Mixed t&period

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Mixed t&period

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Mixed T&period Mixed-Frequency Signals Video

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Explain how both these instruments are similar in function to a spectrum analyzer, and also how both differ. Spectrum analyzers capable of analyzing radio-frequency signals are very expensive, but low-cost personal computer hardware and software does a good job of analyzing complex audio signals.

It would be a benefit to your class to have a low-frequency spectrum analyzer setup available for student use, and possible demonstration during discussion.

Suppose an amplifier circuit is connected to a sine-wave signal generator, and a spectrum analyzer used to measure both the input and the output signals of the amplifier:.

The input signal is clean: a single peak at the 1 kHz mark. The purpose of this question is to get students to realize the presence of harmonics means a departure from a once-perfect sinusoidal wave-shape.

What used to be free of harmonics now contains harmonics, and this indicates distortion of the sine wave somewhere within the amplifier. What causes harmonics to form in the output of a transistor amplifier circuit, if the input waveform is perfectly sinusoidal free from harmonics?

Be as specific as you can in your answer. Any feature or fault of the circuit causing imperfect signal reproduction will necessarily create harmonics, for it will turn a perfectly sinusoidal input signal into a distorted non-perfect-sinusoidal signal.

Discuss with your students the nature of harmonics: how multiple sinusoidal waveforms are necessarily contained in any periodic waveform that is not perfectly sinusoidal itself.

What causes harmonics to form in the output of a transistor oscillator circuit such as a Colpitts or a Hartley, which is designed to produce a sinusoidal signal?

Any feature or fault of the amplifier portion of the oscillator circuit causing imperfect signal reproduction will necessarily create harmonics, for it will turn a perfectly sinusoidal input signal from the LC network into a distorted non-perfect-sinusoidal output signal.

Explain why. A clever way to produce sine waves is to pass the output of a square-wave oscillator through a l ow-pass filter circuit :.

The LP filter blocks all harmonics of the square wave except the fundamental 1st harmonic , resulting in a sinusoidal output.

Ask your students what they think about the rolloff requirement for this LP filter. Will any LP filter work, or do we need something special?

My answer to this question is intentionally vague. Each series LC section is a resonant band-pass filter , tuned to successive harmonics of a 60 Hz sine wave.

The selector switch enables a single voltmeter to measure the RMS amplitude of each harmonic. Follow-up question: calculate the exact inductance values necessary for precise tuning of the five LC filters, for the first five harmonics of a 60 Hz waveform.

Challenge question: the voltmeter in this circuit would not have to be a true-RMS meter. It could simply be an average-responding RMS-calibrated voltmeter and it would work the same.

This question provides students with some review of passive filter circuit theory, as well as insight into a practical circuit they could conceivably build as a project.

High Q values for each filter section ensure that the meter will only register the particular harmonic that is selected for measurement.

If a particular electronic system such as an AC power system has a fundamental frequency of 60 Hz, calculate the frequencies of the following harmonics:.

An octave is a type of harmonic frequency. Suppose an electronic circuit operates at a fundamental frequency of 1 kHz.

Calculate the frequencies of the following octaves:. Ask your students if they can determine the mathematical relationship between octave number, octave frequency, and fundamental frequency.

This is a bit more difficult to do than for integer harmonics, but not beyond reason if students are familiar with exponents. Electrically, we might represent a square-wave voltage source as a circle with a square-wave symbol inside, like this:.

Knowing the Fourier series of this voltage, however, allows us to represent the same voltage source as a set of series-connected voltage sources, each with its own sinusoidal frequency.

Draw the equivalent schematic for a 10 volt peak , Hz square-wave source in this manner showing only the first four harmonics, labeling each sinusoidal voltage source with its own RMS voltage value and frequency:.

To be honest, the four-harmonic equivalent circuit is a rather poor approximation for a square wave. The real purpose of this question, though, is to have students relate the sinusoidal terms of a common Fourier series for a square wave to a schematic diagram, translating between angular velocity and frequency, peak values and RMS values.

Please note that the voltage magnitudes shown in the answer are RMS and not peak! If you were to calculate peak sinusoid source values, you would obtain these results:.

Draw the equivalent schematic in this manner, labeling each voltage source with its RMS voltage value, frequency in Hz , and phase angle:.

The purpose of this question is to have students relate the sinusoidal terms of a particular Fourier series to a schematic diagram, translating between angular velocity and frequency, peak values and RMS values.

To calculate this power figure, students have to determine the RMS value of the square wave. Thankfully, this is not difficult. Just calculate the power dissipated in the resistor by the 1st, 3rd, 5th, and 7th harmonics only.

To calculate this power figure, students have to research the Fourier series for a square wave. Many textbooks use square waves to introduce the subject of Fourier series, so this should not be difficult for students to find.

Ask your students how the real power dissipated by this resistor compares with the final figure of 5. Is the real power dissipation more, less, or equal to this figure?

If not equal, what would we have to do to arrive at a more precise figure? Ideally, a sinusoidal oscillator will output a signal consisting of a single fundamental frequency, with no harmonics.

Realistically, though, sine-wave oscillators always exhibit some degree of distortion, and are therefore never completely harmonic-free.

Describe what the display of a spectrum analyzer would look like when connected to the output of a perfect sinusoidal oscillator. The purpose of this question is to get students to think about how a spectrum analyzer would be used in a practical scenario, and what the spectrum would look like for a couple of different scenarios.

Really, it focuses more on the harmonic analysis instrument the spectrum analyzer more than the oscillator circuit. An electronics technician connects the input of a spectrum analyzer to the secondary winding of an AC power transformer, plugged into a power receptacle.

He sets the spectrum analyzer to show 60 Hz as the fundamental frequency, expecting to see the following display:.

Explain what this pattern means, in practical terms. What this pattern means is the power-line voltage waveform is distorted from what should be a perfect sine-wave shape.

Realistically, though, amplifiers always exhibit some degree of distortion. Describe how harmonic analysis - either with a spectrum analyzer or some other piece of test equipment capable of measuring harmonics in a signal - is used to quantify the distortion of an amplifier circuit.

The answer given is purposefully vague. All I did is describe what gets connected to the amplifier, not how to interpret the measurements.

Ask your students to explain why a pure sine wave is chosen as the test signal, and what sort of response would be considered ideal to see on the spectrum analyzer.

Under certain conditions, harmonics may be produced in AC power systems by inductors and transformers. How is this possible, as these devices are normally considered to be linear?

And of those devices, are they always linear, or are they capable of nonlinear behavior under special conditions?

Use the discussion time to review B-H curves for ferromagnetic materials with your students, asking them to draw the curves and point out where along those curves inductors and transformers normally operate.

What conditions, specifically, would make an iron-core device act nonlinearly? On a similar note, the slightly nonlinear nature of ferromagnetic core transformers is known to permit signals to modulate each other in certain audio amplifier designs, to produce a specific kind of audio signal distortion known as intermodulation distortion.

Normally, modulation is a function possible only in nonlinear systems, so the fact that modulation occurs in a transformer is proof positive of at least some degree of nonlinearity.

Identify some ways in which harmonics may be mitigated in AC power systems, since they tend to cause trouble for a variety of electrical components.

Filter circuits may be employed to block harmonic frequencies from reaching certain sensitive components.

The answer given here is correct, but vague. I did not specify the type of filter or exactly how it might be connected to a load.

These are questions to ask your students during discussion. If both these circuits are energized by an AC sine-wave source providing a perfectly undistorted signal, the resulting output waveforms will differ in phase and possibly in amplitude, but not in shape:.

If, however, the excitation voltage is slightly distorted, one of the outputs will be more sinusoidal than the other. Explain whether it is the differentiator or the integrator that produces the signal most resembling a pure sine wave, and why.

Hint: I recommend building this circuit and powering it with a triangle wave, to simulate a mildly distorted sine wave.

The differentiator circuit will output a much more distorted waveshape, because differentiation magnifies harmonics:. As an interesting footnote, this is precisely why differentiation is rarely performed on real-world signals.

For a practical example of this, tell your students about vibration measurement, where it is more common to calculate velocity based on time-integration of an acceleration signal than it is to calculate acceleration based on time-differentiation of a velocity signal.

Note the effect of adding the second harmonic of a waveform to the fundamental, and compare that effect with adding the third harmonic of a waveform to the fundamental:.

Examine these sets of harmonic sums, and indicate the trend you see with regard to harmonic number and symmetry of the final Sum waveforms. Specifically, how does the addition of an even harmonic compare to the addition of an odd harmonic, in terms of final waveshape?

The addition of an even harmonic introduces asymmetry about the horizontal axis. The addition of odd harmonics does not.

Although the sequence of images presented in the question by no means constitutes a formal proof, it should lead students to observe a trend: that odd harmonics do not make a waveform unsymmetrical about the horizontal axis, whereas even harmonics do.

Given these two facts, we may make qualitative judgments about the harmonic content of a waveform simply by visually checking for symmetry about the horizontal axis.

Incidentally, some students have a difficult time grasping the concept of symmetry about the horizontal axis of a waveform. Take this simple example, which is symmetrical about its horizontal centerline:.

Some students will protest that this waveform is not symmetrical about its centerline, because it does not look exactly the same as before after flipping.

They must bear in mind, though, that this is just one cycle of a continuous waveform. In reality, the waveform looks like this before and after flipping:.

All one needs to do to see that these two waveforms are indeed identical is to do a degree phase shift shifting either to the left or to the right :.

By contrast, a waveform without symmetry about the horizontal axis cannot be made to look the same after flipping, no matter what subsequent phase shift is given to it:.

Discuss ways to identify this type of asymmetry, and what it means in terms of harmonic content. When technicians and engineers consider harmonics in AC power systems, they usually only consider odd-numbered harmonic frequencies.

Explain why this is. Even-numbered harmonics can and do appear in AC power systems, although they are typically much lower in amplitude than the odd-numbered harmonics due to the nature of most nonlinear loads.

If you ever wish to prove the existence of even-numbered harmonics in a power system, all you have to do is analyze the input current waveform of a half-wave rectifier!

By visual inspection, determine which of the following waveforms contain even-numbered harmonics:. Note that only one cycle is shown for each waveform.

The following waveforms contain even-numbered harmonics: B , C , D , F , and I. The rest only contain odd harmonics of the fundamental.

Ask your students how they were able to discern the presence of even-numbered harmonics by visual inspection. This typically proves difficult for some of my students whose spatial-relations skills are weak.

These students need some sort of algorithmic step-by-step procedure to see what other students see immediately, and discussion time is a great opportunity for students to share technique.

The following waveforms contain even-numbered harmonics: C , D , G , and I. A crude measurement circuit for harmonic content of a signal uses a notch filter tuned to the fundamental frequency of the signal being measured.

Examine the following circuit and then explain how you think it would work:. This test circuit relies on the assumption that the notch filter is perfect i.

Since no filter is perfect, it would be a good idea to ask your students what effect they think an imperfect notch filter would have on the validity of the test.

Radio communication functions on the general principle of high-frequency AC power being modulated by low-frequency data. Two common forms of modulation are Amplitude Modulation AM and Frequency Modulation FM.

In both cases, the modulation of a high frequency waveform by a lower-frequency waveform produces something called sidebands. On a spectrum analyzer, they show up as peaks to either side of the main carrier peak.

Their quantity, frequencies, and amplitudes are all a function of the data signals modulating the carrier. The answer makes frequent use of the word carrier without defining it.

Ask them to define this word, in addition to their description of sidebands. The following circuit is a simple mixer circuit, combining three AC voltage signals into one, to be measured by an oscilloscope:.

How does the superposition principle relate to the operation of a mixer circuit like this? What if the mixing circuit contains capacitors and inductors rather than resistors?

Does the same principle apply? Why or why not? The question, then, is really: what constitutes a linear network? When I changed BOTH my Language to English US AND the Keyboard to US then the Emoji panel appeared.

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Then it works! FYI I live in New Zealand In the South Pacific and My Language was set to English New Zealand with a US Keyboard.

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I have the same question Subscribe Subscribe Subscribe to RSS feed. Previous Next. Princess Mat Microsoft Agent. Hi, New Emoji and a Useful Shortcut is one of the new features in the Windows 10 fall Creators Update.

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