage response moved away from the origin.___ e. The pole of the resistor voltage response moved away from the origin.___ f. The pole of the capacitor voltage response moved towards the origin.___ g. The pole of the resistor voltage response moved towards the origin.___ h. The bandwidth (ωc) of the lowpass filter increased.___ i. The bandwidth (ωc) of the highpass filter increased.

Part III. Pole-Zero Patterns and Bandwidth. The placement of the lowpass and highpass filter's polesand zeros in the s-plane affects their frequency-domain bandwidth. Using the screen displays from Part II,examine the relationship between the filter's PZP and their Bode plots.

8. How does moving a lowpass filter's pole in the s- plane affect its bandwidth?___ a. Moving the pole towards from the origin increases its bandwidth.___ b. Moving the pole towards from the origin decreases its bandwidth.___ c. Moving the pole away from the origin increases its bandwidth.___ d. Moving the pole away from the origin decreases its bandwidth.

9. How does moving a highpass filter's pole in the s-plane affect its bandwidth?___ a. Moving the pole towards from the origin increases its bandwidth.___ b. Moving the pole towards from the origin decreases its bandwidth.___ c. Moving the pole away from the origin increases its bandwidth.___ d. Moving the pole away from the origin decreases its bandwidth.

Part IV. DC and High Frequency gain. A filter's affect on the magnitude and phase of DC and highfrequency signals is of general interest. For the circuit values given in Part II:

10. What is the lowpass filter's DC gain?

KDC = ______ (Primary) KDC = ______ (Secondary)


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11. What is the lowpass filter's high frequency gain?

K∞ = ______ (Primary) K∞ = ______ (Secondary)

12. What is the highpass filter's DC gain?

KDC = ______ (Primary) KDC = ______ (Secondary)

13. What is the highpass filter's high frequency gain?

K∞ = ______ (Primary) K∞ = ______ (Secondary)

14. What is the lowpass filter's phase shift at DC?

θDC = ______ (Primary) θDC = ______ (Secondary)

15. What is the lowpass filter's phase shift at high frequency?

θ∞ = ______ (Primary) θ∞ = ______ (Secondary)

16. What is the highpass filter's phase shift at DC?

θDC = ______ (Primary) θDC = ______ (Secondary)

17. What is the highpass filter's phase shift at high frequency?

θ∞ = ______ (Primary) θ∞ = ______ (Secondary)

Part V. Signal Attenuation The relative attenuation provided by low and high pass filter's is apparent in theslopes of their Bode response plots on log frequency scale. For the capacitor voltage response, the slopesindicate how the input/output sinusoidal amplitude ratio |V1/Vin| varies as ω → 0 and as ω → ∞.

18. Using the zoom feature of the Circuit Works system, examine the Bode response plots to determinetheir slopes at low and high frequency.a. mDC = ______ dB/Decade (Lowpass, Primary)b. m∞ = ______ dB/Decade (Lowpass, Primary)c. mDC = ______ dB/Decade (Highpass, Primary)d. m∞ = ______ dB/Decade (Highpass, Primary)e. mDC = ______ dB/Decade (Lowpass, Secondary)f. m∞ = ______ dB/Decade (Lowpass, Secondary)g. mDC = ______ dB/Decade (Highpass, Secondary)h. m∞ = ______ dB/Decade (Highpass, Secondary)

Part VI. Filter Action. To demonstrate the filter action of the series RC circuit apply a periodic sinusoid(Signal #8) to the primary circuit with amplitude A = 10 V and period T = 3.1416 ms (ω = 2000 rad/s).Observe the steady-state capacitor and resistor voltages, v1 and v2, and measure each signal's magnitudeand phase shift (τph) with respect to the input signal.


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19. Using the observed phase shift of the steady-state waveform of the output signal, calculate the phaseangle (φ) of each output signal relative to the input signal. Compare the results to those determined by theBode response of the filter.

Measured Valuesv1 = ______ τph = ______ φ = ______v2 = ______ τph = ______ φ = ______

Bode Response Valuesv1 = ______ τph = ______ φ = ______v2 = ______ τph = ______ φ = ______

20. Repeat with ω = 4000 rad/s.

v1 = ______ τph = ______ φ = ______v2 = ______ τph = ______ φ = ______

Part VII. Now apply a periodic rectangular pulse train (Signal #9) to the series RC having the primarycomponent values given in Part II. Let the pulse train have parameters A = 10 and T = 31.416 ms. For thesecondary circuit:

21. What is the bandwidth of the capacitor voltage response? ωc = _____

22. What is the fundamental frequency of the pulse train? ωo = ______

23. Observe the capacitor voltage waveform, v1(t). Estimate its average value:

vDC = ______

24. Observe the zero-state response of the resistor voltage. Estimate its average value:

vDC = ______

25. At what frequency is the lowpass filter's Bode response 3 dB below its DC value?ω = ______

26. At what frequency is the highpass filter's Bode response 3 dB below its high frequency value? ω =______

Part VIII. Repeat Part VII with T = 1.5708 ms.

27. What is the bandwidth of the capacitor voltage response? ωc = _____

28. What is the fundamental frequency of the pulse train? ωo = ______

29. Observe the capacitor voltage waveform. Estimate its average value: vDC = ______

30. Observe the resistor voltage waveform. Estimate its average value: vDC = ______


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CircuitWorks

Circuit Works Virtual Lab - Exercise 18Periodic and Aperiodic Signal Spectra

Objectives: A periodic signal can be represented by a discrete sum of complex exponentialsignals, called its Fourier series. Each term in the series consists of a unit phasor whosefrequency is an integer multiple of the fundamental frequency, ωωo, and whose amplitude is acomplex Fourier coefficient, Fn, where n is the Fourier index. The frequency ωωo is determined byωωo = 2ππ/T, where T s the smallest interval over which the signal is periodic. Fn is uniquelydetermined by the signal's time-domain waveform. The Fourier series spectra is a pair of graphswith lines drawn at multiples of the fundamental frequency. The heights of the lines indicate therelative frequency content of the signal. Any physical aperiodic signal can be represented as acontinuous sum of complex exponential signals. The Fourier Transform F(j ωω) of a signal f(t)defines the relative amplitude of the signal e j ωω t within f(t). In general, F(j ωω) is a complex quantityhaving magnitude |F(j ωω)| and angle θθ(jωω). The magnitude spectrum of F(j ωω) is a graph of |F(j ωω)| vsωω, and the phase spectrum is the graph of θθ(jωω) vs ωω. The Fourier magnitude and phase spectra ofa signal provide intuitive insight about the relationship between the time and frequency domaincharacterizations of a circuit. This exercise will examine the relationship between the spectra ofthe unit step, the damped exponential, and the damped cosine signals.

Part I. Select the series RC circuit (#1) and assign the primary component values shown below:


R: 12.75 kΩ noneC: 0.005 µF none

Output: v1

1. What is the time constant of this RC circuit lowpass filter? τ = ______

Part II. Fourier Series - Squarewave. Select the periodic squarewave signal (#1) and assign amplitude A= 10 and period = 5 ms. Display the spectra of the signal. To see how the spectra is used to synthesizethe input signal, generate the screen image of the steady-state response of the capacitor voltage for Fourierindex choices of 1, 3, 5, and 15. Notice how the inclusion of more terms in the series progressivelyimproves the synthesized representation of the waveform.

Part III. Fourier Transform - Step Input. Select the unit step input signal with amplitude A = 1 and delaytime τ = 0, and generate a screen image copy of the pole-zero pattern (PZP), Bode plot and input/outputsignal of the capacitor voltage's zero-state response. Print the screen image (using a white background)showing the PZP, Bode plot, and input/output spectra of the response.

Part IV. Fourier Transform - Damped Exponential. Now select the damped exponential signal (#23) andagain set A = 1, τ = 0. Generate screen image copies showing the zero-state response waveforms and theinput/output spectra for σ = 5 ms-1, 10 ms-1, 60 ms-1, 100 ms-1 and 500 ms-1. Compare to the step responseobtained in Part II.

2. How did changing σ affect the input/output response?___ a. It wasn't affected.


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___ b. When 1/σ is smaller than the circuit's time constant the response grows faster.___ c. When 1/σ is smaller than the circuit's time constant the response grows slower.___ d. When 1/σ is much greater than the circuit's time constant the response is like the step response until the input signal decays.___ e. When 1/σ is much smaller than the circuit's time constant the response is like the step response until the input signal decays.

3. What happens to the spectra of the exponential signal as σ decreases?___ a. It vanishes.___ b. It becomes concentrated near ω = π/2.___ c. It becomes concentrated at the origin.

Part V. Now select the damped cosine signal (signal #25) and set A = 1, ωo = 200 rad/s, τ = 0. Generatescreen image copies showing the zero-state response waveforms and the input/output spectra for σ = 5 ms-

1, 10 ms-1, 60 ms-1, 100 ms-1 and 500 ms-1. Compare to the step response obtained in Part II.

4. How did changing σ affect the input/output response?___ a. When 1/σ is smaller than the circuit's time constant the response grows faster.___ b. When 1/σ is smaller than the circuit's time constant the response grows slower.___ c. When 1/σ is much greater than the circuit's time constant the response is like the step response until the input signal decays.___ d. When 1/σ is much smaller than the circuit's time constant the response is like the step response until the input signal decays.

Part VI. Now set ωo = 5 rad/s and repeat Part IV.

5. How did the change in ωo affect the response?___ a. Decreasing ωo caused the output to oscillate.___ b. Decreasing ωo caused the waveform to more closely resemble a sinusoid.___ c. Decreasing ωo caused the response to more closely resemble the response to the damped exponential.

6. What happens to the spectra of the exponential signal as ω decreases?___ a. It vanishes.___ b. It becomes concentrated near ω = π/2.___ c. It becomes concentrated at the origin.___ d. It approaches the spectrum of the damped exponential.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 19Filters and Rectified Signals

Objectives: Halfwave and fullwave rectified sinusoidal signals can be used to create DC signalsfrom AC signals. These rectified signals are not themselves DC, but contain a significant DCcomponent compared to the original unrectified sinusoid. Passing them through an appropriatefilter creates a new signal whose DC content is even greater than that of the input. This exerciseexamines the time domain and frequency domain performance of a second-order passive lowpassfilter in the role of a DC enhancement filter driven by halfwave and fullwave rectified sinusoids.


R1: 5 kΩ noneR2: 15 kΩ noneC: 0.1 µF noneL: 62.5 mH none

Output: v1

Part I. Lowpass Filter. For these component values the transfer function governing the voltage responseacross the load resistor, v1, behaves like a generic second-order lowpass filter. Examine the filter's pole-zero pattern and note the location of its poles and its Bode plot.

1. What type of time-domain response would you expect this filter to exhibit?

___ Overdamped ___ Critically damped ___ Underdamped

2. What are the poles of the input/output transfer function?

s1 = ______ s2 = ______

3. How long does it take this circuit to reach a steady-state operating condition? tss = ______

4. What are the DC and high frequency gains of the filter?

KDC = ______ K∞ = ______

Part II. Halfwave Rectified Signal. Now drive the circuit with a periodic halfwave rectified signal (Signal#12) having amplitude A = 10 V and a period corresponding to a 60 Hz unrectified signal.

5. What is the fundamental frequency of the input signal? ωo = ______

6. Generate the Bode plot of the transfer function governing the load response (v1), and measure its cutofffrequency and high frequency slope (rolloff).

ωc = ______ m∞ = ______ dB/decade

Examine the PZP, bode plot input magnitude spectra, and output magnitude spectra of the response to thehalfwave rectified input signal. Consider whether the output signal will be nearly DC.


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7. How many spectral lines of the input signal fit within the passband of the filter?

N = ______

8. Estimate the fraction of the input signal's power that is at DC: PDC = ______

9. Estimate the fraction of the output signal's power that is at DC: PDC = ______

Part III. Lowpass Filter. Now assign secondary component values of R1 = 5 kΩ, R2 = 15 kΩ, L = 5 H, andC = .75 µF. Repeat Parts I and II.

10. What type of time-domain response would you expect this filter to exhibit?

___ Overdamped ___ Critically damped ___ Underdamped

11. What are the filter's time constant and undamped frequency of oscillation?

τ = ______ ωn = ______

12. How long does it take this circuit to reach a steady-state operating condition?

tss = ______

13. What are the DC and high frequency gains of the filter?

KDC = ______ K∞ = ______


15. What are the cutoff frequency and high frequency rolloff of the filter?

ωc = ______ m∞ = ______ dB/decade

16. How many spectral lines of the input signal fit within the passband of the filter?

N = ______

17. Estimate the fraction of the input signal's power that is at DC: PDC = ______

18. Estimate the fraction of the output signal's power that is at DC: PDC = ______

19. Explain why the output signal of the filter having the secondary component values has a greaterproportion of DC signal output.

___ a. It has smaller component values.___ b. It passes more of the harmonics of the input signal.___ c. It has a smaller bandwidth.___ d. It rejects more of the harmonics of the input signal.

Part IV. Fullwave Rectified Sinusoid. To each of the circuits apply a periodic fullwave rectified signal(Signal #13) having amplitude A = 10 V and period T= 8.333 msec. Examine the PZP, the input signal, theBode plots and the output signal of the response to the fullwave rectified input signal. Also examine the


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pole-zero pattern, input signal magnitude spectrum and output signal magnitude spectrum. Consider therelative performance of the circuits.


21. Estimate the DC power content of the input and output signals.

Input: PDC = ______ (Primary) PDC = ______ (Secondary)Output: PDC = ______ (Primary) PDC = ______ (Secondary)

22. Explain why the output signal of the filter having the secondary component values has a greaterproportion of DC signal output.

___ a. It has smaller component values.___ b. It passes more of the harmonics of the input signal.___ c. It has a smaller bandwidth.___ d. It rejects more of the harmonics of the input signal.

23. Why is the filter output's DC power content significantly greater when the full wave rectified signal is theinput signal?

___ a. The fullwave signal has fewer harmonics___ b. The fullwave signal has more harmonics___ c. The fullwave signal has more DC content.___ d. The halfwave signal has more harmonics in the passband.___ e. The second order filter has a sharper rolloff than the first order filter.___ f. The halfwave signal has more harmonics in the passband.


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CircuitWorks

Circuit Works Virtual Lab - Exercise 20Active Bandpass Filter Design

Objectives: Bandpass filters are used to selectively pass signals in a range of frequencies ωω1 < ωω <ωω2, where ωω1 < ωω2, ωω1 > 0 and ωω2 < ∞∞. For example, bandpass filters are used to pass a signalspectra located about the intermediate frequency (IF) in a superheterodyne AM radio, therebyblocking noise and other undesirable signals. The magnitude response of a bandpass filtertypically has a relatively large value near ωωp, the peak, or resonant, frequency of the filter, and arelatively small value for ωω < ωω1 and ωω > ωω2, with ωω1 < ωωp < ωω2. ωω1 and ωω2 are called the "half-power" frequencies of the filter, and ωωb = ωω2 - ωω1 is the filter's bandwidth. A sinusoidal inputsignal whose frequency lies outside the bandwidth will produce a steady-state output sinusoidhaving less than one-half the average power that a sinusoid with the same amplitude wouldproduce at the peak frequency. The selectivity of a bandpass filter is Q = ωωp/ωωb.

Part I. Bandpass Filter Design. Design an active bandpass filter (Circuit #100) to operate with fp = 5 kHz,Q = 10 and Kp = 5. Hint: The gain, bandwidth, and peak frequency of the filter are given by: |K| = 1/(ωbR1C1),ωb = (C1 + C2)/(R3C1C2), ωp

2 = (R1 + R2)/(R1R2R2C1C2). These equations can be solved simultaneously tomeet a given specification by choosing values of the resistors and then solving for the values of thecapacitors, or by choosing C1, and fixing C2 = βC1, for a chosen capacitor ratio, β, then solving for the valuesof the resistors.

1. First choose convenient values for the resistors and the peak frequency, say 10 Ω and 100 Hz. List yourdesign component values below.

R1 = ______ R2 = ______ R3 = ______ C1 = ______ C2 = ______

2. Examine the pole-zero pattern and the Bode plot of the filter. What is the time constant of the circuit?

τ = ______

3. What is the circuit's damped frequency of oscillation? ωd = ______

4. What is the circuit's undamped frequency of oscillation? ωn = ______

5. If a step input signal is applied to the circuit how many cycles of damped oscillation will occur before theresponse reaches steady-state? N = ______

6. How long will it take for the circuit's response to reach a steady-state condition, i.e. reach and remainwithin 98% of its final value? tss = ______

7. Measure the circuit's bandwidth and its gain at ωp. ωb = ______ Kp = ______


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8. Measure the slope of the circuit's Bode response (magnitude) at ω1 and at ω2.

ω1 = ______ ω2 = ______

Part II. Scaled Design. Now frequency scale the design to operate at fc = 5 kHz, and magnitude scalethe components to have resistors that are multiples of 10 kΩ.

9. List the scale factors: Kf = ______ Km = ______

10. List the scaled component values below.

R1 = ______ R2 = ______ R3 = _______ C1 = _______ C2 = ______

11. Based on the pole-zero pattern, what is the time constant of the circuit? τ = ______

12. What is the circuit's damped frequency of oscillation? ωd = ______

13. How long will it take for the circuit's response to reach a steady-state condition?

tss = ______

14. If a step input signal is applied to the circuit how many cycles of damped oscillation will occur before theresponse reaches steady-state?

N = ______

15. Examine the PZP and the Bode plot of the filter. Measure the circuit's bandwidth.

ωb = ______

16. Measure the circuit's gain at the frequency ωp. Kp = ______

17. Measure the slope of the circuit's Bode response at ω1. m1 = ______

18. Measure the slope of the circuit's Bode response at its upper half-power frequency. m2 = ______

19. Compute the ratio of ωp to ωb for the filter. ωc/ωn = ______

20. If all of the resistor values increase by 10% from their design value, what will be the effect?

___ a. The bandwidth will increase.___ b. The bandwidth will decrease.___ c. The circuit's damped frequency of oscillation will increase.___ d. The circuit's damped frequency of oscillation will decrease.___ e. The circuit's damping factor will increase.___ f. The circuit's damping factor will decrease.___ g. The circuit's gain at ωp will increase.___ h. The circuit's gain at ωp will decrease.

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CIRCUIT WORKS - Wiley · V-Lab #20 Active Bandpass Filter Design. ... Display both circuit’s pole-zero patterns and compare the measured values of ... Circuit Works Virtual Lab