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!
An excellent way to introduce students to the mathematical analysis of real circuits is to have them first determine component values (L and C) from measurements of AC voltage and current. The simplest circuit, of course, is a single component connected to a power source! Not only will this teach students how to set up AC circuits properly and safely, but it will also teach them how to measure capacitance and inductance without specialized test equipment.
A note on reactive components: use high-quality capacitors and inductors, and try to use low frequencies for the power supply. Small step-down power transformers work well for inductors (at least two inductors in one package!), so long as the voltage applied to any transformer winding is less than that transformer's rated voltage for that winding (in order to avoid saturation of the core).
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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Notes:
It is essential, of course, that students understand the operational principle of this circuit before they may even attempt to diagnose possible faults. You may find it necessary to discuss this circuit in detail with your students before they are ready to troubleshoot it.
In case anyone asks, the symbolism R_pot >> R means "potentiometer resistance is much greater than the fixed resistance value."

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Notes:
I suggest you let your students discover how to use the complex number facilities of their scientific calculators on their own. My experience has been that students both young and old take to this challenge readily, because they realize learning how to use their calculators will save them a tremendous amount of hand calculations!

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Ask your students to calculate how fast the voltages from these two alternators "roll" in and out of phase with each other.

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Discuss the consequences of closing the breaker when there is such a large phase shift between the two alternator output voltages. What will likely happen in the circuit if the breaker is closed under these conditions?
Ask your students whether or not the discrepancy in output voltage (218 VAC versus 216.5 VAC) is of any consequence in closing the breaker. Why is phase shift the only factor mentioned in the answer as a reason not to close the breaker?
This question serves to illustrate alternator theory as well as AC network analysis principles. The "phase-locking" phenomenon of two paralleled alternators is very important for students to understand if they are to do any work related to AC power generation systems.

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Such "power-plus-data" strategies are made possible by the Superposition Theorem and the linearity of resistors, capacitors, and inductors. If time permits, this would be a good opportunity to discuss "power-line carrier" systems, where high-frequency data is transmitted over power line conductors. The venerable X10 network system is an example for residential power wiring, while power distribution utilities have been using this "PLC" technology (the acronym not to be confused with Programmable Logic Controllers) for decades over high-voltage transmission lines.

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Explain to your students that AC bridges, while fundamentally the same as DC bridges, do have their differences. Their knowledge of AC circuit calculations (with complex numbers) should be enough for them to see why this is so.

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So long as complex quantities are used, AC bridge circuits "balance" just the same way that DC bridge circuits balance. Consequently, this is really nothing new for your students if they've already studied DC Wheatstone bridge circuits.

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Notes:
So long as complex quantities are used, AC bridge circuits "balance" just the same way that DC bridge circuits balance. Consequently, this is really nothing new for your students if they've already studied DC Wheatstone bridge circuits.

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There is actually a way to solve for the values of L_x and R_x in a Maxwell-Wien bridge circuit without using complex numbers at all. If one or more of your students find out how to do so through their research, don't consider it "cheating." Rather, applaud their research, because they found a quicker path to a solution.
This, of course, doesn't mean you don't ask them to work through the problem using complex numbers! The benefit of students researching other means of solution simply provides more alternative solution strategies to the same problem, which is a very good thing.

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Impedance of a single capacitor (C) = -j [1/(wC)]

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Impedance of a series resistor-capacitor (R, C) network = R -j [1/(wC)]

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Admittance of a parallel inductor-resistor (L, R) network = ¹/_R -j [1/(wL)]

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Admittance of a parallel resistor-capacitor (R, C) network = ¹/_R +j wC

Notes:
One possible point of confusion here is the sign of j after inversion. If it is not evident from the answers, ¹/_j is equal to -j, so that the impedance of an inductor (j wL) becomes -j [1/(wL)] when converted into an admittance.

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Notes:
A rule that students will need to be aware of in order to follow all the algebra shown here is ¹/_j = -j. A brief proof is given here:

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Notes:
Yes, it is out of character for me to show two pages of solution in the änswer" section of one of my questions! I usually do not provide this much information in my answers. However, in this case I believe there is still much to be learned from examining a proof like this shown step-by-step.
You may wish to ask your students to explain the rationale behind each step, especially in the first part where we deal with real and imaginary terms. One point that may be especially confusing is where I separate the real and imaginary terms, setting the imaginary quantity equal to j0. Some students may not see where the j0 comes from, since the preceding (complex) expression was simply equal to 1. Remind them that "1" is a real quantity, possessing an (implied) imaginary component of 0, and that it could very well have been written as 1 + j0.

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This is a very interesting circuit to built and test. You may build one using 1 mF capacitors and 2.7 kW resistors that will successfully operate on 60 Hz power-line excitation.

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�

The potentiometer wiper is fully left: Q = -45^o

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The potentiometer wiper is fully right: Q = 45^o

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The potentiometer wiper is perfectly centered: Q = 0^o

Notes:
This is a very interesting circuit to built and test. You may build one using 1 mF capacitors, 2.7 kW resistors, and a 100 kW potentiometer that will successfully operate on 60 Hz power-line excitation.
An interesting thing to note about using line power is that any distortions in the excitation sine-wave will become obvious when the potentiometer wiper is turned toward the differentiating position (where Q is positive). If listened to with an audio detector, you may even hear the change in timbre while moving the wiper from one extreme to the other. If excited by a "clean" sine-wave, however, no change in timbre should be heard because there are no harmonics present.

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Notes:
It is essential, of course, that students understand the operational principle of this circuit before they may speculate at the effects of various component faults. You may find it necessary to discuss this circuit in detail with your students before they are ready to troubleshoot it.
In case anyone asks, the symbolism R_pot >> R means "potentiometer resistance is much greater than the fixed resistance value."

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Notes:
It is essential, of course, that students understand the operational principle of this circuit before they may even attempt to diagnose possible faults. You may find it necessary to discuss this circuit in detail with your students before they are ready to troubleshoot it.
In case anyone asks, the symbolism R_pot >> R means "potentiometer resistance is much greater than the fixed resistance value."

Hide Answer

Notes:
It is essential, of course, that students understand the operational principle of this circuit before they may even attempt to diagnose possible faults. You may find it necessary to discuss this circuit in detail with your students before they are ready to troubleshoot it.
In case anyone asks, the symbolism R_pot >> R means "potentiometer resistance is much greater than the fixed resistance value."

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Notes:
While rectangular notation is mathematically useful, it does not apply directly to measurements taken with real instruments. Some students might suggest that the 3.000 volt reading and the 4.000 volt reading on the other two voltmeters represent the rectangular components (real and imaginary, respectively) of voltage, but this is a special case. In cases where resistance and reactance are mixed (e.g. a practical inductor with winding resistance), the voltage magnitude will be neither the real nor the imaginary component, but rather the polar magnitude.

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Notes:
Ask your students why polarity markings need to be provided in DC electrical networks, as an essential part of the voltage figures. Why is an answer for a voltage drop incomplete if not accompanied by polarity markings in a DC circuit? Discuss this with your students, then ask them to extrapolate this principle to AC circuits. When we are accounting for the phase shift of a voltage drop in our answer, does the "polarity" of the voltage drop matter?

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Notes:
This question is highly effective in demonstrating why polarity markings are important in AC circuit analysis. Without the polarity marks as "frames of reference" for the phase angles, it is impossible to determine the resultant line voltage from the two 120 VAC phase voltages.

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Of course, complex numbers may also be used to calculate V_AB, but the phasor approach has the benefit of graphical intuitiveness.

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Notes:
Ask your students to describe the effect of the transformer's step-up ratio on impedance magnitude and on impedance phase angle. They may do this by comparing the impedance of the C-R-C circuit on the transformer's secondary side versus the impedance ßeen" at the voltage source terminals. Ask them to explain why one of these parameters is affected but the other is not.
Then, pose the scenario of a real transformer, complete with leakage inductance. Ask your students to explain what effect would leakage inductance have on the load impedance of this circuit.

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Notes:
The real änswer" to this question is the circuit analysis technique employed. Ask your students, perhaps working teams, to explain how to arrive at the different voltage drop figures. Note any similarities in analysis technique to that of DC circuit analysis.

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