Practice: Parallel Circuits

EET-137
Fall 2024

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In this unit we analyze parallel circuits. We are going to learn about Kirchoff’s Current Law. It a main law in determining how circuits work and how we can analyze them. Parallel circuits have the same voltage, but the current is divided up between the elements. We will analyze and exam how this behaves in different circuits. This unit will also introduce a new element, the ideal current source.

This is a two week unit, which means this worksheet is bigger than most.

Problems 1-25 are for primary practice. You should feel comfortable with your ability to solve all of these problems. We will talk about many of them in class. You do not need to work all of them. The knowledge from them will be tested on the quiz.

Problem 27 and 28 are your Demo Problems for this unit.

Problems 29+ are for you if you want extra practice. If you think there is a topic or concept you want more practice on, these problems are for that.

Questions

Question 1

Identify which of these circuits is a parallel circuit (there may be more than one shown!):

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Question 2

In a parallel circuit, certain general rules may be stated with regard to quantities of voltage, current, resistance, and power. Express these rules, using your own words:

"In a parallel circuit, voltage . . ."

"In a parallel circuit, current . . ."

"In a parallel circuit, resistance . . ."

"In a parallel circuit, power . . ."

For each of these rules, explain why it is true.

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Question 3

Draw connecting wires that will create a parallel circuit, such that current (conventional flow notation) will follow the directions shown by the arrows near each resistor:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 8 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 8 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 4

Qualitatively compare the voltage and current for each of the three light bulbs in this circuit (assume the three light bulbs are absolutely identical):

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Question 5

Determine the amount of voltage impressed across each resistor in this circuit:

Hint: locate all the points in this circuit that are electrically common to one another!

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Question 6

According to Ohm’s Law, how much current goes through each of the two resistors in this circuit?

Draw the paths of all currents in this circuit.

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Question 7

What will happen in this circuit as the switches are sequentially turned on, starting with switch number 1 and ending with switch number 3?

Describe how the successive closure of these three switches will impact:

The voltage drop across each resistor
The current through each resistor
The total amount of current drawn from the battery
The total amount of circuit resistance "seen" by the battery

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Question 8

Use Kirchhoff’s Current Law to calculate the magnitudes and directions of currents through all resistors in this circuit:

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Question 9

Use Kirchhoff’s Current Law to calculate the magnitude and direction of the current through resistor \(R_4\) in this resistor network:

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Question 10

Calculate and label the currents at each node (junction point) in this circuit:

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Question 11

Calculate and label the currents at each node (junction point) in this circuit:

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Question 12

Suppose you are asked to determine whether or not each battery in this circuit is charging or discharging:

To do so, of course, you need to determine the direction of current through each battery. Unfortunately, your multimeter does not have the current rating necessary to directly measure such large currents, and you do not have access to a clamp-on (magnetic) ammeter capable of measuring DC current.

You are about to give up when a master technician comes along and uses her multimeter (set to measure millivolts) to take these three voltage measurements:

"There," she says, "it’s easy!" With that, she walks away, leaving you to figure out what those measurements mean. Determine the following:

How these DC millivoltage measurements indicate direction of current.
Whether or not each battery is charging or discharging.
Whether or not these millivoltage measurements can tell you how much current is going through each battery.
How this general principle of measurement may be extended to other practical applications.

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Question 13

Complete the table of values for this circuit:

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Question 14

What will happen to the current through R1 and R2 if resistor R3 fails open?

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Question 15

Predict how all component voltages and currents in this circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults):

Voltage source \(V_1\) fails with increased output:
Resistor \(R_1\) fails open:
Resistor \(R_3\) fails open:
Solder bridge (short) across resistor \(R_2\):

For each of these conditions, explain why the resulting effects will occur.

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Question 16

In an effort to obtain greater overcurrent ratings than a single fuse can provide, an engineer decides to wire two 100 amp fuses in parallel, for a combined rating of 200 amps:

However, after a few years of operation, the system begins blowing fuses even when the ammeter registers less than 200 amps of load current. Upon investigation, it is found that one of the fuse holders had developed corrosion on a terminal lug where one of the wire connects:

Explain how a small accumulation of corrosion led to this condition of fuses blowing when there was no overcurrent condition (load current less than 200 amps), and also why connecting fuses in parallel like this is generally not a good idea.

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Question 17

Although the voltage divider and current divider equations are very useful in circuit analysis, they are easily confused for one another because they look so similar:

\[V_R = V_{total}\left( {R \over R_{total}} \right) {\hskip 100pt} I_R = I_{total}\left( {R_{total} \over R} \right)\]

Specifically, it is easy to forget which way the resistance fraction goes for each one. Is it \({R \over R_{total}}\) or is it \({R_{total} \over R}\) ? Simply trying to memorize which fraction form goes with which equation is a bad policy, since memorization of arbitrary forms tends to be unreliable. What is needed is recognition of some sort of principle that makes the form of each equation sensible. In other words, each equation needs to make sense why it is the way it is.

Explain how one would be able to tell that the following equations are wrong, without referring to a book:

\[V_R = V_{total}\left( {R_{total} \over R} \right) {\hskip 20pt {\bf Wrong equations!} \hskip 20pt} I_R = I_{total}\left( {R \over R_{total}} \right)\]

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Question 18

The circuit shown here is commonly referred to as a current divider. Calculate the voltage dropped across each resistor, the current drawn by each resistor, and the total amount of electrical resistance "seen" by the 9-volt battery:

Current through the 2 k\(\Omega\) resistor =
Current through the 3 k\(\Omega\) resistor =
Current through the 5 k\(\Omega\) resistor =
Voltage across each resistor =
\(R_{total}\) =

Can you think of any practical applications for a circuit such as this?

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Question 19

Calculate the proper value of resistance R₂ needs to be in order to draw 40% of the total current in this circuit:

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Question 20

A student is trying to use the "current divider formula" to calculate current through the second light bulb in a three-lamp lighting circuit (typical for an American household):

The student uses Joule’s Law to calculate the resistance of each lamp (240 \(\Omega\)), and uses the parallel resistance formula to calculate the circuit’s total resistance (80 \(\Omega\)). With the latter figure, the student also calculates the circuit’s total (source) current: 1.5 A.

Plugging this into the current divider formula, the current through any one lamp turns out to be:

\[I = I_{total} \left( {R_{total} \over R} \right) = 1.5 { A} \left( {80 \> \Omega \over 240 \> \Omega} \right) = 0.5 { A}\]

This value of 0.5 amps per light bulb correlates with the value obtained from Joule’s Law directly for each lamp: 0.5 amps from the given values of 120 volts and 60 watts.

The trouble is, something doesn’t add up when the student re-calculates for a scenario where one of the switches is open:

With only two light bulbs in operation, the student knows the total resistance must be different than before: 120 \(\Omega\) instead of 80 \(\Omega\). However, when the student plugs these figures into the current divider formula, the result seems to conflict with what Joule’s Law predicts for each lamp’s current draw:

\[I = I_{total} \left( {R_{total} \over R} \right) = 1.5 { A} \left( {120 \> \Omega \over 240 \> \Omega} \right) = 0.75 { A}\]

At 0.75 amps per light bulb, the wattage is no longer 60 W. According to Joule’s Law, it will now be 90 watts (120 volts at 0.75 amps). What is wrong here? Where did the student make a mistake?

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Question 21

Ideal voltage sources and ideal current sources, while both being sources of electrical power, behave very differently from one another:

Explain how each type of electrical source would behave if connected to a variable-resistance load. As this variable resistance were increased and decreased, how would each type of source respond?

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Question 22

A voltage source is a source of electricity that (ideally) outputs a constant voltage. That is, a perfect voltage source will hold its output voltage constant regardless of the load imposed upon it:

In real life, there is no such thing as a perfect voltage source, but sources having extremely low internal resistance come close.

Another type of electricity source is the current source, which (ideally) outputs a constant current regardless of the load imposed upon it. A common symbol for a current source is a circle with an arrow inside (always pointing in the direction of conventional flow, not electron flow!). Another symbol is two intersecting circles, with an arrow nearby pointing in the direction of conventional flow:

Predict how an ideal current source would behave for the following two load scenarios:

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Question 23

Calculate the total current output to the load resistor by this set of parallel-connected current sources:

Also, calculate the voltage dropped across \(R_{load}\).

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Question 24

Calculate the total current output to the load resistor by this set of parallel-connected current sources:

Also, calculate the voltage dropped across \(R_{load}\).

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Question 25

Predict how all component voltages and currents in this circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults):

Current source \(I_1\) fails with increased output:
Resistor \(R_1\) fails open:
Resistor \(R_3\) fails open:
Solder bridge (short) across resistor \(R_2\):

For each of these conditions, explain why the resulting effects will occur.

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Question 26

Demo Problems

These next problems are your Demonstration Problems. These are graded Problems.

Question 27

Determine the current through battery #2 in this power system, if the generator is outputting 50 amps, battery #1 is charging at a rate of 22 amps, and the light bulbs draw 5 amps of current each. Be sure to indicate whether battery #2 is charging or discharging:

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Question 28

Suppose an ammeter has a range of 0 to 1 milliamp, and an internal resistance of 1000 \(\Omega\):

Show how a single resistor could be connected to this ammeter to extend its range to 0 to 10 amps. Calculate the resistance of this "range" resistor, as well as its necessary power dissipation rating.

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Question 29

Additional Practice Problems

The remaining problems in this worksheet are for additional practice. These are good problems that will help you if you have struggled with the earlier problems in the worksheet.

Question 30

Calculate the percentage of total current for each resistor in this parallel circuit:

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Question 31

In this circuit there is at least one ammeter that is not reading correctly:

Apply Kirchhoff’s Current Law to this circuit to prove why all three current measurements shown here cannot be correct.

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Question 32

Calculate the voltage dropped across the load resistor, and the current through the load resistor, for the following load resistance values in this circuit:

R_{load}

E_{load}

I_{load}

1 kΩ

2 kΩ

5 kΩ

8 kΩ

10 kΩ

Do the "boxed" components in this circuit behave more like a constant voltage source, or a constant current source? Explain your answer.

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Question 33

Choose two resistor values such that one resistor passes 25% of the total current, while the other resistor passes 75% of the total current:

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Question 34

A common saying about electricity is that "it always takes the path of least resistance." Explain how this proverb relates to the following circuit, where electric current from the battery encounters two alternate paths, one being less resistive than the other:

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Question 35

Explain how an analog ammeter could be used to measure the current output of the generator in this circuit, as it charges the two batteries and energizes the three light bulbs. Be sure to connect the ammeter in the circuit in such a way that the meter needle does not drive "downscale"!

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Question 36

How much water must flow out of the pipe with the question-mark symbols next to it?

Explain how this hydraulic example relates to Kirchhoff’s Current Law (KCL) in an electric circuit.

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Question 37

Draw connecting wires that will create a parallel circuit, such that current (conventional flow notation) will follow the directions shown by the arrows near each resistor:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 7 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 7 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 38

Draw connecting wires that will create a parallel circuit with all the components shown:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 6 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 6 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 39

Draw connecting wires that will create a parallel circuit, such that current (conventional flow notation) will follow the directions shown by the arrows near each resistor:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 3 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 3 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 40

Draw connecting wires that will create a parallel circuit with the three resistors and battery:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 12 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 12 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 41

Draw connecting wires that will create a parallel circuit, such that voltage will drop across each resistor in the polarity shown by the (+) and (\(-\)) symbols:

Suggestions for Socratic discussion

Supposing the battery has a voltage of 1.5 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the voltage dropped by each resistor.
Supposing the battery has a voltage of 1.5 volts, and all resistors are 1 k\(\Omega\) in resistance value, calculate the current passing through each resistor as well as the current passing through the battery.

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Question 42

Calculate the necessary resistor values to produce the following percentage splits in current:

Hint: one resistor carries three times the current of the other.

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Question 43

Predict how all component voltages and currents in this circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults):

Current source \(I_1\) fails with increased output:
Solder bridge (short) across resistor \(R_1\):
Solder bridge (short) across resistor \(R_2\):
Resistor \(R_3\) fails open:

For each of these conditions, explain why the resulting effects will occur.

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Question 44

The generator in this power system is disconnected for routine servicing by removing its fuse, leaving the two batteries to supply all power to the bank of light bulbs:

The two batteries are rated such that they are supposed to provide at least 10 hours of back-up power in the event of a generator "outage" such as this. Unfortunately, the light bulbs begin to dim much sooner than expected. Something is wrong in the system, and you are asked to figure out what.

Explain what steps you would take to diagnose the problem in this circuit, commenting on any relevant safety measures taken along the way.

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Question 45

Draw the connecting wires on this terminal strip so that the three light bulbs are wired in parallel with each other and with the battery.

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Question 46

Complete the table of values for this circuit:

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Question 47

Calculate the total amount of current that the battery must supply to this parallel circuit:

Now, using Ohm’s Law, calculate total resistance (\(R_{total}\)) from total (source) voltage \(V_{total}\) and total (source) current \(I_{total}\).

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Question 48

In the following circuit, an adjustable voltage source is connected in series with a resistive load and another voltage source:

Determine what will happen to the current in this circuit if the adjustable voltage source is increased.

In this next circuit, an adjustable voltage source is connected in series with a resistive load and a current source:

Now determine what will happen to the current in this second circuit if the adjustable voltage source is increased.

One way to define electrical resistance is by comparing the change in applied voltage (\(\Delta V\)) to the change in resultant current (\(\Delta I\)). This is mathematically expressed by the following ratio:

\[R = {\Delta V \over \Delta I}\]

From the perspective of the adjustable voltage source (\(V_{adjust}\)), and as defined by the above equation, which of these two circuits has the greatest resistance? What does this result suggest about the equivalent resistance of a constant-voltage source versus the equivalent resistance of a constant-current source?

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Question 49

Determine the amount of current conducted by each resistor in this circuit, if each resistor has a color code of Org, Org, Red, Gld (assume perfectly precise resistance values – 0/ error):

Also, determine the following information about this circuit:

Voltage across each resistor
Power dissipated by each resistor
Ratio of each resistor’s current to battery current (\(I_R \over I_{bat}\))
Ratio of total circuit resistance to each resistor’s resistance (\(R_{total} \over R\))

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Question 50

Calculate one possible set of resistor values that would produce the following percentage splits in current:

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Question 51

We know that the voltage in a parallel circuit may be calculated with this formula:

\[E = {I_{total} R_{total}}\]

We also know that the current through any single resistor in a parallel circuit may be calculated with this formula:

\[I_R = {E \over R}\]

Combine these two formulae into one, in such a way that the \(E\) variable is eliminated, leaving only \(I_R\) expressed in terms of \(I_{total}\), \(R_{total}\), and \(R\).

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Answers

Answer 1

Circuits D and E are parallel circuits.

Answer 2

"In a parallel circuit, voltage is equal across all components."

"In a parallel circuit, current_s add to equal the total_."

"In a parallel circuit, resistance_s diminish to equal the total_."

"In a parallel circuit, power dissipations add to equal the total."

Answer 3

Bear in mind that this is not the only possible circuit solution:

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 4

The voltage dropped across each of the lights bulbs is guaranteed to be equal. The current through each of the light bulbs, in this particular case (with identical bulbs), happens to be equal.

Answer 5

Each resistor has 15 volts across it in this circuit.

Answer 6

\(I_{R(2.2k)} = 10.91 { mA}\)

\(I_{R(4.7k)} = 5.11 { mA}\)

Follow-up question: how much total current does the battery supply to the circuit, given these individual resistor currents?

Answer 7

I won’t explain what happens when each of the switches is closed, but I will describe the effects of the first switch closing:

As the first switch (SW1) is closed, the voltage across resistor R1 will increase to full battery voltage, while the voltages across the remaining resistors will remain unchanged from their previous values. The current through resistor R1 will increase from zero to whatever value is predicted by Ohm’s Law (full battery voltage divided by that resistor’s resistance), and the current through the remaining resistors will remain unchanged from their previous values. The amount of current drawn from the battery will increase. Overall, the battery "sees" less total resistance than before.

Answer 8

It is not necessary to know anything about series-parallel or even parallel circuits in order to solve the \(R_4\)’s current – all one needs to know is how to use Kirchhoff’s Current Law.

Answer 9

Answer 10

Answer 11

Answer 12

The master technician exploited the inherent resistance of each fuse as a current-indicating shunt resistor. The measurements indicate both batteries are charging.

Answer 13

Answer 14

If you think the currents through R1 and R2 would increase, think again! The current through R1 and the current through R2 both remain the same as they were before R3 failed open.

Answer 15

Voltage source \(V_1\) fails with increased output: All resistor currents increase, all resistor voltages increase, current through \(V_1\) increases.
Resistor \(R_1\) fails open: All resistor voltages remain unchanged, current through \(R_1\) decreases to zero, currents through \(R_2\) and \(R_3\) remain unchanged, current through \(V_1\) decreases by the amount that used to go through \(R_1\).
Resistor \(R_3\) fails open: All resistor voltages remain unchanged, current through \(R_3\) decreases to zero, currents through \(R_1\) and \(R_2\) remain unchanged, current through \(V_1\) decreases by the amount that used to go through \(R_3\).
Solder bridge (short) across resistor \(R_2\): Theoretically, all resistor voltages remain unchanged while a near-infinite amount of current goes through the shorted \(R_2\). Realistically, all resistor voltages will decrease to nearly zero while the currents through \(R_2\) and \(V_1\) will increase dramatically.

Answer 16

Here is an electrical calculation to help explain the fuse-blowing problem. Calculate the current through each resistor in this circuit:

Answer 17

The resistance fraction must always be less than 1.

Answer 18

Current through the 2 k\(\Omega\) resistor = 4.5 mA
Current through the 3 k\(\Omega\) resistor = 3 mA
Current through the 5 k\(\Omega\) resistor = 1.8 mA
Voltage across each resistor = 9 volts
\(R_{total}\) = 967.74 \(\Omega\)

How much current is drawn from the battery in this circuit? How does this figure relate to the individual resistor currents, and to the total resistance value?

Answer 19

R₂ = 1.5 kΩ

Follow-up question: explain how you could arrive at a rough estimate of \(R_2\)’s necessary value without doing any algebra. In other words, show how you could at least set limits on \(R_2\)’s value (i.e. "We know it has to be less than . . ." or "We know it has to be greater than . . .").

Answer 20

The student incorrectly assumed that total current in the circuit would remain unchanged after the switch opened. By the way, this is a very common conceptual misunderstanding among new students as they learn about parallel circuits!

Answer 21

An ideal voltage source will output as much or as little current as necessary to maintain a constant voltage across its output terminals, for any given load resistance. An ideal current source will output as much or as little voltage as necessary to maintain a constant current through it, for any given load resistance.

Answer 22

Follow-up question: identify the polarity of the voltage drops across the resistors in the circuits shown above.

Answer 23

The total current in this circuit is 27 mA, and the load voltage is 40.5 volts.

Follow-up question: trace the direction of current through all three sources as well as the load resistor. Compare these directions with the polarity of their shared voltage. Explain how the relationship between voltage polarity and current direction relates to each component’s identity as either a source or a load.

Answer 24

The total current in this circuit is 4 mA, and the load voltage is 18.8 volts.

Follow-up question: indicate the polarity of the voltage across the load resistor with "+" and "-" symbols.

Answer 25

Current source \(I_1\) fails with increased output: All resistor currents increase, all resistor voltages increase, voltage across \(I_1\) increases.
Resistor \(R_1\) fails open: All resistor voltages increase, current through \(R_1\) decreases to zero, currents through \(R_2\) and \(R_3\) increase, current through \(I_1\) remains unchanged, voltage across \(I_1\) increases.
Resistor \(R_3\) fails open: All resistor voltages increase, current through \(R_3\) decreases to zero, currents through \(R_1\) and \(R_2\) increase, current through \(I_1\) remains unchanged, voltage across \(I_1\) increases.
Solder bridge (short) across resistor \(R_2\): All resistor voltages decrease to zero, currents through \(R_1\) and \(R_3\) decrease to zero, current through shorted \(R_2\) increases to equal \(I_1\), current through \(I_1\) remains unchanged, voltage across \(I_1\) decreases to zero.

Answer 26

Demo Problems

Answer 27

Graded Question

Answer 28

This is a graded question.

You will have to go beyond 3 sig figs for the value of your resistor. Make sure you draw a schematic or picture showing your final circuit.

Answer 29

Additional Practice Problems

Answer 30

R₁ = 50.3% of total current

R₂ = 27.6% of total current

R₃ = 22.1% of total current

Answer 31

Kirchhoff’s Current Law renders this scenario impossible: "the algebraic sum of all currents at a node must be zero."

Answer 32

I’ll let you do the calculations on your own! Hint: there is a way to figure out the answer without having to calculate all five load resistance scenarios.

Follow-up question #1: would you say that voltage sources are typically characterized as having high internal resistances or low internal resistances? What about current sources? Explain your answers.

Follow-up question #2: although it is difficult to find real devices that approximate ideal current sources, there are a few that do. An AC device called a "current transformer" is one of them. Describe which scenario would be the safest from a perspective of shock hazard: an open-circuited current transformer, or a shirt-circuited current transformer. Why is this?

Answer 33

There are many combinations of resistor values that will satisfy these criteria.

Answer 34

The 250 \(\Omega\) resistor will experience a current of 40 mA, while the 800 \(\Omega\) resistor will experience a current of 12.5 mA.

Answer 35

To measure generator output current, remove the fuse and connect the ammeter test leads to the two "clips" of the empty fuse holder, the red lead on the clip connecting to the generator, and the black lead on the clip connecting to the battery.

Follow-up question: describe any relevant safety procedures and considerations necessary prior to performing this task.

Answer 36

550 gallons per minute ("GPM"), assuming no leaks in the pipe.

Answer 37

Bear in mind that this is not the only possible circuit solution:

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 38

Bear in mind that this is not the only possible circuit solution!

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 39

Bear in mind that this is not the only possible circuit solution:

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 40

Bear in mind that this is not the only possible circuit solution:

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 41

Bear in mind that this is not the only possible circuit solution:

Challenge yourself by designing a different circuit to meet the same criteria!

Answer 42

There are many different sets of resistor values that will achieve this design goal!

Answer 43

Current source \(I_1\) fails with increased output: All resistor currents increase, all resistor voltages increase, voltage across \(I_1\) increases.
Solder bridge (short) across resistor \(R_1\): All resistor currents remain unchanged, voltage across resistor \(R_1\) decreases to zero, voltage across \(I_1\) decreases by the amount \(R_1\) previously dropped.
Solder bridge (short) across resistor \(R_2\): All resistor currents remain unchanged, voltage across resistor \(R_2\) decreases to zero, voltage across \(I_1\) decreases by the amount \(R_2\) previously dropped.
Resistor \(R_3\) fails open: Theoretically, all resistor currents remain unchanged while a large arc jumps across the failed-open \(R_3\) (and possibly across \(I_1\) as well). Realistically, all resistor currents will decrease to zero while the voltages across \(R_3\) and \(I_1\) will increase to maximum.

Answer 44

One simple procedure would be to connect a voltmeter in parallel with the light bulb bank, then remove the fuse for battery #1 and note the decrease in bus voltage. After that, replace the fuse for battery #1 and then remove the fuse for battery #2, again noting the decrease in bus voltage. If there is a problem with one of the batteries, it will be evident in this test. If both batteries are in good condition, but simply low on charge, that will also be evident in this test.

Answer 45

Answer 46

Answer 47

\(I_{total} = 40.0 { mA}\)

\(R_{total} = 250 \> \Omega\)

Follow-up question: without appealing to Ohm’s Law, explain why the total resistance is one-half as much as either of the individual resistances.

Answer 48

In the first circuit, current will increase as \(V_{adjust}\) is increased, yielding a finite total resistance. In the second circuit, current will remain constant as \(V_{adjust}\) is increased, yielding an infinite total resistance.

Follow-up question: calculate \(R\) as defined by the formula \({\Delta V \over \Delta I}\) for these two circuits, assuming \(V_{adjust}\) changes from 15 volts to 16 volts (1 volt \(\Delta V\)):

Answer 49

Current through each resistor = 3.33 mA

Voltage across each resistor = 11 V

Power dissipated by each resistor = 36.67 mW

Current ratio = \(1 \over 3\)

Resistance ratio = \(1 \over 3\)

Answer 50

There are many different sets of resistor values that will achieve this design goal! I’ll let you try to determine one of your own.

Answer 51

\[I_R = I_{total} \Bigl({R_{total} \over R}\Bigr)\]

How is this formula similar, and how is it different, from the "voltage divider" formula?

Kuphaldt, Tony. "Socratic Electronics." Socratic Electronics. Ibiblio.org, n.d. Web. 28 Dec. 2014.

Kuphaldt, Tony. "Socratic Instrumentation." Socratic Instrumentation. Ibiblio.org, n.d. Web. 29 Jan. 2016.

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