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Red lead on Ä", black lead on ground (Digital voltmeter reads +15 volts)

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Red lead on "B", black lead on ground (Digital voltmeter reads -15 volts)

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Red lead on Ä", black lead on "B" (Digital voltmeter reads +30 volts)

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Red lead on "B", black lead on Ä" (Digital voltmeter reads -30 volts)

Notes:
This question may be easily answered with only a voltmeter, two batteries, and a single "jumper" wire to connect the two batteries in series. It does not matter if the batteries are 15 volts each! The fundamental principle may still be investigated with batteries of any voltage, so this is a very easy demonstration to set up during discussion time.

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Notes:
Here, students must apply Kirchhoff's Voltage Law to determine what the voltmeter's indication will be. This question works well as a prelude to determining comparator (open-loop opamp) output polarities.

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Notes:
It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. While this approach makes students proficient in circuit theory, it fails to fully educate them.
Students don't just need mathematical practice. They also need real, hands-on practice building circuits and using test equipment. So, I suggest the following alternative approach: students should build their own "practice problems" with real components, and try to mathematically predict the various voltage and current values. This way, the mathematical theory "comes alive," and students gain practical proficiency they wouldn't gain merely by solving equations.
Another reason for following this method of practice is to teach students scientific method: the process of testing a hypothesis (in this case, mathematical predictions) by performing a real experiment. Students will also develop real troubleshooting skills as they occasionally make circuit construction errors.
Spend a few moments of time with your class to review some of the "rules" for building circuits before they begin. Discuss these issues with your students in the same Socratic manner you would normally discuss the worksheet questions, rather than simply telling them what they should and should not do. I never cease to be amazed at how poorly students grasp instructions when presented in a typical lecture (instructor monologue) format!
A note to those instructors who may complain about the "wasted" time required to have students build real circuits instead of just mathematically analyzing theoretical circuits:
                    What is the purpose of students taking your course?
If your students will be working with real circuits, then they should learn on real circuits whenever possible. If your goal is to educate theoretical physicists, then stick with abstract analysis, by all means! But most of us plan for our students to do something in the real world with the education we give them. The "wasted" time spent building real circuits will pay huge dividends when it comes time for them to apply their knowledge to practical problems.
Furthermore, having students build their own practice problems teaches them how to perform primary research, thus empowering them to continue their electrical/electronics education autonomously.
In most sciences, realistic experiments are much more difficult and expensive to set up than electrical circuits. Nuclear physics, biology, geology, and chemistry professors would just love to be able to have their students apply advanced mathematics to real experiments posing no safety hazard and costing less than a textbook. They can't, but you can. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits!

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Voltage across R1 (between top to bottom): 3.17 volts

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Polarity (+ and -) of voltage across R1: (-) on top, (+) on bottom

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Current (magnitude) through R1: 67.45 mA

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Direction of current through R1: upward, following conventional flow

Notes:
This is a good example of how Kirchhoff's Voltage Law is more than just an abstract tool for mathematical analysis - it is also a powerful technique for practical circuit diagnosis. Students must apply KVL to determine the voltage drop across R1, and then use Ohm's Law to calculate its current.
If students experience difficulty visualizing how KVL plays a part in the solution of this problem, show them this illustration:

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Notes:
Nothing here but series-parallel calculation practice, combined with KVL. Ask your students to explain how they calculated this voltage, because there is definitely more than one way to do it!

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Notes:
Nothing here but series-parallel calculation practice, combined with KVL. Ask your students to explain how they calculated this voltage, because there is definitely more than one way to do it!

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Notes:
The answer to this question is fairly trivial, but the real point of it is to get students thinking about how and why it is the way it is. One thing I've noticed as an instructor of electronics is that most students tend to follow the rule of proximity: the resistor's voltage drop polarity is determined by proximity to poles of the battery. The resistor terminal closest to the battery's negative terminal must be negative as well, or so the thinking goes. While this is sound reasoning in a simple circuit such as this, it can lead to confusion in circuits where there is more than one power source at work!

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Notes:
The answer to this question is fairly simple, but the real point of it is to get students thinking about how and why it is the way it is. One thing I've noticed as an instructor of electronics is that most students tend to follow the rule of proximity: the resistor's voltage drop polarity is determined by proximity to poles of the battery. The resistor terminal closest to the battery's negative terminal must be negative as well, or so the thinking goes.
In this particular circuit, though, the rule of proximity does not hold very well, and a different rule is necessary.

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Notes:
The answer to this question is fairly simple, but the real point of it is to get students thinking about how and why it is the way it is. One thing I've noticed as an instructor of electronics is that most students tend to follow the rule of proximity: the resistor's voltage drop polarity is determined by proximity to poles of the battery. The resistor terminal closest to the battery's negative terminal must be negative as well, or so the thinking goes.
In this particular circuit, though, the rule of proximity does not hold very well, and a different rule is necessary.

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Red lead on A, black lead on H = +24 volts

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Red lead on C, black lead on G = +12 volts

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Red lead on F, black lead on B = -12 volts

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Red lead on F, black lead on A = -18 volts

Notes:
Kirchhoff's Voltage Law (KVL) is very easily explored in real life with a set of batteries and "jumper wire" connections. Encourage your students to build battery circuits like the one shown in this question, to be able to see the results for themselves!
One really nice feature of digital multimeters (DMMs) is the ability to register negative as well as positive quantities. This feature is very useful when teaching Kirchhoff's Laws, with the algebraic (sign-dependent) summation of voltages and currents.

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Red lead on A, black lead on H = +12 volts

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Red lead on C, black lead on G = 0 volts

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Red lead on F, black lead on B = 0 volts

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Red lead on F, black lead on A = -6 volts

Notes:
Kirchhoff's Voltage Law (KVL) is very easily explored in real life with a set of batteries and "jumper wire" connections. Encourage your students to build battery circuits like the one shown in this question, to be able to see the results for themselves!
One really nice feature of digital multimeters (DMMs) is the ability to register negative as well as positive quantities. This feature is very useful when teaching Kirchhoff's Laws, with the algebraic (sign-dependent) summation of voltages and currents.
Students may arrive at more than one method for determining voltmeter indications in problems like these. Encourage this type of creativity during discussion time, as it both helps students gain confidence in being able to approach problems on their own terms, as well as educates those students who might be confused with the concept. Quite often the explanation of a peer is more valuable than the explanation of an instructor. When students are given the freedom to explore problem-solving methods, and then share those methods with their classmates, substantial learning always results.

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Notes:
Physical height (and depth) is a very useful analogy for electrical potential, helping students relate this abstract thing called "voltage" to more common differential measurements.

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Notes:
Determining differential voltages is a skill that many students find frustrating to attain. There is more than one way to explain how to arrive at +3 volts as the answer for V_AB, and it is good for students to see more than one way presented.

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Notes:
It may be good to remind your students that voltage is and forever will be a quantity between two points and never defined at a single point. The only reason we can say V_A = +9 volts and V_B = -6 volts is because there is a ground point in the circuit, which by convention is the point of reference for any voltages defined at other, single points in the circuit.

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Notes:
Discuss with your students multiple techniques of solving for these voltages, asking them first for their solution strategies.

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Notes:
Discuss with your students multiple techniques of solving for these voltages, asking them first for their solution strategies.

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Notes:
Discuss with your students multiple techniques of solving for these voltages, asking them first for their solution strategies.

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Notes:
Discuss with your students multiple techniques of solving for these voltages, asking them first for their solution strategies.

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Notes:
Discuss with your students multiple techniques of solving for these voltages, asking them first for their solution strategies.

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Notes:
This is a good example of how Kirchhoff's Voltage Law is more than just an abstract tool for mathematical analysis - it is also a powerful technique for practical circuit diagnosis. Students must apply KVL to determine the voltage drop across R2, and then use Ohm's Law to calculate its current.
If students experience difficulty visualizing how KVL plays a part in the solution of this problem, show them this illustration:

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Notes:
This is a simple circuit, yet practical in its function as a telegraph and as an example of KVL in action.

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V_BA = +10.8 volts

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V_CB = +7.2 volts

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V_DC = +18 volts

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V_AD = -36 volts

Notes:
Ask your students this question: "Will the algebraic sum of voltage measurements ever be other than zero in a loop?" Ask them to explain why this is, as best they can.

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V_BA = +10.8 volts

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V_CA = +18.0 volts

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V_DA = +36.0 volts

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V_AA = 0 volts

Notes:
Building on the knowledge of series voltages being additive, this example serves as a simple proof of Kirchhoff's Voltage Law. At the final step of the sequence, the voltage indicated by the meter must be zero volts because the two test leads are now electrically common to each other.

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V_A = +16 volts

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V_B = +16 volts

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V_C = +10 volts

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V_D = 0 volts

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V_E = -2 volts

Notes:
This question helps to familiarize students with the concept of "ground" as the default reference point for taking voltage measurements, as well as being an application of Kirchhoff's Voltage Law.

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Notes:
In your discussion, be sure to explore more than one "loop" when using KVL. Not only does this demonstrate the arbitrary nature of your loop choice, but it also serves as a double-check for your work!
It is not necessary to know anything about series-parallel or even parallel circuits in order to solve the R₄'s voltage - all one needs to know is how to use Kirchhoff's Voltage Law.

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Notes:
Potentiometric voltage measurement is not only practical, but for quite a while in the early field of electrical measurements it was the only way to accurately measure voltage from "weak" (high-impedance) sources. It is still used today in electrical metrology, and serves as a simple application of KVL for students to examine.

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Notes:
Ask your students to explain how they obtained their answers. There is more than one correct way to answer this question!

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V_BA = +10.8 volts

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V_DB = +25.2 volts

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V_FD = -24.0 volts

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V_AF = -12.0 volts

Notes:
Ask your students this question: "Will the algebraic sum of voltage measurements ever be other than zero in a loop?" Ask them to explain why this is, as best they can.

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Notes:
Ask your students where the increased voltage drop(s) might be located in this circuit. It might be helpful to draw the test points, power cable, and power supply as a circuit, using standard schematic symbols. Ask your students, "What type of fault might cause such voltage drop increases in this loop?"

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Notes:
The question of "where will we ever see this?" is too often ignored by teachers, who forget the lack of context present in their new students' understanding. Remember that this is all new to most of your students, so they lack the years of experience you have working with circuits in practical scenarios. A question like this deserves to be answered, and answered well.
Truth be known, the equivalent circuit for electrode-to-skin contact is far more complex than what is shown here (impedances everywhere!), but the point here is to simplify it enough so that students studying DC circuits and KVL would be able to grasp it.
Electrode half-cell potentials (E₁ and E₂ in the equivalent schematic diagram) caused by electrochemical action are not the only source of stray potentials in measurement circuits by any means! Noise voltage is a consideration in many circumstances (whether induced by outside sources or generated by physical action such as Johnson noise), thermal voltages caused by junctions of dissimilar metals, and others. If all we are doing is making crude measurements, these stray voltages will be of little concern. If precision is necessary, which is often the case in medical and scientific measurements, these spurious voltages can be devastating.

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Notes:
Ask your students to draw an electric circuit schematic showing how the same principle illustrated in the hydraulic system would apply to electric current, with "flow rates" of 700 amps, 150 amps, and 550 amps, respectively.

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Notes:
Another way of stating KCL is to say, "What goes in must come out." When continuous DC currents are involved, this law is really nothing more than a restatement of the Conservation of Charge. Of course, there are transient exceptions to this law (static electric charge and discharge, for example) as well as interesting AC exceptions (current "through" a capacitor), so a hard-literal interpretation of KCL may cause confusion later one. However, it is a fairly intuitive Law to grasp, and consequently I seldom find students experiencing difficulty with it.

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Notes:
Let students know that there really is such a thing as a perfect current source, just as there is no such thing as a perfect voltage source. However, there are devices the closely approximate ideal current sources (current transformers in AC circuits and "current mirror" DC transistor circuits, for example).

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Notes:
Have your students collectively agree on a procedure they may use to accurate discern series voltage sums and polarities. Guide their discussion, helping them identify principles that are true and valid for all series circuits.

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Notes:
Have your students collectively agree on a procedure they may use to accurate discern series voltage sums and polarities. Guide their discussion, helping them identify principles that are true and valid for all series circuits.

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Notes:
It is not necessary to know anything about series-parallel or even parallel circuits in order to solve the R₄'s current - all one needs to know is how to use Kirchhoff's Current Law.

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Notes:
It is not necessary to know anything about series-parallel or even parallel circuits in order to solve the R₄'s current - all one needs to know is how to use Kirchhoff's Current Law.

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Voltage source V₁ fails with increased output: All resistor currents increase, all resistor voltages increase, current through V₁ increases.

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Solder bridge (short) across resistor R₁: All resistor currents increase, voltage across resistors R₂ and R₃ increase, voltage across resistor R₁ decreases to zero, current through V₁ increases.

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Solder bridge (short) across resistor R₂: All resistor currents increase, voltage across resistors R₁ and R₃ increase, voltage across resistor R₂ decreases to zero, current through V₁ increases.

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Resistor R₃ fails open: All resistor currents decrease to zero, voltages across R₁ and R₂ decrease to zero, voltage across R₃ increases to equal V₁.

Notes:
The purpose of this question is to approach the domain of circuit troubleshooting from a perspective of knowing what the fault is, rather than only knowing what the symptoms are. Although this is not necessarily a realistic perspective, it helps students build the foundational knowledge necessary to diagnose a faulted circuit from empirical data. Questions such as this should be followed (eventually) by other questions asking students to identify likely faults based on measurements.

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Current source I₁ fails with increased output: All resistor currents increase, all resistor voltages increase, voltage across I₁ increases.

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Solder bridge (short) across resistor R₁: All resistor currents remain unchanged, voltage across resistor R₁ decreases to zero, voltage across I₁ decreases by the amount R₁ previously dropped.

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Solder bridge (short) across resistor R₂: All resistor currents remain unchanged, voltage across resistor R₂ decreases to zero, voltage across I₁ decreases by the amount R₂ previously dropped.

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Resistor R₃ fails open: Theoretically, all resistor currents remain unchanged while a large arc jumps across the failed-open R₃ (and possibly across I₁ as well). Realistically, all resistor currents will decrease to zero while the voltages across R₃ and I₁ will increase to maximum.

Notes:
The purpose of this question is to approach the domain of circuit troubleshooting from a perspective of knowing what the fault is, rather than only knowing what the symptoms are. Although this is not necessarily a realistic perspective, it helps students build the foundational knowledge necessary to diagnose a faulted circuit from empirical data. Questions such as this should be followed (eventually) by other questions asking students to identify likely faults based on measurements.

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Voltage source V₁ fails with increased output: All resistor currents increase, all resistor voltages increase, current through V₁ increases.

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Resistor R₁ fails open: All resistor voltages remain unchanged, current through R₁ decreases to zero, currents through R₂ and R₃ remain unchanged, current through V₁ decreases by the amount that used to go through R₁.

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Resistor R₃ fails open: All resistor voltages remain unchanged, current through R₃ decreases to zero, currents through R₁ and R₂ remain unchanged, current through V₁ decreases by the amount that used to go through R₃.

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Solder bridge (short) across resistor R₂: Theoretically, all resistor voltages remain unchanged while a near-infinite amount of current goes through the shorted R₂. Realistically, all resistor voltages will decrease to nearly zero while the currents through R₂ and V₁ will increase dramatically.

Notes:
The purpose of this question is to approach the domain of circuit troubleshooting from a perspective of knowing what the fault is, rather than only knowing what the symptoms are. Although this is not necessarily a realistic perspective, it helps students build the foundational knowledge necessary to diagnose a faulted circuit from empirical data. Questions such as this should be followed (eventually) by other questions asking students to identify likely faults based on measurements.

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Current source I₁ fails with increased output: All resistor currents increase, all resistor voltages increase, voltage across I₁ increases.

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Resistor R₁ fails open: All resistor voltages increase, current through R₁ decreases to zero, currents through R₂ and R₃ increase, current through I₁ remains unchanged, voltage across I₁ increases.

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Resistor R₃ fails open: All resistor voltages increase, current through R₃ decreases to zero, currents through R₁ and R₂ increase, current through I₁ remains unchanged, voltage across I₁ increases.

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Solder bridge (short) across resistor R₂: All resistor voltages decrease to zero, currents through R₁ and R₃ decrease to zero, current through shorted R₂ increases to equal I₁, current through I₁ remains unchanged, voltage across I₁ decreases to zero.

Notes:
The purpose of this question is to approach the domain of circuit troubleshooting from a perspective of knowing what the fault is, rather than only knowing what the symptoms are. Although this is not necessarily a realistic perspective, it helps students build the foundational knowledge necessary to diagnose a faulted circuit from empirical data. Questions such as this should be followed (eventually) by other questions asking students to identify likely faults based on measurements.

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Notes:
Discuss with your students techniques for calculating the node currents. What Laws did your students apply, and more importantly, in what order did they apply them?

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Notes:
Discuss with your students techniques for calculating the node currents. What Laws did your students apply, and more importantly, in what order did they apply them?

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Notes:
Perhaps the most difficult portion of this question for students to figure out is which way current goes when a battery is charging, versus when it is discharging. Thus, it is a good review for students on secondary battery theory!
Another challenge for students might be where to label a "node" for the algebraic summation of currents. Do not simply tell them, but rather let them choose the node.

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Notes:
Be sure to reinforce the safety aspects of this task. Although the primary focus of this question is most definitely Kirchhoff's Current Law, that is no excuse to avoid discussing electrical safety!

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Notes:
This is a very practical (and handy!) method of qualitative current measurement I've used many times on the job. Fuses, as well as other current-handling components, always have some non-zero amount of electrical resistance between their terminals, which means they are usable as crude shunt resistors. Their actual resistance will most likely be unknown to you, which is why they are useful in this capacity as qualitative measurement devices only, not quantitative.
Some students may point out that the measurement taken across the generator's fuse is not necessary to determine battery charging status. Technically, this is true, but doing so helps to confirm the validity of this technique. The polarity of that measurement does indeed show (conventional flow) current to be going upward through the generator fuse, versus downward through the battery fuses.
Students may also point out that the battery fuse millivoltage measurements do not add up to equal the generator fuse millivoltage measurement, as Kirchhoff's Current Law (KCL) would suggest. This is one example showing how the technique is strictly qualitative, not quantitative. We have no idea how much resistance lies within each fuse, and so we cannot rely on the millivoltage measurements as being proportional to current. This is especially true if the fuses are not identical!
Other circuit elements lending themselves to this same sort of application include overload "heater" elements for motor control circuits, long lengths of wire (provided you can stretch your millivolt-meter leads to reach both ends), circuit breakers, (closed) switches, and wire connections (particularly if the connection is old and possibly corroded).

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Notes:
I purposely avoided giving away explicit answers in the Answer section of this question, electing instead to simply provide a sound procedure. The point here is for students to figure out on their own what the voltmeter indications would mean with regard to battery condition.

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Notes:
Note that I do not explain why the fraction is less than 1 for each equation. I leave it to the students to figure out that R < R_total for series circuits and R_total < R for parallel circuits.
Many students have the unfortunate tendency to memorize in favor of understand, especially when it comes to equations. Comprehension is, of course, far superior, and exercises like this one help students build comprehension of the formulae they use in electronics.

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