1 Prelab 2 Frequency Modulation EE133 Prof Dutton Winter 2004 EE133 Prelab 2 Using Oscillators to Generate FM Signals 1 Introduction Now that we have a firm understanding of how amplitude modulation works we will explore the real core of our project frequency modulation In this lab we will see how oscillators can be made to produce FM signals We will see that we can either produce modulation at an intermediate frequency that can then be multiplied up to a higher carrier or we can modulate directly at the carrier frequency Last week we used two signal generators to supply the multiplier with carrier and modulating signals In a portable system we must find a way to produce these signals without the help of bulky equipment This is where oscillators come in These will be the backbone of your FM transmitter There are a variety of flavors of oscillators and about as many names Many of the different oscillator types are named for their inventors who found new and clever ways to vary the feedback path that makes an otherwise perfectly good amplifier into an oscillator In this lab we will use a chip based VCO or VoltageControlled Oscillator to generate a modulated FM signal We will also build a Colpitts Oscillator and use it as a VCO as well Part of this oscillator will include an LC tank As you will see this discrete based design can be used in the direct creation of FM signals at a carrier frequency In addition to these variable oscillators we ll investigate the properties of a crystal stabilized oscillator We will be building this using the extra transistor located on the SA612 This will serve as the local oscillator 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 implement one discrete based design The rest are chip based for which we will only need to add a few datasheet defined components Transmitter Receiver Audio Amp XO BNC to ANT BNC to ANT XO VCO LM566 Mixer SA602 Power Amp LNA Mixer SA602 IF Amp BNC to Speaker PLL LM565 Tank Colpitts Osc Figure 1 Roadmap for Lab 2 You are extremely advised to solder up the majority of components before your assigned lab period 2 Mathematics of Frequency Modulation In the first part of this lab you will be investigating some of the mathematical properties of FM and trying to understand the significance of the various sidebands the Bessel functions and the bandwidth of an FM signal Prelab 2 Frequency Modulation EE133 Prof Dutton Winter 2004 2 1 2 Bessel Functions Consider a frequency modulated signal of the following form F t Vc cos c t m sin m t 1 This signal can be expressed via trigonometric identities and series representations of Bessel functions as an infinite series of the following form F V c J0 m cos c t J1 m cos c 1 m t cos c 1 m t J2 m cos c 2 m t cos c 2 m t J3 m cos c 3 m t cos c 3 m t J4 m cos c 4 m t cos c 4 m t 2 The first term is the carrier and each of the subsequent terms represents a side frequency pair This representation can aid in predicting the magnitude of modulated carrier and sideband signals It turns out that for certain values of m the carrier or side frequency pair can be deleted from the signal spectrum You will see more of what this means in the Matlab exercises 2 2 Carson s Rule Although the representation given above is comprised of an infinite number of signals in reality the higher order terms die off quickly It is because of this that FM signals can be bandlimited without serious distortion An approximation to the bandwidth required by an angle modulated signal is known as Carson s Rule B 2fm m 1 2 f fm 3 where B is the bandwidth fm is the modulating frequency m is the modulation index and f is the frequency deviation of the sideband from the carrier 2 3 Matlab Plots Now we will look at some FM signals in Matlab You will need to use two scripts which are both posted on the web page under Spice Decks Copy the two scripts into the same directory because one needs to call on the other You will specify your parameters in fm script m but you should leave get fm spect m untouched you can peruse the file if you want to see the Matlab commands for creating an FM spectrum The parameters of the signal are as follows fs the frequency of the modulating signal fc the carrier frequency Vc the carrier voltage m the modulation index To generate a signal just start Matlab open the script and fill in the correct values for your parameters Run the script and the correct frequency content should 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 have to tweak yours 3 Prelab 2 Frequency Modulation EE133 Prof Dutton Winter 2004 1 Plotting FM frequency spectra Obtain plots of the frequency spectrum of an FM signal with these parameters fs 1kHz fc 30kHz m 0 Vc 1V Observe the outputs you obtain for the following 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 Bessel Functions found in the notes on FM Modulation You can also try comparing your results to the plots of the Bessel functions in the lecture handouts Remember that Bessel functions are supposed to predict the amplitude of the carrier and sidebands for a certain modulation index Is there a reason why we gave you these particular values of m to play around with 3 Carson s rule For each of the signals calculate the bandwidth of the transmission using Carson s rule in Equation 3 above Comparing the values obtained with the spectra observed is Carson s rule a good indicator of bandwidth Is there variation of the accuracy of the rule with the value of m 3 The LC Tank We will be building a Colpitts oscillator which requires a narrow band resonator as we will see below Our resonator will consist of an LC parallel filter or an LC tank In this section you will be designing an LC tank to resonate at 24 3MHz Figure 2 shows the ideal and actual implementations for this circuit In the actual circuit notice that L is replaced by a series combination of L1 and L2 as well as the parasitic series resistance Rs We would like our oscillator to drive a 50 load but to do this we need to tap the inductor so as not to load the output too much You can think of this as an impedance transformation of the load resistance to some higher equivalent resistance that appears at …