Physics 3330 Experiment #5 Fall 2006 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active LC filter, add positive feedback to make it oscillate, and then remove the negative feedback to make a Schmitt trigger. Introduction By far the most common problem with op-amp circuits, and amplifiers in general, is unwanted spontaneous oscillations caused by positive feedback. Just as negative feedback reduces the gain of an amplifier, positive feedback can increase the gain, even to the point where the amplifier may produce an output with no input. Unwanted positive feedback is usually due to stray capacitive or inductive couplings, couplings through power supply lines, or poor feedback loop design. An understanding of the causes of spontaneous oscillation is essential for debugging circuits. On the other hand, positive feedback has its uses. Essentially all signal sources contain oscillators that use positive feedback. Examples include the quartz crystal oscillators used in computers, wrist watches, and electronic keyboards, traditional LC oscillator circuits like the Colpitts oscillator and the Wien bridge, and lasers. Positive feedback is also useful in trigger and logic circuits that must determine when a signal has crossed a threshold, even in the presence of noise. In this experiment we will try to understand quantitatively how positive feedback can cause oscillations in an active LC filter, and how much feedback is necessary before spontaneous oscillations occur. We will also construct a Schmitt trigger to see how positive feedback can be used to detect thresholds. Readings The readings in H&H and Bugg are optional for this experiment. 1. Horowitz and Hill, Section 5.12 to 5.19. If you are designing a circuit and want to include an oscillator, look here for advice. Amplifier stability is discussed in Sections 4.33-4.34. 2. Bugg discusses the theory of spontaneous oscillations in Chapter 19. You may want to read Section 19.4 on the Nyquist diagram after you read the theory section below. Experiment #5 5.1 Fall 2006Theory LC ACTIVE BANDPASS FILTER The circuit for the active LC filter is shown in Figures 5.1 and 5.2. Recall from the theory section of Experiment #4 that the gain of an inverting amplifier is G = –RF/R when the open loop gain is large. The basic idea of this filter is to replace RF with a resonant circuit whose impedance becomes very large at the resonant frequency. Then there will be a sharp peak in the gain at the resonant frequency. If we replace RF with the impedance ZF shown in Fig. 5.2 and do a fair bit of algebra, we can show that Figure 5.1 Active Bandpass Filter+–VinVoutZF RA CLFigure 5.2 Parallel Resonant Circuitr ZF G(ω) =−ZFR=−Z0Riωω0+1Q−ω2ω02+iωω0Q+1, where we have defined the resonant frequency ω0, the characteristic impedance Z0, and the Q: ω0=1LC, Z0=LC, Q=Z0r. With quite a bit more work you can show that the peak in the magnitude of G occurs at the frequency ωpeak=ω01+2Q2−1Q2, which is very close to ω0 when Q is large. The gain at the peak is G(ωpeak) =−QZ0R. This last formula is only approximate, but corrections to the magnitude are smaller by a factor of 1/Q2 and thus not usually important. There is quick way to get this final formula if you already know that a parallel tank circuit has an impedance of ZF(ω0) = Q2·r at resonance, and that ωpeak is very close to ω0. Then we get G(ω0) = − ZF(ω0)/R = − Q2·r/R, which is the same result. Experiment #5 5.2 Fall 2006GAIN EQUATION WITH POSITIVE FEEDBACK To turn our band-pass filter into an oscillator, we add positive feedback as shown in Fig. 5.3. This circuit now has both positive and negative feedback paths, so we need a more general gain equation than we have derived before. The two divider ratios are defined as before Figure 5.3 Positive Feedback LC Oscillator+–VinVoutZFRAR 1 R 2 10 k 10-turnRR2R1+ R2B+= , B−=R + ZF. The voltage Vout at the output is related to the voltages V+ and V– at the op-amp inputs by Vout= AV+− V−(). These voltages are themselves determined by the divider networks V−= Vin+ B−Vout− Vin(),V+= B+Vout. Combining all three relations yields the gain equation G =VoutVin=−A 1− B−()1− AB+− B−(). In the absence of positive feedback (B+ = 0, pot wiper at bottom) this reduces to the gain equation for the inverting amplifier discussed in Experiment #4. OSCILLATION THRESHOLD If we increase B+ by moving the pot wiper up, the denominator in the gain equation will decrease and the gain will increase, resulting in a sharper and sharper peak in the filter response near ω0. The system will oscillate when the gain becomes infinite, since infinite gain implies that there is an output even with zero input. The gain will be infinite when the denominator in the gain equation is zero. If the value of the loop gain A is very large at the resonant frequency then, to a good approximation, the denominator will be zero when B+ = B−, or when FZRRRRR+=+212 Experiment #5 5.3 Fall 2006If the oscillation occurs near ω0 we can replace ZF with ZF(ω0) = Q2·r = Z02/r. With this substitution we find that the oscillation threshold occurs at .20212ZrRrRRRRBthresholdthreshold+=+=+ This derivation is correct, but it leaves several questions unanswered. What does it really mean to have infinite gain? What does the system do if we increase B+ past the threshold for oscillations? How do we analyze a system for stability when we don’t know what frequency it will oscillate at? See the Appendix to this experiment for a more complete treatment of stability that addresses these questions. Experiment #5 5.4 Fall 2006Problems 1. Design an active bandpass filter with a resonant frequency of 16 kHz, a Q of 10, and a closed loop gain of one at the peak of the resonance. Choose suitable component values for the parallel LC circuit shown in Figures 5.1 and 5.2, using the inductor that you made in Experiment #3. Use the value of the inductance that you measured earlier. The series resistor shown in Figure 5.2 will have two contributions, one from the losses in your inductor, and one from an actual resistor that you must choose to get the correct Q. If you do not know what the loss of your inductor is, assume it is zero for now. 2. To make an LC oscillator you will add positive feedback