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DEPARTMENT OF ELECTRICAL, ELECTRONIC AND COMPUTER ENGINEERING

ELECTRICAL ENGINEERING EIR 211/221 PRACTICAL 1: DYNAMIC RESPONSES OF RL-, RC- AND RLC CIRCUITS

Name: Diederick Johannes Taljaard

Student no. 29031096

AIM OF THE PRACTICAL The aim of this practical is to experimentally investigate the dynamic responses of passive RC-, RL- and RLC-circuits and the familiarization with the following: - PSPICE simulation of electric circuits - Construction and testing of electric circuits REQUIREMENTS Each and every student is required to individually complete the pre-practical assignment before the start of the practical session. Any student that fails to comply with this requirement will receive an incomplete mark for this practical. The practical must be performed in groups of two students. All students must be present during, and participate actively in all of the measurements.

Table of Contents 1. Introduction ..................................................................................................................... 1 2. Problem statement ........................................................................................................... 1 3. Approach and suggested method .................................................................................... 1 4. Theoretical preparation ................................................................................................... 2 4.1. RC-Circuit simulations ............................................................................................ 2 4.2. RL-Circuit simulations............................................................................................. 4 4.3. RLC-Circuit simulations part 1 ................................................................................ 6 4.4. RLC-Circuit simulations part 2 ................................................................................ 8 5. Experimental results...................................................................................................... 10 5.1. RC-Circuit results .................................................................................................. 10 5.2. RL-Circuit results................................................................................................... 12 5.3. RLC-Circuit results part 1 ...................................................................................... 14 5.4. RLC-Circuit results part 2 ...................................................................................... 16 6. Discussion ..................................................................................................................... 18 7. Conclusion .................................................................................................................... 18 8. Bibliography ................................................................................................................. 18

1. Introduction The following report concerns the application in practise of RLC circuits. These circuits are the basis of modern electronics. The report will show the differences between theoretical, simulation and practical results. This experiment covers only DC circuits but is operational in AC circuits as well.

2. Problem statement As with all new concepts a thorough examination and practise of this concept is of great importance to fully grasp and understand the concepts in question. In this practical the concepts were RL-, RC- and RLC circuits and there simulation in the program PSpice. As these were new theories one could not fully understand the practical analysis through theory alone and thus a practical implementation would clear up any queries on the new concepts. To achieve a true understanding of the core concepts, it is asked of the group to calculate time constant (τ) in different circuits as well as, but not limited to, the damping constant (α).

3. Approach and Method All simulations are to be done theoretically and in practise. RL-, RC, and RLC circuits will be built on the theoretical basis through means of the simulation program PSpice. With all three circuits a graph must be created which shows the potential differents [V] and / or current [A] to time [s]. With these graphs one can see the maximum and minimum values as well as the influence of a capacitor and an inductor to the circuit. With these values a theoretical calculation can be made to calculate the time constants and the damping factor were necessary. With the practical approach, the circuits will be built on a National Instruments electronic board (NI Elvis) and the true graphs will be established through means of electrical equipment and a computer station. When both of these methods are concluded one will be able to see the differents between theoretical calculations and practical values.

1

4. Theoretical preparation Before any practical experiments are to commence, one should prepare with theoretical expectations to evaluate the practical results to. The time constant (in context to a capacitor) in seconds, “the time required for the p.d. across C to increase from zero to its final value…” (Hughes, 2008) The time constant (τ), is measured in seconds [s]. With respect to an inductor the time constant is the time required by the inductor to reach the maximum current. The theoretical preparation was done by means of the module EIR 221’s outcome.

4.1. RC-Circuit simulations The following circuit was simulated (using R = 1000 Ohm, C = 0.1 F and Rs = 6 Ohm) in PSpice:

Figure 1 : RC - Circuit

The signal generator (shown inside the red line acts as a switched Thevenin-equivalent voltage source with an internal resistance, Rs) delivers a square wave to the load.

2

The theoretical expression and value of the time constant τ:

(

)

Figure 2 shows the transient response voltage (vertical axis) waveform across the capacitor, the horizontal axis is time measured in seconds [s] with a 0.1ms interval.

Figure 2 : Voltage [V] vs time [s]

The following table was completed using the transient voltage response simulated across the capacitor. The value of V0 and the time constant is obtained from the graph. 0.368 V0 was calculated.

3

Table 1 : Theoretical time constant for RC-Circuit

V0

5V

0.368 V0

1.84 V

τ

1.0955 ms

4.2. RL-Circuit simulations The following circuit was simulated (using R = 1000 Ohm, L = 47 mH and Rs = 6 Ohm) in PSpice:

Figure 3 : RL - Circuit

The theoretical expression and value of the time constant τ.

4

Figure 4 is a plot of the transient response current waveform simulated in the circuit.

Figure 4 : Current [A] vs time [s]

The following table was completed using the transient current response simulated across the inductor. The value of I0 and the time constant is obtained from the graph. 0.368 I0 was calculated. Table 2 : Theoretical time constant for RL-Circuit

I0

0.005 A

0.368 I0

0.00184 A

τ

50.1 s

5

4.3. RLC-Circuit simulations part 1 The following circuit was simulated ( using R = 100 Ohm, C = 0.1 F, L = 47 mH and Rs = 6 Ohm) in PSpice:

Figure 5 : RLC-Circuit

In the case of an under damped circuit, a decaying oscillatory current will be observed as shown in figure 6.

Figure 6 : Current waveform of under damped circuit.

The peak values of the oscillation can be connected by a line with equation

y = Aet

6

The theoretical expression and value of the damping constant .

(

)

Attached figure 7 is a plot of the transient response current waveform in the circuit.

Figure 7 : Current [I] vs time [s]

7

The table below was completed using the simulated transient current in the sircuit. The value T was read from figure 7 and used to calculate fd. The first and second peak from the graph was used to calculate α.

Table 3 : Theoretical under damped RLC-Circuit attenuation factor

Damping period T

485 s

Oscillation frequency (fd = 1/T)

2061.86 Hz

The value of the first peak (= Ae t )

5.863 mA

The value of the second peak (= Ae (t+T) )

3.690 mA

Solve the two equations for α

953.37

4.4. RLC-Circuit simulations part 2 The following circuit was simulated (using R = 1000 Ohm, C = 0.1 F, L = 47 mH, Rs = 6 Ohm):

Figure 8 : RLC-Circuit

8

The theoretical damping constant α, for the values of R, L and C in the circuit:

(

)

Figure 9 is a plot of the transient response current waveform in the circuit.

Figure 9 : Current [A] vs time [s]

9

5. Experimental results The theoretical results showed a close relation to the results obtained by PSpice’ simulation. This was expected as the simulation does not account for any wire resistance or any other faults that may be present in the electronic equipment. These results were a good standard to measure the practical results to. The practical experiments will follow. The experiments were done with ordinary electronic equipment available in any electronic store and the NI Elvis equipment which have been used before.

5.1. RC-Circuit results Build the following circuit (using R = 1000 Ohm and C = 0.1 F);

Figure 10 : RC-Circuit

The signal generator (shown inside the red line acts as a switched Thevenin-equivalent voltage source with an internal resistance, Rs) delivers a square wave to the load. The theoretical expression and value of the time constant τ:

(

)

10

Figure 11 shows the transient response voltage waveform measured across the capacitor. Using NI Elvis.

Figure 11 : NI Elvis Graph, Voltage [V] vs time [s]

The time constant was determined using he transient voltage response measured across the capasitor. Table 4 : Practical time constant for RC circuit

V0

4.34 V

0.368 V0

1.6 V

τ

0.1 ms

11

5.2. RL-Circuit results The following circuit was built (using R = 1000 Ohm and L = 47 mH);

Figure 12 : RL-Circuit

The theoretical expression and value of the time constant τ:

12

Figure 13 shows the transient voltage waveform, it is sufficient because voltage across a resistor is proportional to current in a circuit like this.

Figure 13 : NI Elvis graph, Voltage [V] vs time [s]

Table 5 shows the time constant determined using the transient voltage response. Table 5 : Practical time constant for RL circuit

I0

4.995 mA

0.368 I0

1.84 mA

τ

45.3

13

5.3. RLC-Circuit results part 1 The following circuit was built (using R = 100 Ohm, C = 0.1 F and L = 42 mH);

Figure 14 : RLC-Circuit

In the case of an under damped circuit, a decaying oscillatory current will be observed as shown in Figure 15.

Figure 15 : Voltage waveform of under damped circuit.

The peak values of the oscillation can be connected by a line with equation

y = Aet 14

The theoretical expression and value of the time constant τ:

(

)

The transient response voltage waveform measured across the resistor is shown in Figure 16:

Figure 16 : NI Elvis graph, Voltage [V] vs time [s]

15

The table below was completed using figure 16.

Table 6 : Theoretical under damped RLC circuit attenuation constant

Damping period T

420

Oscillation frequency (fd = 1/T)

2.38 kHz

The value of the first peak (= Ae t )

280 mV

The value of the second peak (= Ae (t+T) )

100 mV

Solve the two equations for α

2451.5

5.4. RLC-Circuit results part 2 The following circuit was built (using R = 1000 Ohm, C = 0.1 F and L = 42 mH);

Figure 17 : RLC-Circuit

16

The theoretical damping constant:

(

)

Figure 18 shows the transient response voltage waveform measured across the resistor.

Figure 18 : NI Elvis graph, Voltage [V] vs time [s]

17

6. Discussion The voltage waveform measured in 5.3 and 5.4 is different because of the resisters being different sizes. This caused different damping constants. The graph in figure 18 from 5.4 shows a graph which is not damped, this is incorrect because it has a higher damping constant than the graph from 5.3. This could be because of a human error or the equipment being faulty.

7. Conclusion In this report a thorough research was done of RLC circuits. The time constant and damping factor was fully investigated. Through this experiment one can see that just through theoretical, simulation or practical experiments or calculations is not enough to achieve a thorough understanding of the concepts like RLC circuits. With all three as done in this report is it possible to get a larger understanding of these perceptions.

8. Bibliography Hughes, E. (2008). Hughes electrical and electronic technology. Edinburgh Gate.

18

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ELECTRICAL ENGINEERING EIR 211/221 PRACTICAL 1: DYNAMIC RESPONSES OF RL-, RC- AND RLC CIRCUITS

Name: Diederick Johannes Taljaard

Student no. 29031096

AIM OF THE PRACTICAL The aim of this practical is to experimentally investigate the dynamic responses of passive RC-, RL- and RLC-circuits and the familiarization with the following: - PSPICE simulation of electric circuits - Construction and testing of electric circuits REQUIREMENTS Each and every student is required to individually complete the pre-practical assignment before the start of the practical session. Any student that fails to comply with this requirement will receive an incomplete mark for this practical. The practical must be performed in groups of two students. All students must be present during, and participate actively in all of the measurements.

Table of Contents 1. Introduction ..................................................................................................................... 1 2. Problem statement ........................................................................................................... 1 3. Approach and suggested method .................................................................................... 1 4. Theoretical preparation ................................................................................................... 2 4.1. RC-Circuit simulations ............................................................................................ 2 4.2. RL-Circuit simulations............................................................................................. 4 4.3. RLC-Circuit simulations part 1 ................................................................................ 6 4.4. RLC-Circuit simulations part 2 ................................................................................ 8 5. Experimental results...................................................................................................... 10 5.1. RC-Circuit results .................................................................................................. 10 5.2. RL-Circuit results................................................................................................... 12 5.3. RLC-Circuit results part 1 ...................................................................................... 14 5.4. RLC-Circuit results part 2 ...................................................................................... 16 6. Discussion ..................................................................................................................... 18 7. Conclusion .................................................................................................................... 18 8. Bibliography ................................................................................................................. 18

1. Introduction The following report concerns the application in practise of RLC circuits. These circuits are the basis of modern electronics. The report will show the differences between theoretical, simulation and practical results. This experiment covers only DC circuits but is operational in AC circuits as well.

2. Problem statement As with all new concepts a thorough examination and practise of this concept is of great importance to fully grasp and understand the concepts in question. In this practical the concepts were RL-, RC- and RLC circuits and there simulation in the program PSpice. As these were new theories one could not fully understand the practical analysis through theory alone and thus a practical implementation would clear up any queries on the new concepts. To achieve a true understanding of the core concepts, it is asked of the group to calculate time constant (τ) in different circuits as well as, but not limited to, the damping constant (α).

3. Approach and Method All simulations are to be done theoretically and in practise. RL-, RC, and RLC circuits will be built on the theoretical basis through means of the simulation program PSpice. With all three circuits a graph must be created which shows the potential differents [V] and / or current [A] to time [s]. With these graphs one can see the maximum and minimum values as well as the influence of a capacitor and an inductor to the circuit. With these values a theoretical calculation can be made to calculate the time constants and the damping factor were necessary. With the practical approach, the circuits will be built on a National Instruments electronic board (NI Elvis) and the true graphs will be established through means of electrical equipment and a computer station. When both of these methods are concluded one will be able to see the differents between theoretical calculations and practical values.

1

4. Theoretical preparation Before any practical experiments are to commence, one should prepare with theoretical expectations to evaluate the practical results to. The time constant (in context to a capacitor) in seconds, “the time required for the p.d. across C to increase from zero to its final value…” (Hughes, 2008) The time constant (τ), is measured in seconds [s]. With respect to an inductor the time constant is the time required by the inductor to reach the maximum current. The theoretical preparation was done by means of the module EIR 221’s outcome.

4.1. RC-Circuit simulations The following circuit was simulated (using R = 1000 Ohm, C = 0.1 F and Rs = 6 Ohm) in PSpice:

Figure 1 : RC - Circuit

The signal generator (shown inside the red line acts as a switched Thevenin-equivalent voltage source with an internal resistance, Rs) delivers a square wave to the load.

2

The theoretical expression and value of the time constant τ:

(

)

Figure 2 shows the transient response voltage (vertical axis) waveform across the capacitor, the horizontal axis is time measured in seconds [s] with a 0.1ms interval.

Figure 2 : Voltage [V] vs time [s]

The following table was completed using the transient voltage response simulated across the capacitor. The value of V0 and the time constant is obtained from the graph. 0.368 V0 was calculated.

3

Table 1 : Theoretical time constant for RC-Circuit

V0

5V

0.368 V0

1.84 V

τ

1.0955 ms

4.2. RL-Circuit simulations The following circuit was simulated (using R = 1000 Ohm, L = 47 mH and Rs = 6 Ohm) in PSpice:

Figure 3 : RL - Circuit

The theoretical expression and value of the time constant τ.

4

Figure 4 is a plot of the transient response current waveform simulated in the circuit.

Figure 4 : Current [A] vs time [s]

The following table was completed using the transient current response simulated across the inductor. The value of I0 and the time constant is obtained from the graph. 0.368 I0 was calculated. Table 2 : Theoretical time constant for RL-Circuit

I0

0.005 A

0.368 I0

0.00184 A

τ

50.1 s

5

4.3. RLC-Circuit simulations part 1 The following circuit was simulated ( using R = 100 Ohm, C = 0.1 F, L = 47 mH and Rs = 6 Ohm) in PSpice:

Figure 5 : RLC-Circuit

In the case of an under damped circuit, a decaying oscillatory current will be observed as shown in figure 6.

Figure 6 : Current waveform of under damped circuit.

The peak values of the oscillation can be connected by a line with equation

y = Aet

6

The theoretical expression and value of the damping constant .

(

)

Attached figure 7 is a plot of the transient response current waveform in the circuit.

Figure 7 : Current [I] vs time [s]

7

The table below was completed using the simulated transient current in the sircuit. The value T was read from figure 7 and used to calculate fd. The first and second peak from the graph was used to calculate α.

Table 3 : Theoretical under damped RLC-Circuit attenuation factor

Damping period T

485 s

Oscillation frequency (fd = 1/T)

2061.86 Hz

The value of the first peak (= Ae t )

5.863 mA

The value of the second peak (= Ae (t+T) )

3.690 mA

Solve the two equations for α

953.37

4.4. RLC-Circuit simulations part 2 The following circuit was simulated (using R = 1000 Ohm, C = 0.1 F, L = 47 mH, Rs = 6 Ohm):

Figure 8 : RLC-Circuit

8

The theoretical damping constant α, for the values of R, L and C in the circuit:

(

)

Figure 9 is a plot of the transient response current waveform in the circuit.

Figure 9 : Current [A] vs time [s]

9

5. Experimental results The theoretical results showed a close relation to the results obtained by PSpice’ simulation. This was expected as the simulation does not account for any wire resistance or any other faults that may be present in the electronic equipment. These results were a good standard to measure the practical results to. The practical experiments will follow. The experiments were done with ordinary electronic equipment available in any electronic store and the NI Elvis equipment which have been used before.

5.1. RC-Circuit results Build the following circuit (using R = 1000 Ohm and C = 0.1 F);

Figure 10 : RC-Circuit

The signal generator (shown inside the red line acts as a switched Thevenin-equivalent voltage source with an internal resistance, Rs) delivers a square wave to the load. The theoretical expression and value of the time constant τ:

(

)

10

Figure 11 shows the transient response voltage waveform measured across the capacitor. Using NI Elvis.

Figure 11 : NI Elvis Graph, Voltage [V] vs time [s]

The time constant was determined using he transient voltage response measured across the capasitor. Table 4 : Practical time constant for RC circuit

V0

4.34 V

0.368 V0

1.6 V

τ

0.1 ms

11

5.2. RL-Circuit results The following circuit was built (using R = 1000 Ohm and L = 47 mH);

Figure 12 : RL-Circuit

The theoretical expression and value of the time constant τ:

12

Figure 13 shows the transient voltage waveform, it is sufficient because voltage across a resistor is proportional to current in a circuit like this.

Figure 13 : NI Elvis graph, Voltage [V] vs time [s]

Table 5 shows the time constant determined using the transient voltage response. Table 5 : Practical time constant for RL circuit

I0

4.995 mA

0.368 I0

1.84 mA

τ

45.3

13

5.3. RLC-Circuit results part 1 The following circuit was built (using R = 100 Ohm, C = 0.1 F and L = 42 mH);

Figure 14 : RLC-Circuit

In the case of an under damped circuit, a decaying oscillatory current will be observed as shown in Figure 15.

Figure 15 : Voltage waveform of under damped circuit.

The peak values of the oscillation can be connected by a line with equation

y = Aet 14

The theoretical expression and value of the time constant τ:

(

)

The transient response voltage waveform measured across the resistor is shown in Figure 16:

Figure 16 : NI Elvis graph, Voltage [V] vs time [s]

15

The table below was completed using figure 16.

Table 6 : Theoretical under damped RLC circuit attenuation constant

Damping period T

420

Oscillation frequency (fd = 1/T)

2.38 kHz

The value of the first peak (= Ae t )

280 mV

The value of the second peak (= Ae (t+T) )

100 mV

Solve the two equations for α

2451.5

5.4. RLC-Circuit results part 2 The following circuit was built (using R = 1000 Ohm, C = 0.1 F and L = 42 mH);

Figure 17 : RLC-Circuit

16

The theoretical damping constant:

(

)

Figure 18 shows the transient response voltage waveform measured across the resistor.

Figure 18 : NI Elvis graph, Voltage [V] vs time [s]

17

6. Discussion The voltage waveform measured in 5.3 and 5.4 is different because of the resisters being different sizes. This caused different damping constants. The graph in figure 18 from 5.4 shows a graph which is not damped, this is incorrect because it has a higher damping constant than the graph from 5.3. This could be because of a human error or the equipment being faulty.

7. Conclusion In this report a thorough research was done of RLC circuits. The time constant and damping factor was fully investigated. Through this experiment one can see that just through theoretical, simulation or practical experiments or calculations is not enough to achieve a thorough understanding of the concepts like RLC circuits. With all three as done in this report is it possible to get a larger understanding of these perceptions.

8. Bibliography Hughes, E. (2008). Hughes electrical and electronic technology. Edinburgh Gate.

18

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