In very simple, qualitative terms, rate the impedance of capacitors and inductors as “seen” by lowfrequency and highfrequency signals alike:
Challenge question: what does a capacitor “appear” as to a DC signal?
Ask your students how they arrived at their answers for these qualitative assessments. If they found difficulty understanding the relationship of frequency to impedance for reactive components, I suggest you work through the reactance equations qualitatively with them. In other words, evaluate each of the reactance formulae (X_{L} = 2 πf L and X_{C} = [1/(2 πf C)]) in terms of f increasing and decreasing, to understand how each of these components reacts to low and highfrequency signals.
Identify these filters as either being “lowpass” or “highpass”, and be prepared to explain your answers:


Lowpass and highpass filter circuit are really easy to identify if you consider the input frequencies in terms of extremes: radio frequency (very high), and DC (f = 0 Hz). Ask your students to identify the respective impedances of all components in a filter circuit for these extreme frequency examples, and the functions of each filter circuit should become very clear to see.
Suppose you were installing a highpower stereo system in your car, and you wanted to build a simple filter for the “tweeter” (highfrequency) speakers so that no bass (lowfrequency) power is wasted in these speakers. Modify the schematic diagram below with a filter circuit of your choice:

Hint: this only requires a single component per tweeter!

Followup question: what type of capacitor would you recommend using in this application (electrolytic, mylar, ceramic, etc.)? Why?
Ask your students to describe what type of filter circuit a seriesconnected capacitor forms: lowpass, highpass, bandpass, or bandstop? Discuss how the name of this filter should describe its intended function in the sound system.
Regarding the followup question, it is important for students to recognize the practical limitations of certain capacitor types. One thing is for sure, ordinary (polarized) electrolytic capacitors will not function properly in an application like this!
It is common in audio systems to connect a capacitor in series with each “tweeter” (highfrequency) speaker to act as a simple highpass filter. The choice of capacitors for this task is important in a highpower audio system.
A friend of mine once had such an arrangement for the tweeter speakers in his car. Unfortunately, though, the capacitors kept blowing up when he operated the stereo at full volume! Tired of replacing these nonpolarized electrolytic capacitors, he came to me for advice. I suggested he use mylar or polystyrene capacitors instead of electrolytics. These were a bit more expensive than electrolytic capacitors, but they did not blow up. Explain why.
The issue here was not polarity (AC versus DC), because these were nonpolarized electrolytic capacitors which were blowing up. What was an issue was ESR (Equivalent Series Resistance), which electrolytic capacitors are known to have high values of.
Your students may have to do a bit of refreshing (or firsttime research!) on the meaning of ESR before they can understand why large ESR values could cause a capacitor to explode under extreme operating conditions.
Suppose a friend wanted to install filter networks in the “woofer” section of their stereo system, to prevent highfrequency power from being wasted in speakers incapable of reproducing those frequencies. To this end, your friend installs the following resistorcapacitor networks:

After examining this schematic, you see that your friend has the right idea in mind, but implemented it incorrectly. These filter circuits would indeed block highfrequency signals from getting to the woofers, but they would not actually accomplish the stated goal of minimizing wasted power.
What would you recommend to your friend in lieu of this circuit design?
Rather than use a “shunting” form of lowpass filter (resistor and capacitor), a “blocking” form of lowpass filter (inductor) should be used instead.
The reason for this choice in filter designs is very practical. Ask your students to describe how a “shunting” form of filter works, where the reactive component is connected in parallel with the load, receiving power through a series resistor. Contrast this against a “blocking” form of filter circuit, in which a reactive component is connected in series with the load. In one form of filter, a resistor is necessary. In the other form of filter, a resistor is not necessary. What difference does this make in terms of power dissipation within the filter circuit?
The superposition principle describes how AC signals of different frequencies may be “mixed” together and later separated in a linear network, without one signal distorting another. DC may also be similarly mixed with AC, with the same results.
This phenomenon is frequently exploited in computer networks, where DC power and AC data signals (onandoff pulses of voltage representing 1and0 binary bits) may be combined on the same pair of wires, and later separated by filter circuits, so that the DC power goes to energize a circuit, and the AC signals go to another circuit where they are interpreted as digital data:

Filter circuits are also necessary on the transmission end of the cable, to prevent the AC signals from being shunted by the DC power supply’s capacitors, and to prevent the DC voltage from damaging the sensitive circuitry generating the AC voltage pulses.
Draw some filter circuits on each end of this twowire cable that perform these tasks, of separating the two sources from each other, and also separating the two signals (DC and AC) from each other at the receiving end so they may be directed to different loads:


Followup question #1: how might the superposition theorem be applied to this circuit, for the purposes of analyzing its function?
Followup question #2: suppose one of the capacitors were to fail shorted. Identify what effect, if any, this would have on the operation of the circuit. What if two capacitors were to fail shorted? Would it matter if those two capacitors were both on either the transmitting or the receiving side, or if one of the failed capacitors was on the transmitting side and the other was on the receiving side?
Discuss with your students why inductors were chosen as filtering elements for the DC power, while capacitors were chosen as filtering elements for the AC data signals. What are the relative reactances of these components when subjected to the respective frequencies of the AC data signals (many kilohertz or megahertz) versus the DC power supply (frequency = 0 hertz).
This question is also a good review of the “superposition theorem,” one of the most useful and easiesttounderstand of the network theorems. Note that no quantitative values need be considered to grasp the function of this communications network. Analyze it qualitatively with your students instead.
The following schematic shows the workings of a simple AM radio receiver, with transistor amplifier:

The “tank circuit” formed of a parallelconnected inductor and capacitor network performs a very important filtering function in this circuit. Describe what this filtering function is.
The “tank circuit” filters out all the unwanted radio frequencies, so that the listener hears only one radio station broadcast at a time.
Followup question: how might a variable capacitor be constructed, to suit the needs of a circuit such as this? Note that the capacitance range for a tuning capacitor such as this is typically in the picoFarad range.
Challenge your students to describe how to change stations on this radio receiver. For example, if we are listening to a station broadcasting at 1000 kHz and we want to change to a station broadcasting at 1150 kHz, what do we have to do to the circuit?
Be sure to discuss with them the construction of an adjustable capacitor (air dielectric).
Draw the Bode plot for an ideal highpass filter circuit:

Be sure to note the “cutoff frequency” on your plot.

Followup question: a theoretical filter with this kind of idealized response is sometimes referred to as a “brick wall” filter. Explain why this name is appropriate.
The plot given in the answer, of course, is for an ideal highpass filter, where all frequencies below f_{cutoff} are blocked and all frequencies above f_{cutoff} are passed. In reality, filter circuits never attain this ideal “squareedge” response. Discuss possible applications of such a filter with your students.
Challenge them to draw the Bode plots for ideal bandpass and bandstop filters as well. Exercises such as this really help to clarify the purpose of filter circuits. Otherwise, there is a tendency to lose perspective of what real filter circuits, with their correspondingly complex Bode plots and mathematical analyses, are supposed to do.
Draw the Bode plot for an ideal lowpass filter circuit:

Be sure to note the “cutoff frequency” on your plot.

Followup question: a theoretical filter with this kind of idealized response is sometimes referred to as a “brick wall” filter. Explain why this name is appropriate.
The plot given in the answer, of course, is for an ideal lowpass filter, where all frequencies below f_{cutoff} are passed and all frequencies above f_{cutoff} are blocked. In reality, filter circuits never attain this ideal “squareedge” response. Discuss possible applications of such a filter with your students.
Challenge them to draw the Bode plots for ideal bandpass and bandstop filters as well. Exercises such as this really help to clarify the purpose of filter circuits. Otherwise, there is a tendency to lose perspective of what real filter circuits, with their correspondingly complex Bode plots and mathematical analyses, are supposed to do.
Identify what type of filter this circuit is, and calculate its cutoff frequency given a resistor value of 1 kΩ and a capacitor value of 0.22 μF:

Calculate the impedance of both the resistor and the capacitor at this frequency. What do you notice about these two impedance values?
This is a lowpass filter.
f_{cutoff} = 723.4 Hz
Be sure to ask students where they found the cutoff frequency formula for this filter circuit.
When students calculate the impedance of the resistor and the capacitor at the cutoff frequency, they should notice something unique. Ask your students why these values are what they are at the cutoff frequency. Is this just a coincidence, or does this tell us more about how the “cutoff frequency” is defined for an RC circuit?
Identify what type of filter this circuit is, and calculate its cutoff frequency:

This is a highpass filter.
f_{cutoff} = 1.061 kHz
Be sure to ask students where they found the cutoff frequency formula for this filter circuit.
Identify what type of filter this circuit is, and calculate its cutoff frequency:

This is a lowpass filter.
f_{cutoff} = 1.026 kHz
Be sure to ask students where they found the cutoff frequency formula for this filter circuit.
Identify what type of filter this circuit is, calculate its cutoff frequency, and distinguish the input terminal from the output terminal:

This is a lowpass filter.
f_{cutoff} = 48.23 Hz
The input terminal is on the right, while the output terminal is on the left.
Be sure to ask students where they found the cutoff frequency formula for this filter circuit. Also, ask them how they were able to distinguish the input and output terminals. What would happen if these terminals were reversed (i.e. if the input signal were applied to the output terminal)?
The formula for determining the cutoff frequency of a simple LR filter circuit looks substantially different from the formula used to determine cutoff frequency in a simple RC filter circuit. Students new to this subject often resort to memorization to distinguish one formula from the other, but there is a better way.
In simple filter circuits (comprised of one reactive component and one resistor), cutoff frequency is that frequency where circuit reactance equals circuit resistance. Use this simple definition of cutoff frequency to derive both the RC and the LR filter circuit cutoff formulae, where f_{cutoff} is defined in terms of R and either L or C.


This is an exercise in algebraic substitution, taking the formula X = R and introducing f into it by way of substitution, then solving for f. Too many students try to memorize every new thing rather than build their knowledge upon previously learned material. It is surprising how many electrical and electronic formulae one may derive from just a handful of fundamental equations, if one knows how to use algebra.
Some textbooks present the LR cutoff frequency formula like this:

If students present this formula, you can be fairly sure they simply found it somewhere rather than derived it using algebra. Of course, this formula is exactly equivalent to the one I give in my answer  and it is good to show the class how these two are equivalent  but the real point of this question is to get your students using algebra as a practical tool in their understanding of electrical theory.
Identify what type of filter this circuit is, and calculate the size of resistor necessary to give it a cutoff frequency of 3 kHz:

This is a highpass filter.
R = 2 πf L
R = 5.65 kΩ
The most important part of this question, as usual, is to have students come up with methods of solution for determining R’s value. Ask them to explain how they arrived at their answer, and if their method of solution made use of any formula or principle used in capacitive filter circuits.
Calculate the power dissipated by this circuit’s load at two different source frequencies: 0 Hz (DC), and f_{cutoff}.

What do these figures tell you about the nature of this filter circuit (whether it is a lowpass or a highpass filter), and also about the definition of cutoff frequency (also referred to as f_{−3 dB})?
P_{load} @ f = 0 Hz = 64 mW
P_{load} @ f_{cutoff} = 32 mW
These load dissipation figures prove this circuit is a lowpass filter. They also demonstrate that the load dissipation at f_{cutoff} is exactly half the amount of power the filter is capable of passing to the load under ideal (maximum) conditions.
If your students have never encountered decibel (dB) ratings before, you should explain to them that 3 dB is an expression meaning “onehalf power,” and that this is why the cutoff frequency of a filter is often referred to as the halfpower point.
The important lesson to be learned here about cutoff frequency is that its definition means something in terms of load power. It is not as though someone decided to arbitrarily define f_{cutoff} as the point at which the load receives 70.7% of the source voltage!
Filter circuits don’t just attenuate signals, they also shift the phase of signals. Calculate the amount of phase shift that these two filter circuits impart to their signals (from input to output) operating at the cutoff frequency:

HP filter: Θ = +45^{o} (V_{out} leads V_{in})
LP filter: Θ = 45^{o} (V_{out} lags V_{in})
Note that no component values are given in this question, only the condition that both circuits are operating at the cutoff frequency. This may cause trouble for some students, because they are only comfortable with numerical calculations. The structure of this question forces students to think a bit differently than they might be accustomed to.
Real filters never exhibit perfect “squareedge” Bode plot responses. A typical lowpass filter circuit, for example, might have a frequency response that looks like this:

What does the term rolloff refer to, in the context of filter circuits and Bode plots? Why would this parameter be important to a technician or engineer?
“Rolloff” refers to the slope of the Bode plot in the attenuating range of the filter circuit, usually expressed in units of decibels per octave (dB/octave) or decibels per decade (dB/decade):

Point students’ attention to the scale used on this particular Bode plot. This is called a loglog scale, where neither vertical nor horizontal axis is linearly marked. This scaling allows a very wide range of conditions to be shown on a relatively small plot, and is very common in filter circuit analysis.
Explain what a bandpass filter is, and how it differs from either a lowpass or a highpass filter circuit. Also, explain what a bandstop filter is, and draw Bode plots representative of both bandpass and bandstop filter types.
A bandpass filter passes only those frequencies falling within a specified range, or “band.” A bandstop filter, sometimes referred to as a notch filter, does just the opposite: it attenuates frequencies falling within a specified band.
Challenge question: what type of filter, bandpass or bandstop, do you suppose is used in a radio receiver (tuner)? Explain your reasoning.
In this question, I’ve opted to let students draw Bode plots, only giving them written descriptions of each filter type.
A common way of representing complex electronic systems is the block diagram, where specific functional sections of a system are outlined as squares or rectangles, each with a certain purpose and each having input(s) and output(s). For an example, here is a block diagram of an analog (“Cathode Ray”) oscilloscope, or CRO:

Block diagrams may also be helpful in representing and understanding filter circuits. Consider these symbols, for instance:

Which of these represents a lowpass filter, and which represents a highpass filter? Explain your reasoning.
Also, identify the new filter functions created by the compounding of low and highpass filter “blocks”:


Aside from getting students to understand that bandfunction filters may be built from sets of low and highpass filter blocks, this question is really intended to initiate problemsolving activity. Discuss with your students how they might approach a problem like this to see how the circuits respond. What “thought experiments” did they try in their minds to investigate these circuits?
What kind of filtering action (highpass, lowpass, bandpass, bandstop) does this resonant circuit provide?

This circuit is a bandpass filter.
As usual, ask your students to explain why the answer is correct, not just repeat the answer that is given!
What kind of filtering action (highpass, lowpass, bandpass, bandstop) does this resonant circuit provide?

This circuit is a bandstop filter.
As usual, ask your students to explain why the answer is correct, not just repeat the answer that is given!
Identify each of these filter types, and explain how you were able to positively identify their behaviors:


Followup question: in each of the circuits shown, identify at least one single component failure that has the ability to prevent any signal voltage from reaching the output terminals.
Some of these filter designs are resonant in nature, while others are not. Resonant circuits, especially when made with highQ components, approach ideal bandpass (or block) characteristics. Discuss with your students the different design strategies between resonant and nonresonant band filters.
The highpass filter containing both inductors and capacitors may at first appear to be some form of resonant (i.e. bandpass or bandstop) filter. It actually will resonate at some frequency(ies), but its overall behavior is still highpass. If students ask about this, you may best answer their queries by using computer simulation software to plot the behavior of a similar circuit (or by suggesting they do the simulation themselves).
Regarding the followup question, it would be a good exercise to discuss which suggested component failures are more likely than others, given the relatively likelihood for capacitors to fail shorted and inductors and resistors to fail open.
Identify the following filter types, and be prepared to explain your answers:


Some of these filter designs are resonant in nature, while others are not. Resonant circuits, especially when made with highQ components, approach ideal bandpass (or block) characteristics. Discuss with your students the different design strategies between resonant and nonresonant band filters.
Although resonant band filter designs have nearly ideal (theoretical) characteristics, band filters built with capacitors and resistors only are also popular. Ask your students why this might be. Is there any reason inductors might purposefully be avoided when designing filter circuits?
The cutoff frequency, also known as halfpower point or 3dB point, of either a lowpass or a highpass filter is fairly easy to define. But what about bandpass and bandstop filter circuits? Does the concept of a “cutoff frequency” apply to these filter types? Explain your answer.
Unlike lowpass and highpass filters, bandpass and bandstop filter circuits have two cutoff frequencies (f_{c1} and f_{c2})!
This question presents a good opportunity to ask students to draw the Bode plot of a typical bandpass or bandstop filter on the board in front of the class to illustrate the concept. Don’t be afraid to let students up to the front of the classroom to present their findings. It’s a great way to build confidence in them and also to help suppress the illusion that you (the teacher) are the Supreme Authority of the classroom!
Plot the typical response of a bandpass filter circuit, showing signal output (amplitude) on the vertical axis and frequency on the horizontal axis:

Also, identify and label the bandwidth of the circuit on your filter plot.

The bandwidth of a bandpass filter circuit is that range of frequencies where the output amplitude is at least 70.7% of maximum:

Bandwidth is an important concept in electronics, for more than just filter circuits. Your students may discover references to bandwidth of amplifiers, transmission lines, and other circuit elements as they do their research. Despite the many and varied applications of this term, the principle is fundamentally the same.
Plot the typical response of a bandstop filter circuit, showing signal output (amplitude) on the vertical axis and frequency on the horizontal axis:

Also, identify and label the bandwidth of the circuit on your filter plot.

The bandwidth of a bandstop filter circuit is that range of frequencies where the output amplitude is reduced to at least 70.7% of full attenuation:

Bandwidth is an important concept in electronics, for more than just filter circuits. Your students may discover references to bandwidth of amplifiers, transmission lines, and other circuit elements as they do their research. Despite the many and varied applications of this term, the principle is fundamentally the same.
Plot the typical frequency responses of four different filter circuits, showing signal output (amplitude) on the vertical axis and frequency on the horizontal axis:

Also, identify and label the bandwidth of the filter circuit on each plot.

Although “bandwidth” is usually applied first to bandpass and bandstop filters, students need to realize that it applies to the other filter types as well. This question, in addition to reviewing the definition of bandwidth, also reviews the definition of cutoff frequency. Ask your students to explain where the 70.7% figure comes from. Hint: halfpower point!
A white noise source is a special type of AC signal voltage source which outputs a broad band of frequencies (“noise”) with a constant amplitude across its rated range. Determine what the display of a spectrum analyzer would show if directly connected to a white noise source, and also if connected to a lowpass filter which is in turn connected to a white noise source:


The purpose of this question, besides providing a convenient way to characterize a filter circuit, is to introduce students to the concept of a white noise source and also to strengthen their understanding of a spectrum analyzer’s function.
In case anyone happens to notice, be aware that the rolloff shown for this filter circuit is very steep! This sort of sharp response could never be realized with a simple oneresistor, onecapacitor (“first order”) filter. It would have to be a multistage analog filter circuit or some sort of active filter circuit.
Predict how the operation of this secondorder passive filter circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults):

For each of these conditions, explain why the resulting effects will occur.
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.
The Q factor of a series inductive circuit is given by the following equation:

Likewise, we know that inductive reactance may be found by the following equation:

We also know that the resonant frequency of a series LC circuit is given by this equation:

Through algebraic substitution, write an equation that gives the Q factor of a series resonant LC circuit exclusively in terms of L, C, and R, without reference to reactance (X) or frequency (f).
\(Q=\frac{1}{R}\sqrt\frac{L}{C}\)
This is merely an exercise in algebra. However, knowing how these three component values affects the Q factor of a resonant circuit is a valuable and practical insight!
Calculate the resonant frequency, bandwidth, and halfpower points of the following filter circuit:

f_{r} = 6.79 kHz
Bandwidth = 289.4 Hz
f_{1} = 6.64 kHz
f_{2} = 6.93 kHz
Followup question: how would a decrease in the Q (“quality factor”) of the circuit affect the bandwidth, or would it at all?
The formulae required to calculate these parameters are easily obtained from any basic electronics text. No student should have trouble finding this information.
Suppose a few turns of wire within the inductor in this filter circuit suddenly became shortcircuited, so that the inductor effectively has fewer turns of wire than it did before:

What sort of effect would this fault have on the filtering action of this circuit?
The resonant frequency of the circuit would increase.
Challenge question: what would happen to the Q of this filter circuit as a result of the fault within the inductor?
Determining the effect on resonant frequency is a simple matter of qualitative analysis with the resonant frequency formula. The effect on Q (challenge question) may be answered just as easily if the students know the formula relating bandwidth to L, C, and R.
An interesting technology dating back at least as far as the 1940’s, but which is still of interest today is power line carrier: the ability to communicate information as well as electrical power over power line conductors. Hardwired electronic data communication consists of highfrequency, low voltage AC signals, while electrical power is lowfrequency, highvoltage AC. For rather obvious reasons, it is important to be able to separate these two types of AC voltage quantities from entering the wrong equipment (especially the highvoltage AC power from reaching sensitive electronic communications circuitry).
Here is a simplified diagram of a powerline carrier system:

The communications transmitter is shown in simplified form as an AC voltage source, while the receiver is shown as a resistor. Though each of these components is much more complex than what is suggested by these symbols, the purpose here is to show the transmitter as a source of highfrequency AC, and the receiver as a load of highfrequency AC.
Trace the complete circuit for the highfrequency AC signal generated by the “Transmitter” in the diagram. How many power line conductors are being used in this communications circuit? Explain how the combination of “line trap” LC networks and “coupling” capacitors ensure the communications equipment never becomes exposed to highvoltage electrical power carried by the power lines, and visaversa.

Followup question #1: trace the path of linefrequency (50 Hz or 60 Hz) load current in this system, identifying which component of the line trap filters (L or C) is more important to the passage of power to the load. Remember that the line trap filters are tuned to resonate at the frequency of the communication signal (50150 kHz is typical).
Followup question #2: coupling capacitor units used in power line carrier systems are specialpurpose, highvoltage devices. One of the features of a standard coupling capacitor unit is a spark gap intended to “clamp” over voltages arising from lightning strikes and other transient events on the power line:

Explain how such a spark gap is supposed to work, and why it functions as an overvoltage protection device.
Although power line carrier technology is not used as much for communication in highvoltage distribution systems as it used to be  now that microwave, fiber optic, and satellite communications technology has superseded this older technique  it is still used in lower voltage power systems including residential (home) wiring. Ask your students if they have heard of any consumer technology capable of broadcasting any kind of data or information along receptacle wiring. “X10” is a mature technology for doing this, and at this time (2004) there are devices available on the market allowing one to plug telephones into power receptacles to link phones in different rooms together without having to add special telephone cabling.
Even if your students have not yet learned about threephase power systems or transformers, they should still be able to discern the circuit path of the communications signal, based on what they know of capacitors and inductors, and how they respond to signals of arbitrarily high frequency.
Information on the coupling capacitor units was obtained from page 452 of the Industrial Electronics Reference Book, published by John Wiley & Sons in 1948 (fourth printing, June 1953). Although power line carrier technology is not as widely used now as it was back then, I believe it holds great educational value to students just learning about filter circuits and the idea of mixing signals of differing frequency in the same circuit.
In this powerline carrier system, a pair of coupling capacitors connects a highfrequency “Transmitter” unit to two power line conductors, and a similar pair of coupling capacitors connects a “Receiver” unit to the same two conductors:

While coupling capacitors alone are adequate to perform the necessary filtering function needed by the communications equipment (to prevent damaged from the highvoltage electrical power also carried by the lines), that signal coupling may be made more efficient by the introduction of two line tuning units:

Explain why the addition of more components (in series, no less!) provides a better “connection” between the highfrequency Transmitter and Receiver units than coupling capacitors alone. Hint: the operating frequency of the communications equipment is fixed, or at least variable only over a narrow range.
The introduction of the linetuning units increases the efficiency of signal coupling by exploiting the principle of resonance between seriesconnected capacitors and inductors.
Challenge question: there are many applications in electronics where we couple highfrequency AC signals by means of capacitors alone. If capacitive reactance is any concern, we just use capacitors of large enough value that the reactance is minimal. Why would this not be a practical option in a powerline carrier system such as this? Why could we not (or why would we not) just choose coupling capacitors with very high capacitances, instead of adding extra components to the system?
Although power line carrier technology is not used as much for communication in highvoltage distribution systems as it used to be  now that microwave, fiber optic, and satellite communications technology has come of age  it is still used in lower voltage power systems including residential (home) wiring. Ask your students if they have heard of any consumer technology capable of broadcasting any kind of data or information along receptacle wiring. “X10” is a mature technology for doing this, and at this time (2004) there are devices available on the market allowing one to plug telephones into power receptacles to link phones in different rooms together without having to add special telephone cabling.
I think this is a really neat application of resonance: the complementary nature of inductors to capacitors works to overcome the lessthanideal coupling provided by capacitors alone. Discuss the challenge question with your students, asking them to consider some of the practical limitations of capacitors, and how an inductor/capacitor resonant pair solves the linecoupling problem better than an oversized capacitor.
The following circuit is called a twintee filter:

Research the equation predicting this circuit’s “notch” frequency, given the component value ratios shown.


Answering this question is simply a matter of research! There are many references a student could go to for information on twintee filters.
Suppose this bandstop filter were to suddenly start acting as a highpass filter. Identify a single component failure that could cause this problem to occur:

If resistor R_{3} failed open, it would cause this problem. However, this is not the only failure that could cause the same type of problem!
Ask your students to explain why an open R_{2} would cause this filter to act as a highpass instead of a bandstop. Then, ask them to identify other possible component failures that could cause a similar effect.
By the way, this filter circuit illustrates the popular twintee filter topology.
Examine the following schematic diagram for an audio tone control circuit:

Determine which potentiometer controls the bass (low frequency) tones and which controls the treble (high frequency) tones, and explain how you made those determinations.

The most important answer to this question is how your students arrived at the correct potentiometer identifications. If none of your students were able to figure out how to identify the potentiometers, give them this tip: use the superposition theorem to analyze the response of this circuit to both lowfrequency signals and highfrequency signals. Assume that for bass tones the capacitors are opaque (Z = ∞) and that for treble tones they are transparent (Z = 0). The answers should be clear if they follow this technique.
This general problemsolving technique  analyzing two or more “extreme” scenarios to compare the results  is an important one for your students to become familiar with. It is extremely helpful in the analysis of filter circuits!
Examine the following audio tone control circuit, used to control the balance of bass and treble heard at the headphones from an audio source such as a radio or CD player:

Suppose that after working just fine for quite a long while, suddenly no more bass tones were heard through the headphones. Identify at least two component or wiring faults that could cause this to happen.
Here are some possibilities:
The circuit shown in the question is not very practical for direct headphone use, unless lowvalue resistors are used. Otherwise, the losses are too great and maximum volume suffers. An improvement over the original circuit is one where a matching transformer is used to effectively increase the impedance of the headphones:

Suppose that the following audio tone control circuit has a problem: the second potentiometer (R_{pot2}) seems to act more like a plain volume control than a tone control. Instead of adjusting the amount of treble heard at the headphones, it seems to adjust the volume of bass and treble tones alike:

What do you think might be wrong with this circuit? Assuming it has been correctly designed and was working well for some time, what component or wire failure could possibly account for this behavior?
Most likely capacitor C_{1} has failed shorted.
Discuss with your students how this circuit functions before they offer their ideas for faulted components or wires. One must understand the basic operating principle(s) of a circuit before one can troubleshoot it effectively!
Suppose that the following audio tone control circuit has a problem: the first potentiometer (R_{pot1}) seems to act more like a plain volume control than a tone control. Instead of adjusting the amount of bass heard at the headphones, it seems to adjust the volume of bass and treble tones alike:

What do you think might be wrong with this circuit? Assuming it has been correctly designed and was working well for some time, what component or wire failure could possibly account for this behavior?
Most likely inductor L_{1} has failed shorted.
Discuss with your students how this circuit functions before they offer their ideas for faulted components or wires. One must understand the basic operating principle(s) of a circuit before one can troubleshoot it effectively!
Controlling electrical “noise” in automotive electrical systems can be problematic, as there are many sources of “noise” voltages throughout a car. Spark ignitions and alternators can both generate substantial noise voltages, superimposed on the DC voltage in a car’s electrical system. A simple way to electrically model this noise is to draw it as an AC “noise voltage” source in series with the DC source. If this noise enters a radio or audio amplifier, the result will be an irritating sound produced at the speakers:

What would you suggest as a “fix” for this problem if a friend asked you to apply your electronics expertise to their noisy car audio system? Be sure to provide at least two practical suggestions.
This is perhaps the easiest solution, to install a very large capacitor (C_{huge}) in parallel with the audio load:

Other, more sophisticated solutions exist, however!
Followup question: use superposition theorem to show why the capacitor mitigates the electrical noise without interfering with the transfer of DC power to the radio/amplifier.
The followup question is yet another example of how practical the superposition theorem is when analyzing filter circuits.
A helical resonator is a special type of bandpass filter commonly used in VHF and UHF radio receiver circuitry. Such a device is made up of multiple metal cavities, each containing a helix (coil) of wire connected to the cavity at one end and free at the other. Slots cut between the cavities permits coupling between the coils, with the input at one extreme end and the output at the other:

The above illustration shows a threestage helical resonator, with adjustable metal plates at the top of each helix for tuning. Draw a schematic representation of this resonator, and explain where the capacitance comes from that allows each of the coils to form a resonant circuit.

Followup question: why do you suppose multiple stages of tuned (“tank”) circuits would be necessary in a highquality tuner circuit? Why not just use a single tank circuit as a filter? Would that not be simpler and less expensive?
If students have a difficult time seeing where the capacitance comes from, remind them that we are dealing with very high frequencies here, and that air between metal parts is a sufficient dielectric to create the needed capacitance.
The coupling between coils may be a bit more difficult to grasp, especially if your students have not yet studied mutual inductance. Suffice it to say that energy is transferred between coils with little loss at high frequencies, permitting an RF signal to enter at one end of the resonator and exit out the other without any wires physically connecting the stages together.
Don’t just sit there! Build something!! 
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.
You will learn much more by actually building and analyzing real circuits, letting your test equipment provide the “answers” instead of a book or another person. For successful circuitbuilding exercises, follow these steps:
For AC circuits where inductive and capacitive reactances (impedances) are a significant element in the calculations, I recommend high quality (highQ) inductors and capacitors, and powering your circuit with low frequency voltage (powerline frequency works well) to minimize parasitic effects. If you are on a restricted budget, I have found that inexpensive electronic musical keyboards serve well as “function generators” for producing a wide range of audiofrequency AC signals. Be sure to choose a keyboard “voice” that closely mimics a sine wave (the “panflute” voice is typically good), if sinusoidal waveforms are an important assumption in your calculations.
As usual, avoid very high and very low resistor values, to avoid measurement errors caused by meter “loading”. I recommend resistor values between 1 kΩ and 100 kΩ.
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 timesaving technique is to reuse the same components in a variety of different circuit configurations. This way, you won’t have to measure any component’s value more than once.
Let the electrons themselves give you the answers to your own “practice problems”!
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, handson 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 highquality capacitors and inductors, and try to use low frequencies for the power supply. Small stepdown 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!
Published under the terms and conditions of the Creative Commons Attribution License
In Partnership with Rohde & Schwarz
by Robert Keim
by Jake Hertz