Experiment No. 2 Pre-Lab Signal Mixing and Amplitude Modulation
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1 Experiment No. 2 Pre-Lab Signal Mixing and Amplitude Modulation Read the information presented in this pre-lab and answer the questions given. Submit the answers to your lab instructor before the experimental procedure is performed. In the study of communications, mixing refers to a non-linear combination of signals. A circuit element having a non-linear relationship between current and voltage is used for this type of mixing. The non-linearity creates harmonics and sum and difference frequencies of the original signals. In the experiment two sinusoidal signals, a high frequency carrier and a low frequency modulation, will be combined in a linear summing amplifier and applied to a semiconductor diode. The diode's non-linearity will add harmonics and sum and difference frequencies to the signals and a band-pass filter will eliminate all but the carrier and sum and difference frequencies. The trapezoidal method will be used to measure properties of the resulting AM signal. The spectrum of the signal products of linear and non-linear mixing will be observed using the FFT function of the oscilloscope and a process called heterodyning, used in transmitters and receivers to change frequencies, will demonstrated. Part A: Amplitude Modulation Objective: 1. To show the difference between linear and non-linear mixing 2. To see that non-linear mixing produces amplitude modulation 3. To use the trapezoidal method to measure modulation depth +15V 10k Colpitts oscillator 1mH A 22k + 100k +15V 1N4148 B C 6.8k 10k 2N uF 0.02uF 100k LM318-15V 2k D 0.01uF 1mH 47k Function Generator 1
2 In the circuit shown, the transistor circuit on the left is a Colpitts oscillator that generates a 50kHz signal at point A. The oscillator and the function generator signals are applied to the op amp summing amplifier. The function generator will be set to 1kHz. The op amp is needed to provide amplification for optimum operation of the diode that is connected between points B and C. Q1. What is the voltage gain of the amplifier for the Colpitts oscillator input signal? Q2. What is the voltage gain of the amplifier for the function generator input? You will observe the waveforms at point A, B, C and D respectively. The output of the summing amplifier at point at B will be the linear sum of the oscillator signal and the function generator signal. It should look something like the following, Waveform at B V (volts) Series t (ms) A non-linear combination of the two signals will appear at point C. The peak-to-peak amplitude of the higher frequency portion will not be constant throughout the low frequency cycle due to the non-linear characteristic of the diode. Refer to the idealized diode curve shown below to see how larger input voltages will produce disproportional large currents, thus larger output voltages. Diode Characteristic Current (A) 0.4 Series Voltage (V) Negative voltages will produce insignificant current and thus no output. Refer to footnote #1 for a more detailed explanation. 2
3 As a result, the waveform at point C should look something like the following, Waveform at C 1.2E-10 1E-10 8E-11 V (volts) 6E-11 4E-11 Series1 2E E-11 t (ms) At point D the LC band pass filter will remove all frequencies produced by non-linear mixing except the frequencies near 50 khz. You should see an AM waveform. The envelope of the waveform can be viewed as a line connecting the positive or negative peaks of the waveform. Q3. Which circuit voltage, if any, should the envelope (as defined above) look like? Q4. What is the difference between the positive and negative envelopes? Next the function generator will be connected to the oscilloscope so that it becomes the horizontal, X input (Ch 1), and the Y input (Ch 2) will be connected to point D, the AM signal. The oscilloscope will be set for XY operation using the Display button. You will see a trapezoidal pattern, but the pattern may be reversed in the X direction. An RC network can be connected between the function generator and the X input to correct for phase-shift. This will minimize phase distortion in the trapezoidal pattern. Q5. What affect should the function generator amplitude and frequency have on the pattern? Q6. If the pattern has curved rather than straight sides, what would that indicate? Q7. What affect should the modulation level have an on the linearity of the sides? Part B: Frequency Spectrums Objective: To show the frequencies produced by non-linear mixing 3
4 The Math/Operation button on the oscilloscope will be used to select FFT in order to view the spectrum of the input signal. The spectrum has a logarithmic (db) vertical scale where db refers to db relative to 1VRMS. To be technically correct the units should be labeled dbv since db alone is a relative unit, not an actual voltage. The Vertical Scale and the Horizontal Scale controls will adjust the db and the frequency scales respectively. You will observe the spectrums at points A, B, C and D. At each point there will be peaks that rise above the background level at the bottom. Ideally there should be only one peak at A, two at B and three at D. At point C there will be additional peaks due to inter-modulation products (sum and difference frequencies) of the fundamental frequencies and their harmonics. See footnote #1 for mathematical justification of the frequencies created. Q8. What frequency peak(s) would you expect at, A? B? C? D? There will be other minor peaks and harmonic peaks that may be ignored. These occur because the waveforms from the oscillator and the function generator are not purely sinusoidal. See footnote #2 for more detail. For the waveform at point D the FFT Zoom feature of the oscilloscope will be used to expand the frequency scale around the center screen position. The carrier frequency should be centered on the screen in order to view the AM spectrum. With maximum zoom, you should be able to clearly see and measure the AM signal sidebands. Q9. What should the frequency separation between the carrier and the sidebands be? You will need to convert the db measurements to voltage values and calculate the modulation percentage based on these values. You will compare this value to the modulation percentage measured using the trapezoidal pattern in Part A. Q10. Calculate the circuit Q of the tuned circuit at point D (loaded Q). The circuit Q will be smaller than the coil Q due to the resistances connected at point D. (See footnote #3). Q11. Calculate the bandwidth of the circuit. Q12. Is the bandwidth wide enough to allow the 1 khz sidebands? Q13. What would happen to the spectrum if the function generator were set to a very large voltage so as to cause overmodulation? Q14. What additional sidebands would you see? 4
5 Q15. Why might this be unacceptable for broadcasting the signal? Part C: Beat Frequencies Objective: To demonstrate products of mixing called heterodyning Amplitude modulation requires non-linear mixing to produce sum and difference frequencies. Producing sum and difference frequencies is also called heterodyning and the difference frequency is called a beat" frequency. Heterodyning is used in transmitters, receivers and frequency synthesizers to generate new frequencies. Stable oscillator circuits are used to prevent frequency drift. The 1mH inductor at point D will be removed for this part of the experiment. This changes the LC band pass filter to an RC low pass filter. The function generator frequency will then be increased until it approaches the frequency of the Colpitts oscillator. Q16. Determine the cut-off frequency of the filter. Q17. What signal frequency would you see at point D with the function generator at 1 khz when the inductor is removed? Q18. What would happen to the signal at point D when the function generator frequency is increased beyond the cut-off frequency. Q19. What would you expect to happen if the function generator frequency approaches the frequency of the Colpitts oscillator? Remember that the diode is still producing sum and difference frequencies as well as harmonics. Footnotes: 1. The non-linear mixing causes distortion that produces harmonics of the signal and also sum and difference frequencies. To show this we can use a mathematical power series expansion for the semiconductor diode characteristic. The ideal characteristic is exponential given by, i d = I 0 (e kv -1) So from the expansion, i d = I 0 [kv + (kv) 2 /2+(kV) 3 /6+(kV) 4 /24+ ] The second equation is an infinite series with increasingly smaller terms. The first term is linear. If the diode voltage is the sum of two input voltages, the second term (ignoring constants) becomes, (v 1 + v 2) = v 1 + v 2 + 2v 1v 2. By using the product of cosines formula from trigonometry you can verify that the term, 2v 1v 2, produces sum and difference frequencies. The squared terms lead to second harmonic frequencies. 2. The higher order terms beyond the squared term in the power series for the diode current help to explain the additional spectral peaks you may observe. These terms lead to combinations of sum and differences of fundamental and harmonic frequencies. Since the higher order terms are small, the unexpected peaks should also be small. Background noise, like the noise you measured in Experiment 1, will also be a source of some of what you observe. Aliasing of the harmonic frequencies may also cause 5
6 spurious peaks. 3. The circuit Q for a practical parallel RLC circuit is determined by dividing the equivalent parallel resistance by the reactance at resonance (the inverse of the formula for a series RLC circuit). The equivalent parallel resistance is the parallel combination of all external parallel resistances including the reflected series coil resistance (which is Q 2 coil x R coil ). It is hard to know the exact value of external parallel resistances in this circuit. However, you may assume it is about 9kΩ (6.8k +2k). The circuit bandwidth is then calculated by dividing the resonant frequency by the circuit Q. Note that the circuit Q is not Q 2 coil x R coil which is only accurate when there are no external parallel resistances. 6
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