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Learning to mathematically analyze circuits requires much study and practice. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. While this is good, there is a much better way. For successful circuit-building exercises, follow these steps:. For AC circuits where inductive and capacitive reactances impedances are a significant element in the calculations, I recommend high quality high-Q inductors and capacitors and powering your circuit with low-frequency voltage power-line frequency works well to minimize parasitic effects.
One way you can save time and reduce the possibility of error is to begin with a very simple circuit and incrementally add components to increase its complexity after each analysis, rather than building a whole new circuit for each practice problem.
Another time-saving technique is to re-use the same components in a variety of different circuit configurations. 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. They also need real, hands-on practice building circuits and using test equipment. 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. 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!
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. 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. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits! In AC circuits, we similarly need a formula to relate voltage, current, and impedance together.
Be prepared to show how you may use algebra to manipulate one of these equations into the other two forms. If using phasor quantities complex numbers for voltage, current, and impedance, the proper way to write these equations is as follows:.
Often it is appropriate to express an AC voltage, current, or impedance as a simple scalar number. Together, their series opposition to alternating current results in a current of 10 mA from the 5 volt source:.
How many ohms of opposition does the series combination of resistor and inductor offer? What name do we give to this quantity, and how do we symbolize it, being that it is composed of both resistance R and reactance X? Students may experience difficulty arriving at the same quantity for impedance shown in the answer.
If this is the case, help them problem-solve by suggesting they simplify the problem : short past one of the load components and calculate the new circuit current. Soon they will understand the relationship between total circuit opposition and total circuit current, and be able to apply this concept to the original problem. Where have we seen this before, especially in the context of electric circuits?
What type of number would be suitable for representing such a quantity, and how might it be written? Together, their series opposition to alternating current results in a current of 1 mA from the 5 volt source:.
How many ohms of opposition does the series combination of resistor and capacitor offer? Then, when studying AC circuit theory, you learned that reactance was another type of opposition to current. Now, a third term is introduced: impedance.
Like resistance and reactance, impedance is also a form of opposition to electric current. Explain the difference between these three quantities resistance, reactance, and impedance using your own words. The fundamental distinction between these terms is one of abstraction: impedance is the most general term, encompassing both resistance and reactance. Here is an explanation given in terms of logical sets using a Venn diagram , along with an analogy from animal taxonomy:. Resistance is a type of impedance, and so is reactance.
The difference between the two has to do with energy exchange. The given answer is far from complete. Be sure to discuss with your students what the fundamental difference is between resistance and reactance, in terms of electrical energy exchange. It is often necessary to represent AC circuit quantities as complex numbers rather than as scalar numbers, because both magnitude and phase angle are necessary to consider in certain calculations.
So, a voltage of 3. But what about impedance Z? Does impedance have a phase angle, too, or is it a simple scalar number like resistance or reactance?
Calculate the amount of current that would go through a mH inductor with 36 volts RMS applied to it at a frequency of Hz. Explain why or why not. This is a challenging question, because it asks the student to defend the application of phase angles to a type of quantity that does not really possess a wave-shape like AC voltages and currents Convert Mathematical Equations Pdf Word 404 do.
Conceptually, this is difficult to grasp. Although it is natural to assign a phase angle of 0 o to the 36 volt supply, making it the reference waveform, this is not actually necessary. Work through this calculation with your students, assuming different angles for the voltage in each instance. You should find that the impedance computes to be the same exact quantity every time. Express the impedance Z in both polar and rectangular forms for each of the following components:.
Follow-up question: what would the phasors look like for resistive, inductive, and capacitive impedances? Real inductors and capacitors are never purely reactive. There will inevitably be some resistance intrinsic to these devices as well. How would this combination be expressed as a single impedance? State your answer in both polar and rectangular forms. This is a very important concept to understand: that reactive components are never purely reactive.
Parasitic resistance is impossible to avoid short of using superconductors. Even then, inductors are bound to have some parasitic capacitance, and capacitors are bound to have some parasitic inductance! For instance, inductors are bound to have a small amount of capacitance built-in, and capacitors are bound to have a small amount of inductance built-in.
These effects are not intentional, but they exist anyway. Describe how a small amount of capacitance comes to exist within an inductor, and how a small amount of inductance comes to exist within a capacitor. Capacitance exists any time there are two conductors separated by an insulating medium.
Inductance exists any time a magnetic field is permitted to exist around a current-carrying conductor. Look for each of these conditions within the respective structures of inductors and capacitors to determine where the parasitic effects originate.
Once students have identified the mechanisms of parasitic reactances, challenge them with inventing means of minimizing these effects. Suppose you were given a component and told it was either a resistor, an inductor, or a capacitor. The component is unmarked, and impossible to visually identify.
Explain what steps you would take to electrically identify what type of component it was, and what its value was, without the use of any test equipment except a signal generator, a multimeter capable of measuring nothing but voltage, current, and resistance , and some miscellaneous passive components resistors, capacitors, inductors, switches, etc.
Demonstrate your technique if possible. This is an excellent opportunity to brainstorm as a group and experiment on real components. There is obviously more than one way to make the determinations of identity and value! Use the class time to engage your students in lively discussion and debate over how to approach this practical problem.
Suppose you were given two components and told one was an inductor while the other was a capacitor. Both components are unmarked, Convert Mathematical Equations Pdf Word Ying and impossible to visually distinguish or identify. Then, explain how you could approximately measure the value of each component using nothing more than a sine-wave signal generator and an AC meter capable only of precise AC voltage and current measurements across a wide frequency range no direct capacitance or inductance measurement capability , and show how the reactance equation for each component L and C would be used in your calculations.
Challenge question: suppose the only test equipment you had available was a 6-volt battery and an old analog volt-milliammeter with no resistance check function. How could you use this primitive gear to identify which component was the inductor and which was the capacitor? The ohmmeter test is based on DC component response, which may be thought of in terms of reactance at a frequency at or near zero.
If a sinusoidal voltage is applied to an impedance with a Convert Mathematical Equations Pdf Word Python phase angle of 0 o , the resulting voltage and current waveforms will look like this:. Ask your students to observe the waveform shown in the answer closely, and determine what sign the power values always are. Note how the voltage and current waveforms alternate between positive and negative, but power does not.
Of what significance is this to us? What does this indicate about the nature of a load with an impedance phase angle of 0 o? If a sinusoidal voltage is applied to an impedance with a phase angle of 90 o , the resulting voltage and current waveforms will look like this:.
Ask your students to observe the waveform shown in the answer closely, and determine what sign the power values are. Note how the power waveform alternates between positive and negative values, just as the voltage and current waveforms do. Ask your students to explain what negative power could possibly mean.
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