electricity fundamentals, 7 solving parallel and mixed ... · kirchhoff’s current law is a law...

© Festo Didactic 89688-00 123

When you have completed this exercise, you will be able to calculate the equivalent resistance of multiple resistors in parallel circuits. You will be introduced to Kirchhoff’s current law and be able to apply this law to electrical circuits. You will also know how to solve mixed circuits.

The Discussion of this exercise covers the following points:

Calculating the equivalent resistance in parallel circuits

Kirchhoff’s current law

Solving mixed circuitsExample 1. Example 2.

Printed circuit boards

Training system modulePrinted Circuit Board module.

Calculating the equivalent resistance in parallel circuits

In the previous exercise, you saw how to calculate the equivalent resistance of series resistors. It is also possible to calculate the equivalent resistance of resistors that are connected in parallel using the following equation:

(10)

where is the equivalent resistance of all resistors in the parallel circuit,

expressed in ohms ( ) is the resistance of each resistor in the circuit, expressed in

ohms ( )

As the equation shows, the reciprocal of the equivalent resistance of any

resistors connected in parallel is equal to the sum of the reciprocals of their resistance values. Consider, for example, the parallel circuit containing three resistors shown in Figure 100a. To calculate the current flowing in this circuit, it is necessary to calculate the equivalent resistance of the circuit, as shown in

Figure 100b. This is calculated using the following equation:

Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law

Exercise 7

EXERCISE OBJECTIVE

DISCUSSION OUTLINE

DISCUSSION

Exercise 7 – Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law Discussion

124 © Festo Didactic 89688-00

Figure 100. Circuits showing how to calculate the equivalent resistance of parallel resistors.

Knowing the equivalent resistance of the circuit, it is now possible calculate the

current flowing in the circuit using Ohm’s law:

Therefore, a current of 1.04 A flows in both circuits in Figure 100, since both have the same equivalent resistance of 23.1 . Note that, as the calculations

show, adding resistors in parallel greatly decreases the equivalent resistance of the circuit. This is contrary to what happens in series, where adding resistors in series increases the equivalent resistance.


Kirchhoff’s current law is a law described by German physicist Gustav Kirchhoff that is used in the study of parallel circuits. It states that the sum of all currents entering a circuit junction is equal to the sum of the currents leaving it. In practice, this means that whenever a current separates into multiple branches, such as in a parallel circuit, the sum of these currents is equal to the current before it separated.

Consider, for example, the parallel circuit containing three resistors shown in Figure 101.

100

50

75

23.1

24 V 24 V

(a) Parallel circuit with three resistors (b) Circuit showing the equivalent resistance of

resistors , , and .



Figure 101. Kirchhoff’s current law applied to a parallel circuit containing three resistors.

Since, in a parallel circuit, the same voltage is applied to all branches of the

circuit, it is possible to use the source voltage to calculate the current flowing in each resistor in the circuit, as shown below:

To verify that Kirchhoff’s circuit law is respected, it is necessary to calculate the

current flowing in the main branch of the circuit. To do so, it is first necessary to calculate the equivalent resistance of the circuit:

Using the equivalent resistance of the circuit, it is then possible to calculate

the current flowing in the circuit:

24 V 50 200 125



Now that we have calculated all currents in the circuit, it is possible to verify

Kirchhoff’s current law. According to this law, the sum of the currents , ,

and flowing in the resistors should be equal to the current flowing in the

main branch of the circuit. This equation is given below:

The above equation thus confiRMS Kirchhoff’s current law.

Solving mixed circuits

In Exercise 6 and in this exercise, you learned how to solve series and parallel circuits. It often happens, however, that a circuit is connected neither entirely in series nor in parallel, but rather is a combination of both. Such a circuit is called a mixed circuit (or series-parallel circuit). Although more complex to solve than simple series or parallel circuits, they can be solved using exactly the same principles.

Example 1

Consider the circuit shown in Figure 102.

Figure 102. Mixed circuit containing four resistors.

To solve this circuit (i.e., to find the values of the different parameters in the circuit), it is first necessary to simplify some resistor arrangements. As the figure shows, resistors and are connected in series. Their equivalent

resistance can thus be calculated, allowing us to replace these

resistors with a single resistor whose resistance is equal to the equivalent resistance calculated as follows:

40

60

150

20

24 V



The circuit in Figure 102 can then be simplified to the circuit in Figure 103.

Figure 103. Simplification of the circuit in Figure 102.

We now observe that and are connected in parallel. Their equivalent

resistance can thus be calculated, allowing us to replace these

resistors with a single resistor whose resistance is equal to the equivalent resistance calculated as follows:



100

150

20

24 V

60

20

24 V



We now observe that and are connected in series. Their

equivalent resistance can thus be calculated, allowing us to

replace these resistors with a single resistor whose resistance is equal to the equivalent resistance calculated as follows:



Now that all resistors have been simplified to a single resistor, it is possible to

calculate that the current flowing in the circuit is equal to:

Knowing the value of the current , it is possible to progress backward through the circuit diagrams to calculate all parameters in the circuit. Therefore, the voltages across the resistors in Figure 104 are calculated as follows:

Since the voltage across two parallel branches is equal, the voltage across each parallel resistor in Figure 103 is equal to voltage . The current flowing

through each parallel resistor can then be calculated as follows:

80

24 V



Finally, knowing the current flowing in the two series resistors in Figure 102, it is possible to calculate the voltage across each resistor:

All circuit parameters have been calculated. It is possible to use Kirchhoff’s voltage and current laws to confirm each parameter calculated above. For example, Kirchhoff’s current law confiRMS that:

Example 2

Consider the circuit shown in Figure 106.

Figure 106. Mixed circuit containing five resistors.

The first step to solve the circuit is to simplify parallel resistors and , as well as and . This is done below:

30

45

20

25

24 V

50



And:



Since all resistors are now in series, it is possible to find their equivalent resistance:

30

16.7

24 V

13.9





The current flowing in the circuit can now be calculated:

From the current , it is then possible to calculate all other parameters in the circuit, in the following order:

60.6 24 V



Printed circuit boards

A printed circuit board, or PCB, is a thin board on which electronic components are mounted. The electronic components connect together using conductive lines printed or etched into the board to form a circuit.

Most printed circuit boards are made from fiberglass with copper traces. They can have traces on one side (single layer) for simple electronic circuits, but they can also have many layers for complex circuits.

Figure 109 shows the two sides of the PCB in the Printed Circuit Board module. As can be seen, this PCB has traces on both sides (double layer).

Figure 109. Front and back views of the PCB in the Printed Circuit Board module.

Observe the size of resistors , , and , capacitors and , as well as

diode on the PCB. These components are specially designed to be mounted on the surface of printed circuit boards, and are called surface-mount components. Surface-mount components are usually small, allow the mounting of many components per unit area, and offer a good resistance to shocks and vibration.

TraceTrace

Front view Back view



Training system module

Printed Circuit Board module

Figure 110. Printed Circuit Board module.

The printed circuit board (PCB) in the Printed Circuit Board module consists of electrical circuits. Each circuit is illustrated with a drawing to show how each component in the circuit is electrically connected. Connections to the circuits are made through 2 mm terminals (note that this type of terminal is often called test points).

The dashed symbols of an ac power source drawn between test points D-E, H-I, and L-M show that an external ac power source is required to power the circuit.

The printed circuit board is divided into three sections. The upper section consists of a circuit of three resistors and a slide switch. This circuit is used in the Procedure of this exercise to familiarize yourself with equivalent resistance calculations in mixed circuits, and resistance measurements on a PCB.

The middle section of the PCB is also used in the Procedure of this exercise. The two circuits in this section of the PCB are designed to familiarize yourself with Kirchhoff's voltage and current laws. The lower section of the PCB contains components that will be learned later in the manual.

The Printed Circuit Board module is also equipped with two ground terminals.

Ground terminal

Middle section

Upper section

Lower section

Exercise 7 – Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law Procedure Outline


The Procedure is divided into the following sections:

Setup

Calculating and measuring the voltages and currents in a parallel circuitSolving the circuit through mathematical calculations. Solving the circuit through circuit measurements.

Calculating and measuring the voltages and currents in a mixed circuitSolving the circuit through mathematical calculations. Solving the circuit through circuit measurements.

Light intensity control circuit connected in parallel with a resistorKirchhoff’s voltage law. Kirchhoff’s current law.

Resistance measurements on a printed circuit board

Voltage measurements on a printed circuit boardCircuit A. Circuit B.

High voltages are present in this laboratory exercise. Do not make or modify any

banana jack connections with the power on unless otherwise specified.

Setup

In this section, you will install the training system modules in the workstation.

1. Refer to the Equipment Utilization Chart in Appendix A to obtain the list of equipment required to perform this exercise.

Install the equipment required in the workstation.

Make sure that all fault switches are set to the O (off) position.

PROCEDURE OUTLINE

PROCEDURE

Exercise 7 – Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law Procedure


Calculating and measuring the voltages and currents in a parallel circuit

In this section, you will connect a parallel circuit containing two resistors. You will solve the circuit by calculating the voltage and current across each resistor, as well as the source current and equivalent resistance of the circuit. You will then measure these parameters and compare the measured values to the calculated values. You will confirm that Kirchhoff’s current law is respected in the circuit, and determine the effect on the circuit parameters of removing one of the resistors.

2. Consider the circuit shown in Figure 111.

Figure 111. Circuit containing two parallel resistors.

Solving the circuit through mathematical calculations

3. Solve the circuit in Figure 111 by determining all of the circuit parameters indicated below.

Equivalent resistance

Current A

Resistor voltage V

Resistor voltage V

Resistor current A

Resistor current A

50

24 V 250



a The equivalent resistance of parallel resistors is calculated using the following equation:

Also, Ohm’s law states that:

Solving the circuit through circuit measurements

4. Set up the circuit shown in Figure 112. Use the 50 and 250 resistors of

the Resistors module to implement resistors and respectively. This circuit allows measurement of the equivalent resistance of the two parallel resistors.

Figure 112. Measuring the equivalent resistance in the circuit of Figure 111.

5. Using an ohmmeter, measure the equivalent resistance of the circuit.

Record the resistance value below.

Measured equivalent resistance

6. Make sure that the main power switch on the Power Source module is set to the O (off) position, then connect it to an ac power outlet.

Set up the circuit shown in Figure 113. For the moment, connect only the voltmeters to the circuit.

50

250



Figure 113. Measuring the voltages and currents in the circuit of Figure 111.

To reduce the risk of electrical shock, connect all ground (green) terminals of the

modules in series with the ground (green) terminal of the power source.

7. Turn the power source on.

Measure the voltages and across each resistor. Record the voltage

values below.

Measured voltage V

Measured voltage V

8. Measure the current flowing in the circuit, as well as the currents

and flowing in each resistor by successively connecting the ammeter at

the locations shown in Figure 113. Record the current values below.

b Make sure to turn the power source off before making any change to the circuit connections. Also make sure to turn it back on before any measurement.

Measured current A

Measured current A

Measured current A

50 250

24 V



9. Compare the circuit parameter values you measured in steps 5, 7, and 8 to the values you calculated in step 3. Are the corresponding values virtually equal?

Yes No

10. Using the current values you measured in step 8, can you confirm through calculations that Kirchhoff’s current law is respected when the current divides into two parallel branches?

Yes No

11. Indicate if the value of the circuit parameters listed below will increase,

decrease, or will not change, if you were to remove resistor from the circuit in Figure 111 without modifying any other component in the circuit.

Equivalent resistance of the circuit:

Current :

Resistor voltage :

Resistor current :

12. Turn the power source off.



Calculating and measuring the voltages and currents in a mixed circuit

In this section, you will connect a mixed circuit containing three resistors. You will solve the circuit by calculating the voltage and current across each resistor, as well as the source current and equivalent resistance of the circuit. You will then measure these parameters and compare the measured values to the calculated values. You will confirm that Kirchhoff’s current law is respected in the circuit.

13. Consider the circuit shown in Figure 114.

Figure 114. Mixed circuit containing three resistors.

Solving the circuit through mathematical calculations

14. Solve the circuit in Figure 114 by determining all of the circuit parameters indicated below.


Current A

Resistor voltage V

Resistor voltage V

Resistor voltage V

Resistor current A

Resistor current A

Resistor current A

250

24 V

500

50




Also, the equivalent resistance of series resistors is calculated using the following equation:

Finally, Ohm’s law states that:

Solving the circuit through circuit measurements

15. Set up the circuit shown in Figure 115. Use the 50 resistor to implement

resistor , and the 250 and 500 resistors to implement resistors

and respectively. This circuit allows measurement of the equivalent resistance of a resistor connected in series with two resistors connected in parallel.

Figure 115. Measuring the equivalent resistance in the circuit of Figure 114.

16. Using an ohmmeter, measure the equivalent resistance of the

circuit. Record the resistance value below.

Measured equivalent resistance

250 500

50



17. Set up the circuit shown in Figure 116. For the moment, do not connect any of the voltmeters and ammeters to the circuit.

Figure 116. Measuring the voltages and currents in the circuit of Figure 114.


Measure the voltages , , and across each resistor by successively

connecting the voltmeter at the locations shown in Figure 116. Record the voltage values below.

Measured voltage V

Measured voltage V

Measured voltage V

19. Measure the current flowing in resistor , as well as the currents

and flowing in resistors and by successively connecting the

ammeter at the locations shown in Figure 116. Record the current values below.

b Make sure to turn the power source off before making any change to the circuit connections. Also make sure to turn it back on before any measurement.

Measured current A

Measured current A

Measured current A

250

500

50

24 V



20. Compare the circuit parameter values you measured in steps 16, 18, and 19 to the values you calculated in step 14. Are the corresponding values virtually equal?

Yes No

21. Using the current values you measured in step 19, can you confirm through calculations that Kirchhoff’s current law is respected when the current divides into two parallel branches?

a According to Kirchhoff’s current law:

Yes No

Light intensity control circuit connected in parallel with a resistor

In this section, you will connect a light intensity control circuit (implemented using a switch and a resistor) connected in parallel with a resistor. You will measure the voltage across each load, and confirm that Kirchhoff’s voltage law is respected, regardless of the light intensity setting. You will then measure the current flowing in the circuit and in each load, and confirm that Kirchhoff’s current law is respected, regardless of the light intensity setting.

22. Set up the circuit shown in Figure 117. Use the 500 and 250 resistors of

the Resistors module to implement resistor and respectively. For the moment, do not connect any of the voltmeters and ammeters to the circuit.



Figure 117. Light intensity control circuit connected in parallel with a resistor.

Kirchhoff’s voltage law

23. Set the toggle switch to position B.

Turn the power source on.

Measure the source voltage , the voltage across the resistor , the

voltage across the resistor , and the voltage across indicator

light , by successively connecting the voltmeter at the locations shown in Figure 117. Record the voltage values below.

b Make sure to turn the power source off before making any change to the circuit connections. Also make sure to turn it back on before taking any measurements.

Measured source voltage V

Measured resistor voltage V


Measured indicator light voltage V

24 V

250

500

A B

Toggle switch



24. Using the voltage values you measured in the previous step when the toggle switch is set to B, can you confirm through calculations that Kirchhoff’s voltage law is respected for each closed loop of the circuit in Figure 117?

Yes No

25. Set the toggle switch to position A.

Repeat the measurement of the source voltage , the resistor voltage ,

and the indicator light voltage . Record the voltage values below.

Measured source voltage V


Measured indicator light voltage V

26. Using the voltage values you measured in the previous step when the toggle switch is set to position A, can you confirm through calculations that Kirchhoff’s voltage law is respected for each closed loop of the circuit in Figure 117?

Yes No




27. Set the toggle switch to position B.

28. Measure the current flowing in the circuit, as well as the current flowing

in resistor , and the current flowing in the indicator light , by

successively connecting the ammeter at the locations shown in Figure 117.


Measured source current A

Measured resistor current A

Measured indicator light current A

29. Using the current values you measured in the previous step when the toggle switch is set to position B, can you confirm through calculations that Kirchhoff’s current law is respected when the current divides into parallel branches?

Yes No

30. Set the toggle switch to position A.

Repeat the measurement of the source current , the resistor current ,

and the indicator light current . Record the current values below.


Measured source current A

Measured resistor current A

Measured indicator light current A



31. Using the current values you measured in the previous step when the toggle switch is set to position A, can you confirm through calculations that Kirchhoff’s current law is respected when the current divides into parallel branches?

Yes No


Resistance measurements on a printed circuit board

In this section, you will determine the value of the resistors of the printed circuit board using an ohmmeter.

33. Consider the circuit in the upper section of the PCB shown in Figure 118.

Figure 118. Circuit used to determine the value of resistors.

34. Determine the resistance of resistors , , and by making the appropriate resistance measurements.

b The identification of the various current paths in the circuit when switch is open and when it is closed may help you to determine the resistance values.

Resistance of resistor



B A

C



35. Resistors , , and are located at the left of switch on the PCB. Using an ohmmeter, determine to which resistor , , and corresponds.

Resistor at the left:

Resistor at the center:

Resistor at the right:

Voltage measurements on a printed circuit board

In this section, you will calculate the voltage at the output of a mixed circuit containing three resistors. You will solve the circuit by calculating the voltage and current across the resistors, as well as the source current and equivalent resistance of the circuit. You will then measure these parameters and compare the measured values to the calculated values.

Circuit A

36. Consider the circuit at the left in the middle section of the PCB shown in Figure 119.

Figure 119. Circuit A used to perform voltage measurements on a PCB.

37. Calculate the voltage across terminals F and G when a 24 V voltage is applied across terminals D and E. To do so, determine the circuit parameters indicated below.


a Since the loop containing resistor is open between terminals F and G, no current flows in this branch. For this reason, the power source is not affected by the presence of resistor and it is not necessary to take account of this resistor in the calculation of the equivalent resistance.

D

E

F

G

24 V

2 k

1 k 1 k



Current A

Resistor voltage V

Voltage across terminals F and G V

a The equivalent resistance of series resistors is calculated using the following equation:


38. Set up the circuit shown in Figure 120 by using the circuit at the left in the middle section of the PCB.

Figure 120. Circuit A used to perform voltage measurements on a PCB.


Measure the voltage across terminals F and G. Record the voltage value below.

Measured voltage across terminals F and G V

40. Compare the voltage you measured across terminals F and G in the previous step to the value you calculated in step 37. Are the values virtually equal?

Yes No


D

E

F

G

24 V

2 k

1 k 1 k



Circuit B

42. Consider the circuit at the right in the middle section of the PCB shown in Figure 121.

Figure 121. Circuit B used to perform voltage measurements on a PCB.

43. Calculate the voltage across terminals J and K when a 24 V voltage is applied across terminals H and I. To do so, determine the circuit parameters indicated below.


Current A

Resistor voltage V

Voltage across terminals J and K V


Also, the equivalent resistance of series resistors is calculated using the following equation:

H

I

J

K

24 V

2 k

1 k

1 k




44. Set up the circuit shown in Figure 124 by using the circuit at the right in the middle section of the PCB.

Figure 122. Circuit B used to perform voltage measurements on a PCB.


Measure the voltage across terminals J and K. Record the voltage value below.

Measured voltage across terminals J and K V

46. Compare the voltage you measured across terminals J and K in the previous step to the value you calculated in step 43. Are the values virtually equal?

Yes No

47. Turn off the power source and the measuring instruments.

Disconnect your circuit.

Return the leads to their storage location.

H

I

J

K

24 V

2 k

1 k

1 k

Exercise 7 – Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law Conclusion


In this exercise, you learned to calculate the equivalent resistance of multiple resistors in parallel circuits. You were introduced to Kirchhoff’s current law and learned to apply it to electrical circuits. You also learned to solve mixed circuits. Finally, you were also introduced to printed circuit boards.

1. Calculate the current flowing in the circuit in Figure 123 from the indicated parameters.

Figure 123. Circuit for review question 1.

2. State the basic principle of Kirchhoff’s current law.

CONCLUSION

REVIEW QUESTIONS

85

25

50 24 V

Exercise 7 – Solving Parallel and Mixed Circuits, and Kirchhoff’s Current Law Review Questions


3. Consider the circuit in Figure 124. Calculate the equivalent resistance

from the indicated parameters.

Figure 124. Circuit for review questions 3, 4, and 5.

4. Consider the circuit in Figure 124. Calculate the voltage across

resistor from the indicated parameters and the values you calculated in the previous question.

5. Consider the circuit in Figure 124. Calculate the current flowing in

resistor from the indicated parameters and the values you calculated in the previous questions.

30

25

90

15

60 V

electricity fundamentals, 7 solving parallel and mixed ... · kirchhoff’s current law is a law...

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