University of California College of Engineering Department of Electrical Engineering and Computer Sciences EECS128 Feedback Control Systems Lab Fall 2001 Hsu Lab 4 Phase Shift Oscillator A phase shift oscillator is shown in the following figure In order to obtain sustained oscillation the phase margin and gain margin of the circuit must be set to zero This zero margin condition is equivalent to the condition that the complex poles of the circuit are on the imaginary axes In this lab we will use the frequency domain stability analysis method to determine the value of the feedback gain K The circuit consists of a 3 stage RC filter a voltage follower and an inverting amplifier The inverting amplifier s gain is R 1k where the resistance R is to be determined The dynamics of the circuit is determined by the 3 stage RC filter Each RC stage can provide up to 90 degrees of phase shift With 3 such stages connected in cascade the total phase shift can reach 270 degrees at high frequency In order to obtain zero phase margin the feedback gain must be set so that the loop gain is unity at the cross over frequency In practice it is impossible to keep the phase margin exactly at zero as required for an oscillator producing constant amplitude output In other words the poles of the circuit cannot be positioned exactly on the imaginary axes If the phase margin becomes negative i e poles shift slightly to the right of the imaginary axes the output signal s amplitude will increase exponentially If the phase margin is positive i e poles shift to the left the output amplitude will decrease exponentially A solution to this problem is to introduce a slight nonlinearity into the circuit and this is the purpose of the Zener diodes The impedance of a set of back to back Zener diodes connected in series is infinite and therefore the gain of the op amp circuit is R 1k Suppose R is selected so that the poles are slightly on the right side of the imaginary axes With this RHP poles the output voltage will gradually increase When the output voltage reaches the Zener voltage the Zener circuit starts to conduct and the op amp feedback impedance drops This lower feedback impedance renders lower loop gain and hence shifts the poles back to the left of the axes Consequently the output amplitude is kept under a certain value Rs R B A 10k 0 1uF 10k 0 1uF 10k 1k 0 1uF 2001 P Hsu Lab 4 1 Prelab 1 Optional If you need to refresh your memory about Nyquist plots and the Nyquist stability criterion covered in EECS120 read section 6 3 pages 390 402 2 Why do we need to add a non linearity to the oscillator circuit 3 Read section 6 4 of the textbook pages 403 411 Give the definitions of the terms gain margin phase margin and crossover frequency Also give a brief explanation about the importance of these values and about how to determine the values from the Bode plot Lab procedure 1 Derive the loop transfer function from A to B In other words find the transfer function from point A to B assuming that there is no direct connection between B and A as in the actual circuit Assume R 1k i e unity gain amplifier and ignore the Zener diodes and the resistor Rs in your derivation 2 Generate a Bode pole for the loop transfer function obtained in step 1 and determine the cross over frequency and the value of R that renders zero phase margin 3 Construct the 3 stage RC filter and experimentally determine its frequency response In particular find the frequency and the loop transfer function magnitude at which the phase shift is 180 degrees Do these values agree with the values obtained in step 2 4 Construct the circuit without the Zener diodes Use a variable resistor for R and preset its value to the value found in step 2 5 Observe the output with a scope If the circuit does not oscillate increase the value of R by 10 Record the output waveform 6 Install the Zener diode and the value of Rs to ten times the value of R Record the waveform 7 What is the purpose of the voltage follower amplifier How will the circuit operation change if this voltage follower is replaced by a straight through wire 2001 P Hsu Lab 4 2