Stanford EE 133 - Using Oscillators to Generate FM Signals

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Prelab 2 - Frequency Modulation - EE133 - Prof. Dutton - Winter 2004 1EE133 - Prelab 2Using Oscillators to Generate FM Signals1 IntroductionNow that we have a firm understanding of how amplitude modulation works, we will explore the real coreof our project, frequency modulation. In this lab, we will see how oscillators can be made to produce FMsignals. We will see that we can either produce modulation at an intermediate frequency that can then bemultiplied up to a higher carrier, or we can modulate directly at the carrier frequency.Last week, we used two signal g e nerators to supply the multiplier with carrier and modulating signals.In a portable system, we must find a way to produce these signals witho ut the help of bulky equipment.This is where oscillato rs come in. These will be the backbone of your FM transmitter.There are a variety of flavors of oscilla tors, and about as many names. Many of the different oscillato rtypes are named for their inventors, who found new and clever ways to va ry the feedback path that makes anotherwise perfectly good amplifier into an oscillator. In this lab, we will use a chip-based VCO, or Voltage-Controlled O scillator, to generate a modulated FM signal. We w ill also build a Colpitts Oscillator and useit as a VCO as well. Part of this oscillator will include an LC tank . As you will see, this discrete-baseddesign can b e used in the direct creation of FM signals at a carrier frequency.In addition to these variable os cillators, we’ll investigate the properties o f a crystal stabilized oscillator .We will be building this using the extra transistor located on the SA612. This will serve as the local oscil-lator on both the transmitter and receiver boards.As you can see, we will be soldering a lot of pieces this week. Fortunately, you really only have to im-plement one discrete-based design. The rest are chip based, for which we will only need to add a few,datasheet defined, components.TransmitterReceiverBNC toANTMixer(SA602)XOAudioAmpVCO(LM566)PowerAmpTank/ColpittsOscPLL(LM565)Mixer(SA602)IFAmpLNABNC toANTBNC toSpeakerXOFigure 1: Roadmap for Lab 2You are extremely a dv ised to s older up the majority of c omponents before your assigned lab period.2 Mathematics of Frequency ModulationIn the first part of this lab, you will be investigating s ome of the mathematical properties of FM, and tryingto understand the significance of the various sidebands, the Bessel functions, a nd the bandwidth of an FMsignal.Prelab 2 - Frequency Modulation - EE133 - Prof. Dutton - Winter 2004 22.1 Bessel FunctionsConsider a frequency modulated signal of the following fo rm:Fθ(t) = Vccos(ωct + mθsinωmt) (1)This signal can be expressed via trigonometric identities and series representations of Be ssel functions as aninfinite series of the following form:Fθ= V c{J0(mθ)cosωct+J1(mθ)[cos(ωc+ 1ωm)t − cos(ωc− 1ωm)t]+J2(mθ)[cos(ωc+ 2ωm)t − cos(ωc− 2ωm)t]+J3(mθ)[cos(ωc+ 3ωm)t − cos(ωc− 3ωm)t]+J4(mθ)[cos(ωc+ 4ωm)t − cos(ωc− 4ωm)t]+...} (2)The first term is the carrier, and each of the subse quent terms repres e nts a side-frequency pair. Thisrepresentation can aid in pr e dicting the magnitude of modulated carrier and sideband signals. It turns outthat for certain values of mθ, the carrier or side-frequency pair can be ‘deleted’ from the signal s pectrum.You will see more of what this means in the Matlab exercises.2.2 Carson’s RuleAlthough the representation given above is comprised of an infinite number of signals, in reality the higherorder terms die off quickly. It is because of this that FM signals can be bandlimited without serious distortion.An approximation to the bandwidth require d by an angle-modulated signal is known as Carson’s Rule:B = 2fm(mθ+ 1) = 2(∆f + fm) (3)where B is the bandwidth, fmis the modulating frequency, mθis the modulation index, and ∆f is thefrequency deviation of the sideband from the carrier.2.3 Matlab PlotsNow we will look at some FM signals in Matlab. You will need to use two scripts, which are both postedon the web page under ”Spice Decks.” Copy the two scripts into the same directory, because one needs tocall on the other. You will specify your parameters in fm script.m, but you should leave get fm spe ct.muntouched (you can pe ruse the file if you want to se e the Matlab commands for creating an FM spectrum).The parameters of the signal are as follows:• fs, the fr e quency of the modulating signal• fc, the carrier frequency• Vc, the carrier voltage• m , the modulation indexTo generate a signal, just start Matlab, open the s c ript, and fill in the correct values for your pa rameters.Run the script, and the correct frequency content s hould appear. The script has been written to give you a”nice” display. There may be some variation on what’s ”nice” for your computer, however, so you may haveto tweak yours.Prelab 2 - Frequency Modulation - EE133 - Prof. Dutton - Winter 2004 31. Plotting FM frequency spectra: Obtain plots of the frequency spectrum of an FM signal withthese parameters: fs= 1kHz, fc= 30kHz, m = 0, Vc= 1 V . Observe the outputs you obtain for thefollowing values of m, keeping all other parameters constant: m = 1, 2.40, 3.83, 5.14.2. Bessel Functions: Do your spectra agree with the Table of Bess e l Functions found in the notes onFM Modulation? You can also try comparing your results to the plots of the Be ssel functions in thelecture handouts. Remember tha t Bessel functions are supposed to predict the amplitude of the carrierand sidebands for a certain modulation index. Is there a reason why we gave you these particular valuesof m to play around with?3. Carson’s rule: For each of the signals, calculate the bandwidth of the tr ansmission using Cars on’srule in Equation 3 above. Comparing the values obtained with the spectra observed, is Carson’s rulea good indicator of bandwidth? Is there variation of the accuracy of the rule with the value of m?3 The LC TankWe will be building a Colpitts oscillator, which requires a narrow band resona tor, as we will see below. Ourresonator will consist of an LC parallel filter, or an ‘LC tank’. In this section, you will be designing an LCtank to resonate at 24.3MHz. Figure 2 shows the ideal and actual implementations for this circuit. In theactual circuit, notice that L is replaced by a series combination of L1and L2, as well a s the parasitic seriesresistance, Rs.We would like our oscillator to drive


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