OSCILLATORS. Introduction

Transcription

1 OSILLATOS Introduction Oscillators are essential components in nearly all branches o electrical engineering. Usually, it is desirable that they be tunable over a speciied requency range, one example being the signal generator in the lab, which allows you to adjust the put requency over several decades, as well as select the wave shape (e.g., sinusoidal, square, sawtooth, etc.). Another application in which oscillators ind utility is in receivers; a voltage-controlled oscillator is programmed by a voltage source to a requency very near the incoming F signal, then multiplied by this F signal resulting in a signal much lower in requency (and hence, easier to process) than the original F signal. In this lab, the basics o global eedback and instability criteria or oscillators will be reviewed, ollowed by the design o some very common oscillators.

2 Theory Oscillators at the System Level To understand the undamentals o oscillators, it is necessary to review the system-level global eedback diagram o Fig.. + v sum v in A v Σ - v Fig. Global eedback diagram. ecall that the transer unction o this system is A + A. Now, imagine or a moment that the eedback network is disconnected rom the summer; under this condition, and assuming v in is a sinusoid, the put o the open-loop block is a sinusoid whose amplitude is A times that o v in, while the eedback network put is a sinusoid o amplitude A times that o v in. I the loop gain A is designed or a magnitude o and a phase o 80 degrees at the requency o v in, then the eedback network put is an inverted copy o v in. econnecting the eedback network to the inverting node o the summer results in a net 360 degree phase shit, and thus the put o the eedback network acts exactly like the external source v in, save or a delay in propagating the signal around the loop beore reconnecting the eedback network. It ollows that i v in is removed, the closed-loop put remains unchanged, as the loop gain is just able to sustain the oscillation instigated by v in. Key problem #: Time-invariance The main point is that eedback and a loop gain o are essential in designing an oscillator. The tricky part is in achieving and maintaining a loop gain o. Even i one is able to get a loop gain o precisely (equivalent to placing a pair o poles EXATLY on the jω-axis), one has to deal with the real-world act that systems are not time-invariant, the result being that system characteristics like pole locations drit ever so slightly with time. In systems where the poles are real, the eect is seldom important or noticeable, but with an oscillator, this time-invariance is crucial: i the poles drit to the right, the positive real part causes the put oscillation to grow unbounded, whereas i the poles shit let, the associated positive damping actor causes the put oscillation to decay with time. onclusion: oscillators need dynamic eedback to sense time-varying luctuations in the pole locations and pull the poles back toward the jω-axis as they drit.

3 Key problem #: Linearity Another problem deals with the external input, v in : an oscillator is supposed to work with an input (think how silly it would be i the signal generator in the lab required a sinusoidal input to get started what would generate that sine wave?). The good news is that reality provides the necessary input in the orm o noise - noise or our purposes contains all spectral components, so the oscillator resonant requency is present). The bad news is that the oscillator described so ar is linear, thus the put amplitude is proportional to the magnitude o the glitch that triggers the oscillation. A well-designed oscillator has a ixed put amplitude, independent o stimulus; such behavior is non-linear. onclusion: oscillators require non-linearity and dynamic eedback to work properly. Though this may sound like two extra design burdens, in reality, there is only one burden, as the appropriate non-linearity can implement the requisite dynamic eedback! The idea is that the non-linearity be transparent to the system while the amplitude is the desired value, but whenever the amplitude deviates rom this value (either because the stimulus is o such a large value that linearity is a problem, or because time-invariance has caused the poles to drit, changing the loop gain and hence the put amplitude) the non-linearity kicks in, limiting the amplitude. Non-linear Square-Wave Oscillator The irst oscillator considered in this lab exploits the positive eedback seen a ew labs back with the Schmitt trigger. The only dierences between the oscillator and the Schmitt trigger are that one o the resistors is replaced by a capacitor (intuition suggests the necessity o a dynamic element, since the put changes with time with no driving input!), and the reerence voltage o the Schmitt trigger is now derived via negative eedback. The oscillator is shown below: v Fig. Square-wave oscillator variant o Schmitt trigger. The nice thing ab this oscillator is that it is easy to understand both qualitatively and quantitatively. Qualitatively, the positive eedback causes the put to limit at one o the power supply rails depending on the sign o the dierential input to the op-amp. onsider the case in which the put resides at cc ; in this case, the non-inverting input to the op-amp is at at cc * /( + ), and the capacitor voltage starts charging up toward cc (think o a series circuit in which the input is v, and the put is the capacitor voltage). Once the capacitor voltage passes cc * /( + ), the sign o the dierential input to the op-

4 amp changes, thus the positive eedback causes the put to limit at cc * /( + ), and the capacitor starts charging toward - cc. When the capacitor voltage reaches cc * /( + ), the process repeats. The qualitative analysis explains the oscillation mechanism, but does not reveal the oscillation requency. Focusing on the negative eedback loop, one may recall that the solution o any irst-order ordinary dierential equation is given by: v = ( I F ) e + F t τ () where I denotes initial value, F inal value, and τ the time constant o the system. For the series- eedback loop, the initial value is determined by the trip voltages, namely +/- cc * /( + ). With the inal value, one must exercise some care. While the capacitor will only charge as ar as the negative o its initial value, it would charge up to the put voltage (i.e., -/+ cc ) i the Schmitt trigger behavior didn t kick in. Thus, or the purposes o this problem, F = -/+ cc. To determine the oscillation requency, we need to consider the time it takes or the capacitor voltage to change rom its initial value to -/+ cc * /( + ). Thus, we wind up solving the ollowing equation or t : t mcc = ± cc ± cc e m + + cc () The result is requency o t ln + = + ln., which corresponds to hal the entire cycle, hence a undamental Wein-Bridge Oscillator The Wein-Bridge oscillator produces a sinusoidal put. In its simplest orm it is a purely linear oscillator, and hence it is diicult to sustain oscillation as described earlier. However, a simple non-linear modiication leads to more sustainable oscillations. The basic Wein-Bridge oscillator ollows: a b sum v

5 Fig. 3 lassical Wein-Bridge oscillator. In Fig. 3, an op-amp has been used as the ampliier around which negative and positive eedback have been connected. While it is not essential (nor necessarily optimal!) that one use an op-amp as the ampliier, it is very nice or instructional purposes, as one can make a one-to-one correspondence between each node in the op-amp oscillator and each node in the classical global eedback diagram o Fig.. Straightorward hand analysis reveals that sum = ( ω ) + jω + ( + ) = + a b jω [ ] (3) For oscillation, the loop gain requirement mandates = =. Since / sum is purely real, it ollows that at the oscillation requency, / sum is real, requiring the denominator be purely imaginary. This happens at ω osc. =, resulting in sum sum + + = (4) It is apparent then that the inal design criterion is: a b = + (5) This allows or some latitude in determing component values, which is very handy given the uncertainty in capacitor values in the lab. For simplicity, it is common to choose =, = and a = b. In the lab, the two capacitors are not likely going to be equal given component tolerances, so the design equation tells us that one may use a potentiometer or one o the resistors, say a or b, to make the open-loop and eedback gains match. Let s say that you have determined the right component values or the desired oscillation requency, and you have also adjusted b to match the open-loop and eedback gains precisely. You view the oscillator put on the oscilloscope and/or spectrum analyzer, and lo and behold, time-invariance kicks in, and the oscillator amplitude starts luctuating, tending to die or limit at the supply voltage (i.e., look like a square wave). What to do? Furthermore, how to make the put amplitude not only stable, but a particular value? The manner in which one may trim a and/or b to match open-loop and eedback gains, in addition to the necessity o non-linear dynamic eedback, provides the necessary insight: have the circuit automatically update the eective value o a or b. I the put amplitude starts to exceed (all below) the desired amplitude, the loop gain must be reduced (increased), necessitating b increase (decrease), or a decrease (increase). One means o eecting this dynamic resistance is shown in Fig. 4:

6 a a b sum v Fig. 4 Wein-Bridge oscillator with non-linear, dynamic eedback. In the prelab, you will analyze the operation o this circuit to determine how it maintains a desired amplitude, as well as develop a ormula or computing the necessary ratio o a / a to achieve a particular put amplitude.

7 Prelab Exercises ) Design a square-oscillator with oscillation requency o 0 khz and an amplitude o cc /3 (that s right, you need to igure a way to limit the amplitude with altering the oscillation requency!). Additionally, constrain your design by choosing a standard value or the capacitor. It is much easier in the lab to synthesize non-standard resistor values via series or parallel combinations than it is with capacitors! ) Assuming that the square-wave oscillator puts a perect square wave, what is the Fourier series o the put waveorm? What is the total harmonic distortion (THD) o a square wave (symbolic expression and numerical result)? What is the ratio (in db) o the undamental amplitude to the next harmonic? One motivated student decides to extract a sinusoidal oscillation rom the squarewave put by iltering the put. Our hero successully designs an op-amp integrator with a D gain o 0-dB and a 3-dB bandwidth equal to the undamental requency o the oscillator put. With neglecting the ilter s eect on the undamental amplitude, what is the new THD (numerical result)? onsider placing the ilter cuto requency between the undamental and the next harmonic - is there an optimum location to minimize distortion? (You don t need to do any calculus here, just write a simple computer program that sweeps the ilter cuto and computes the resulting THD.) What is the lowest-order Butterworth ilter that results in a THD o < %? Hint: assume that the correct Butterworth has such high attenuation that only one harmonic contributes signiicant distortion. 3) onsider the Wein-Bridge oscillator o Fig. 4. Assuming the diodes turn on at ab 700m, at which they become perect shorts, what is the range o voltage over which a conducts current? I the voltage is such that one o the diodes is on, what happens to / sum (again, assume the diode is a perect short)? ome up with an equation or / sum, and in one sentence explain how it compares to the case when the diodes are both o. 4) I you answered the previous question correctly, you should understand how the circuit maintains the put at a particular amplitude. Now we shall consider how to select a / a to achieve such an amplitude. In terms o and sum, derive expressions or the diode voltage or the case o > 0, sum > 0 and the case o < 0, sum < 0. Solve each equation or the ratio, a / a. Assuming that the diodes begin to conduct at approximately 700 m, and that the desired rms value o is., what does your ormula predict or a / a? 5) Design a Wein-Bridge oscillator to oscillate at 0 khz with a peak-to-peak voltage o 6. (You may assume that the power supplies or the op-amp are >> 3).

8 Lab Exercises ) Build the square-wave oscillator you designed in the prelab. eriy that the amplitude is what you predict (i it diers, check your values or and ). Measure the oscillation requency as well as the precise values or each resistor. By what actor is the oscillation requency o? Based on these measurements, iner the actual capacitor value. With altering the put amplitude, what change must you make to get the correct put requency? ompute the necessary component value, and veriy with measurement. ) Build the Wein-Beidge oscillator you designed in the prelab. Try to match and as closely as possible. Test your circuit with the diodes to veriy the oscillation requency, and i need be, adjust the value o b (this is easier than adjusting a, since the ratio o a and a needs to be preserved) with a pot to compensate or mismatches in the capacitor ratio. Append the diodes and veriy the put amplitude.