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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Notes:
In my experience, most American high school graduates are extremely weak in logarithms. Apparently this is not taught very well at the high school level, which is a shame because logarithms are a powerful mathematical tool. You may find it necessary to explain to your students what a logarithm is, and exactly why it ün-does" the exponent.
When forced to give a quick presentation on logarithms, I usually start with a generic definition:

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Notes:
I am usually not a fan of discussing calculator keystroke sequences to college-level students, but I have resorted to this at the college where I teach because so many of my students have no idea of how to evaluate exponential expressions! Of course, the actual keystrokes one must push to evaluate this expression depend on the brand of calculator used.

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Notes:
While some may wonder what this question has to do with electronics, it is an exercise in qualitative analysis. This skill is very important for students to master if they are to be able to distinguish between the equations e^-[t/(t)] and 1 - e^-[t/(t)], both used in time-constant circuit analysis.
The actual procedure for determining the nature of the equation is simple: consider what happens as t begins at 0 and as it increases to some arbitrary positive value. Some students may rely on their calculators, performing actual calculations to see whether the percentage increases or decreases with increasing t. Encourage them to analyze the equation qualitatively rather than quantitatively, though. They should be able to tell which way the percentage changes with time without having to consider a single numerical value!

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Notes:
It is very important for students to understand what this expression means and how it works, lest they rely solely on memorization to use it in their calculations. As I always tell my students, rote memorization will fail you! If a student does not comprehend why the expression works as it does, they will be helpless to retain it as an effective "tool" for performing calculations in the future.

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Notes:
Have your students explain why the voltage and current curves are shaped as they are.

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Notes:
Here, students must choose which equation(s) to use for the calculation, calculate the time constant for the circuit, and put all the variables in the right place to obtain the correct answers. Discuss all these steps with your students, allowing them to explain how they approached the question.
If anyone asks, let them know that the capacitor symbol shown represents a polarized (electrolytic) capacitor.

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Notes:
Here, students must choose which equation(s) to use for the calculation, calculate the time constant for the circuit, and put all the variables in the right place to obtain the correct answers. Discuss all these steps with your students, allowing them to explain how they approached the question.
If anyone asks, let them know that the capacitor symbol shown represents a polarized (electrolytic) capacitor.

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Notes:
The purpose of this question is to preview the concept of ïnitial" and "final" values in RC circuits, before they learn to use the üniversal time constant formula."

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Notes:
The purpose of this question is to preview the concept of ïnitial" and "final" values in RC circuits, before they learn to use the üniversal time constant formula."

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Notes:
The main purpose of this question is to get students thinking in terms of ïnitial" and "final" values for LR circuits, and how one might calculate them. It is largely a conceptual question, with just a bit of calculation necessary.
One practical application of this circuit is for ßtepping up" DC voltage. The circuit topology shown in the question is that of an inverting converter circuit. This form of DC-DC converter circuit has the ability to step voltage up or down, depending on the duty cycle of the switch's oscillation.

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Notes:
This is advice I always give my students, after seeing so many students get themselves into trouble by blindly plugging numbers into equations. Think before you act, is the motto here!
Actually, this general advice applies to most all physics-type problems: identify what it is you're trying to solve and what you have to work with before jumping into calculations.

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�

1 time constant (t) after connecting the resistor: V_C = 36.79 volts

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2 time constants (2t) after connecting the resistor: V_C = 13.53 volts

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3 time constants (3t) after connecting the resistor: V_C = 4.979 volts

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4 time constants (4t) after connecting the resistor: V_C = 1.832 volts

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5 time constants (5t) after connecting the resistor: V_C = 0.6738 volts

Notes:
Although students should be able to look up approximate answers to this question from almost any beginning electronics textbook, the point here is to get them to relate the question to an actual formula so they may calculate this on their own.

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Notes:
Here, students must choose which equation to use for the calculation, calculate the time constant for the circuit, and put all the variables in the right place to obtain the correct answer. Discuss all these steps with your students, allowing them to explain how they approached the question.

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Notes:
Here, students must choose which equation to use for the calculation, calculate the time constant for the circuit, and put all the variables in the right place to obtain the correct answer. Discuss all these steps with your students, allowing them to explain how they approached the question.

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Time       VC (volts)       0 ms       0       30 ms       12.29       60 ms       19.71       90 ms       24.19       120 ms       26.89       150 ms       28.52

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class.

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Notes:
I intentionally left the graph unscaled in the problem, so that students might determine their own scales to plot the points in. The scaling shown in the answer is obviously not ideal, as the graphs have reached their terminal values (for all practical purposes) well before the horizontal axis is complete.

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Notes:
I intentionally left the graph unscaled in the problem, so that students might determine their own scales to plot the points in. The scaling shown in the answer is obviously not ideal, as the graphs have reached their terminal values (for all practical purposes) well before the horizontal axis is complete.

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Notes:
In order for students to solve this problem, they must algebraically manipulate the "normal" time-constant formula to solve for time instead of solving for voltage.

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VC       Time       0 volts       0 ms       10 volts       297.5 ms       20 volts       716.7 ms       30 volts       1.433 s       40 volts       > 5 s

Notes:
Some students may write 5.17 seconds as the time required to charge to 40 volts (5 time constants' worth of time). If so, remind them that the ßtandard" of 5t is arbitrary, and that theoretically the capacitor never actually reaches full charge.

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Notes:
In order for students to solve this problem, they must algebraically manipulate the "normal" time-constant formula to solve for time instead of solving for voltage.

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Notes:
In developing equations, students often feel äbandoned" if the instructor does not provide an answer for them. How will they ever know if their equation is correct? If the phenomenon the equation seeks to predict is well-understood, though, this is not a problem.

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Notes:
Once students grasp the concept of initial and final values in time-constant circuits, they may calculate any variable at any point in time for any RC or LR circuit (not for RLC circuits, though, as these require the solution of a second-order differential equation!). All they need is the universal time-constant equation:

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Notes:
Once students grasp the concept of initial and final values in time-constant circuits, they may calculate any variable at any point in time for any RC or LR circuit (not for RLC circuits, though, as these require the solution of a second-order differential equation!). All they need is the universal time-constant equation:

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Notes:
This so-called üniversal time-constant formula" is my own (Tony R. Kuphaldt's) invention. A product of frustration from trying to teach students to calculate variables in RC and LR time-constant circuits using one formula for decay and another one for increasing values, this equation works for all conditions. Vitally important to this formula's accuracy, however, is that the student correctly assesses the initial and final values. This is the biggest problem I see students having when they go to calculate voltages or currents in time-constant circuits.
In my Lessons In Electric Circuits textbook, I introduce this formula in a slightly different form:

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Notes:
This problem is unique in that the capacitor does not discharge all the way to 0 volts when the switch is moved to the lower position. Instead, it discharges down to a (final) value of 3 volts. Solving for the answer requires that students be a bit creative with the common time-constant equations (e^-[t/(t)] and 1 - e^-[t/(t)]).
The follow-up question is simply an exercise in troubleshooting theory.

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Notes:
There is quite a lot to calculate in order to reach the solutions in this question. There is more than one valid way to approach it, as well. An important fact to note: the voltage across the switch contacts, in both examples, is greater than the battery voltage! Just as capacitive time-constant circuits can generate currents in excess of what their power sources can supply, inductive time-constant circuits can generate voltages in excess of what their power sources can supply.
The follow-up question is simply an exercise in troubleshooting theory.

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t = 1 second ; e^-[t/(t)] = 36.788%

t = 2 seconds ; e^-[t/(t)] = 13.534%

t = 3 seconds ; e^-[t/(t)] = 4.979%

t = 4 seconds ; e^-[t/(t)] = 1.832%

t = 5 seconds ; e^-[t/(t)] = 0.6738%

t = 6 seconds ; e^-[t/(t)] = 0.2479%

t = 7 seconds ; e^-[t/(t)] = 0.09119%

t = 8 seconds ; e^-[t/(t)] = 0.03355%

t = 9 seconds ; e^-[t/(t)] = 0.01234%

t = 10 seconds ; e^-[t/(t)] = 0.004540%

Notes:
The purpose of this question is for students to learn the significance of the expression e^-[t/(t)] by "playing" with the numbers. The negative exponent may confuse some students, so be sure to discuss its significance with all students, so that all understand what it means.
Another concept for students to grasp in this question is that of an asymptotic function: a function that approaches a final value in incrementally smaller intervals.

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t = 1 second ; e^-[t/(t)] = 60.65% ; increasing variable = 39.35%

t = 2 seconds ; e^-[t/(t)] = 36.79% ; increasing variable = 63.21%

t = 3 seconds ; e^-[t/(t)] = 22.31% ; increasing variable = 77.69%

t = 4 seconds ; e^-[t/(t)] = 13.53% ; increasing variable = 86.47%

t = 5 seconds ; e^-[t/(t)] = 8.208% ; increasing variable = 91.79%

t = 6 seconds ; e^-[t/(t)] = 4.979% ; increasing variable = 95.02%

t = 7 seconds ; e^-[t/(t)] = 3.020% ; increasing variable = 96.98%

t = 8 seconds ; e^-[t/(t)] = 1.832% ; increasing variable = 98.17%

t = 9 seconds ; e^-[t/(t)] = 1.111% ; increasing variable = 98.89%

t = 10 seconds ; e^-[t/(t)] = 0.6738% ; increasing variable = 99.33%

Notes:
Do not simply tell your students how to calculate the values of the increasing variable. Based on their qualitative knowledge of time-constant circuit curves and their ability to evaluate the downward (decay) expression, they should be able to figure out how to calculate the increasing variable's value over time as well.
Some students will insist that you give them an equation to do this. They want to be told what to do, rather than solve the problem on their own based on an observation of pattern. It is very important that students of any science learn to recognize patterns in data, and that they learn to fit that data into a mathematical equation. If nothing else, these figures given in the answer for both decreasing and increasing variables should be plain enough.

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Notes:
Being able to derive an equation from numerical data is a complex, but highly useful, skill in all the sciences. Sure, your students will be able to find this mathematical expression in virtually any basic electronics textbook, but the point of this question is to derive this expression from an examination of data (and, of course, an examination of the other time-constant expression: e^-[t/(t)]).
Be sure to challenge your students to do this, by asking how they obtained the answer to this question. Do not ßettle" for students simply telling you what the equation is - ask them to explain their problem-solving techniques, being sure that all students have contributed their insights.

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t = 0 seconds ; E_C = 10 V

t = 1 second ; E_C = [10/3] V = 3.33 V

t = 2 seconds ; E_C = [10/9] V = 1.11 V

t = 3 seconds ; E_C = [10/27] V = 0.370 V

t = 4 seconds ; E_C = [10/81] V = 0.123 V

t = 5 seconds ; E_C = [10/243] V = 0.0412 V

Notes:
Calculating the voltage for the first few time constants' worth of time should be easy without a calculator. I strongly encourage your students to develop their estimation skills, so that they may solve problems without being dependent upon a calculator. Too many students depend heavily on calculators - some are even dependent on specific brands or models of calculators!
Equally important as being able to estimate is knowing whether or not your estimations are over or under the exact values. This is especially true when estimating quantities relevant to safety and/or reliability!

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t = 0 seconds ; E_C = 0 V

t = 1 second ; E_C = [20/3] V = 6.67 V

t = 2 seconds ; E_C = [80/9] V = 8.89 V

t = 3 seconds ; E_C = [260/27] V = 9.63 V

t = 4 seconds ; E_C = [800/81] V = 9.88 V

t = 5 seconds ; E_C = [2420/243] V = 9.96 V

Notes:
Calculating the voltage for the first few time constants' worth of time should be easy without a calculator. I strongly encourage your students to develop their estimation skills, so that they may solve problems without being dependent upon a calculator. Too many students depend heavily on calculators - some are even dependent on specific brands or models of calculators!
Equally important as being able to estimate is knowing whether or not your estimations are over or under the exact values. This is especially true when estimating quantities relevant to safety and/or reliability!

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�

RC circuit; R = 10 kW, C = 220 mF ; 7.5 sec = 3.41 t

�

RC circuit; R = 33 kW, C = 470 mF ; 7.5 sec = 0.484 t

�

RC circuit; R = 1.5 kW, C = 100 mF ; 7.5 sec = 50.0 t

�

RC circuit; R = 790 W, C = 9240 nF ; 7.5 sec = 1027 t

�

RC circuit; R = 100 kW, C = 33 pF ; 7.5 sec = 2,272,727 t

�

LR circuit; R = 100 W, L = 50 mH ; 7.5 sec = 15,000 t

�

LR circuit; R = 45 W, L = 2.2 H ; 7.5 sec = 153.4 t

�

LR circuit; R = 1 kW, L = 725 mH ; 7.5 sec = 10,345 t

�

LR circuit; R = 4.7 kW, L = 325 mH ; 7.5 sec = 108,462 t

�

LR circuit; R = 6.2 W, L = 25 H ; 7.5 sec = 1.86 t

Notes:
An interesting thing to note here is the span of time constant values available from common capacitor/inductor/resistor sizes. As students should notice, the capacitor-resistor combinations (all very practical values, I might add) create both longer and shorter time constant values than the inductor-resistor combinations, and that is even including the 25 Henry - 6.2 Ohm combination, which would be difficult (read: expensive) to achieve in real life.

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Notes:
While some may wonder what this question has to do with electronics, it is an exercise in qualitative analysis. This skill is very important for students to master if they are to be able to distinguish between the equations e^-[t/(t)] and 1 - e^-[t/(t)], both used in time-constant circuit analysis.
The actual procedure for determining the nature of the equation is simple: consider what happens as t begins at 0 and as it increases to some arbitrary positive value. Some students may rely on their calculators, performing actual calculations to see whether the percentage increases or decreases with increasing t. Encourage them to analyze the equation qualitatively rather than quantitatively, though. They should be able to tell which way the percentage changes with time without having to consider a single numerical value!

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Notes:
It is very important for students to understand what this expression means and how it works, lest they rely solely on memorization to use it in their calculations. As I always tell my students, rote memorization will fail you! If a student does not comprehend why the expression works as it does, they will be helpless to retain it as an effective "tool" for performing calculations in the future.
A good way to suggest students approach a problem such as this is to imagine t increasing in value. As t grows larger, what happens to the expression's overall value? Then, compare which of the two percentages (percentage traversed, or percentage remaining) follow the same trend. One not need touch a calculator to figure this out!

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Time       VC (volts)       IC (mA)       0 ms       0       4.255       30 ms       6.932       2.781       60 ms       11.46       1.817       90 ms       14.42       1.187       120 ms       16.35       0.7757       150 ms       17.62       0.5069

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class.

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Time       VL (volts)       IL (mA)       0 ms       20       0       10 ms       11.89       6.238       20 ms       7.069       9.947       30 ms       4.203       12.15       40 ms       2.499       13.46       50 ms       1.485       14.24

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class.

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Time       VL (volts)       IL (mA)       0 ns       12       0       250 ns       6.110       21.82       500 ns       3.111       32.92       750 ns       1.584       38.58       1.00 ms       0.8065       41.46       1.25 ms       0.4106       42.93

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class.

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Time (ms)       VL (V)       IL (mA)       0       5.00       5.00       50       3.22       5.81       100       2.07       6.33       150       1.34       6.67       200       0.860       6.88       250       0.554       7.02       300       0.357       7.11       350       0.230       7.17       400       0.148       7.21

Notes:
This circuit demands careful pre-analysis of the initial and final values. If students experience difficulty calculating the voltage and current figures here, it is probably due to incorrect initial and final values for voltage and/or current.

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Notes:
Although I have revealed a problem-solving technique in this question, it does not show the students exactly how to separate the inductor's 2.5 henrys of inductance and 14 ohms of resistance into two components, nor does it give away the answer. Discuss the analytical technique of drawing idealized components ("lumped parameters") as a problem-solving technique, and encourage students to use it whenever they are faced with analyzing a component exhibiting parasitic characteristics.
An excellent example of this technique is in "modeling" transformers. Transformers exhibit much more than just mutual inductance. They also exhibit self-inductance, leakage inductance, capacitance, resistance, and hysteretic losses. A comprehensive model for a transformer is a very complex thing, and it appears on a schematic to be a whole network of components connected together:

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Notes:
In order for students to solve this problem, they must algebraically manipulate the "normal" time-constant formula to solve for time instead of solving for voltage.

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VC       Time       0 volts       0 ms       -5 volts       29.75 ms       -10 volts       71.67 ms       -15 volts       143.3 ms       -19 volts       309.8 ms

Notes:
Ask your students to show how they algebraically solved the standard time constant equation for t using logarithms.

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VC       Time       10 volts       588.9 ms       8 volts       1.31 s       6 volts       2.24 s       4 volts       3.55 s       2 volts       5.79 s

Notes:
Ask your students to explain how they set up each calculation.

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Notes:
The follow-up question illustrates an important mathematical principle regarding logarithmic decay functions: for every passing of a fixed time interval, the system decays by the same factor. This is most clearly (and popularly) seen in the concept of half-life for radioactive substances, but it is also seen here for RC (or LR) circuits.

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Notes:
Although it may seem premature to introduce the 555 timer chip when students are just finishing their study of DC, I wanted to provide a practical application of RC circuits, and also of algebra in generating useful equations. If you deem this question too advanced for your student group, by all means skip it.
Incidentally, I simplified the diagram where I show the capacitor discharging: there is actually another current at work here. Since it wasn't relevant to the problem, I omitted it. However, some students may be adept enough to catch the omission, so I show it here:

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Notes:
Some students may think that a rate of change of 4.44 kilovolts per second harbors shock hazard, because, well, 4.44 thousand volts is a lot of voltage! Remind them that this is a rate of change and not an actual voltage figure. This number simply tells us how fast the voltage is changing, not how far it will rise to. It is the difference between saying that a car travels at 75 miles per hour and that a car will travel 75 miles.

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Notes:
Some students may think that a rate of change of 280 amps per second might burn up the wiring, because 280 amps seems like a lot of current. Remind them that this is a rate of change and not an actual current figure. This number simply tells us how fast the current is changing, not how far it will rise to. It is the difference between saying that a car travels at 75 miles per hour and that a car will travel 75 miles.

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Notes:
RC time constant circuits are an excellent example of how to apply simple differential equations. In this case, we see that the differential equation is first-order, with separable variables, making it comparatively easy to solve.
It should also be evident to students that any initial condition of capacitor charge may be set into the general solution (by changing the value of the constant).

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Notes:
L/R time constant circuits are an excellent example of how to apply simple differential equations. In this case, we see that the differential equation is first-order, with separable variables, making it comparatively easy to solve.
It should also be evident to students that any initial condition for current may be set into the general solution (by changing the value of the constant).

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Time       Switch motion       VC (volts)       0 s       discharge � charge       0 volts       5 s       charge � discharge       6.549 volts       10 s       discharge � charge       2.260 volts       15 s       charge � discharge       7.329 volts       20 s       discharge � charge       2.529 volts       25 s       charge � discharge       7.422 volts

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class. See how many of them notice that the exponential portion of the equation (e^[t/(t)]) is the same for each calculation, and if they find an easy way to manage the calculations by storing charge/discharge percentages in their calculator memories!

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Transition       vout       #1 (0 � 5 volts)       0 volts       #2 (5 � 0 volts)       3.395 volts       #3 (0 � 5 volts)       1.090 volts       #4 (5 � 0 volts)       3.745 volts       #5 (0 � 5 volts)       1.202 volts       #6 (5 � 0 volts)       3.781 volts       #7 (0 � 5 volts)       1.214 volts       #8 (5 � 0 volts)       3.785 volts

Notes:
Be sure to have your students share their problem-solving techniques (how they determined which equation to use, etc.) in class. See how many of them notice that the exponential portion of the equation (e^[t/(t)]) is the same for each calculation, and if they find an easy way to manage the calculations by storing charge/discharge percentages in their calculator memories!

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Notes:
This question incorporates electrical safety into an RC circuit time calculation. Safety is something you should periodically revisit with your students, because it is so important. Ask your students how a ßafe" value of voltage may be determined, and if there are any environmental circumstances impacting this determination.

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