EECS-100 Astable, Monostable, and Bistable op amp circuits 1EE-100 Lab: Astable, Monostable, and Bistable op amp circuits - Theory 1. Objective In this laboratory measurement you will learn about oscillation mechanism and nonlinear wave shaping. You will measure simple oscillatory and bi-stable flip-flop circuits. 2. Introduction: oscillators Oscillation is a very natural phenomenon and you can see many different examples including physics, biology, chemistry, and electronics as well. Oscillators are used in many electronic devices (computers, radios, quartz watches, wireless devices, etc). Their common purposes are to generate a periodic signal. Every oscillator has at least one active device acting as an amplifier. All rely on the same basic principle: employing an amplifier whose output is fed back to the input in phase. Thus, the signal regenerates itself. This is known as a positive feedback in contrast to previous laboratory experiments where we used always negative feedback in op-amp circuits. In this experiment we will focus on a special type of oscillators called relaxation oscillator. A relaxation oscillator is a circuit that repeatedly alternates between two states with a period that depends on the charging of a capacitor. The capacitor voltage may change exponentially when charged or discharged through a resistor from a constant voltage, or linearly through a constant current source. Next we will examine a simple circuit that can be easily transformed to an oscillator or a bi-stable flip-flop. 3. Negative resistance converter Consider the following circuit shown in Figure 1: Figure 1 (a) A negative resistance converter and (b) its driving-point transfer characteristic. This circuit realizes a negative resistance converter incorporating both a negative feedback path (via Rf) and a positive feedback path (via RA). We will derive its drivingEECS-100 Astable, Monostable, and Bistable op amp circuits 2characteristic by inspecting both the linear and the saturation regions as well. First assuming that the circuit works in the linear region, we note that the RA and RB form a voltage divider so that outoutBABinvvRRRv ⋅=⋅+=β Eq 1 We assumed ideal op-amp model so the voltage across RB is equal to vin. Applying KVL we obtain outfinviRv +⋅= Eq 2 Substituting Eq1 into Eq2 and solving for i, we obtain infBAvRRRi ⋅ −=1 Eq 3 Eq 3 is drawn as the middle segment in Figure 1-b). The boundary of this segment can be obtained by substituting Eq 1 into the validating inequality satoutEv < so that satinsatEvEββ<<− Eq 4 By inspection of the positive saturation region (vout = Esat), we find that fsatinfsatfinREvRiEiRv −⋅=+⋅=1getweso, Eq 5 Eq 5 defines the lower segment in Figure 1-b). By inspection of the negative saturation region and following the same procedure as above, we obtain fsatinfREvRi +⋅=1 Eq 6 This equation defines the upper segment in Figure 1-b). This circuit is called a negative-resistance converter because it converts positive resistance RA, RB, and Rf into a negative resistance equal to Ω⋅−ABfRRR in the linear region. Next we show how this circuit can be easily transformed into an oscillator. 4. Relaxation oscillator Let us connect a capacitor across the input terminals of the negative resistance converter. Such a circuit is shown in Figure 2. Its driving point characteristic was derived earlier in Figure 1. Let us consider the four different initial points Q1, Q2, Q3, and Q4 (corresponding to four different initial capacitor voltages at t = 0) on this characteristic. Since Ctitvtvcin/)()()( −==and C > 0, we have 0)( >tvin for all t such that i(t) < 0, and 0)( <tvin for all t such that i(t) > 0.EECS-100 Astable, Monostable, and Bistable op amp circuits 3 Figure 2 (a) An astable RC op-amp circuit and (b) its driving-point characteristic Hence the dynamic route from any initial point must move toward the left in the upper half plane, and toward the right in the lower half plane, as indicated by the arrow heads in Figure 2 (b). Observe that there is no stable equilibrium point in the circuit because the zero i current belongs to an unstable equilibrium (opposite arrowheads diverging from zero). This equilibrium point cannot be observed in practice, because any small amount of noise will drive out the circuit from this point. Also, the breakpoints QA, and QB cannot be equilibrium points because 0≠i . Since, arrowheads towards QA, and QB are oppositely directed it is impossible to continue drawing the dynamic route beyond QA, or QB. In other words, an impasse is reached whenever the solution reaches QA, or QB.1 The dotted arrows show that a sudden instantaneous transition will occur (also called jump). For an RC circuit this transition should be always a vertical jump (assuming in the v-i plane that i is the vertical axis) because the voltage across a capacitance cannot be changed suddenly such that )()(−+= TvTvcc. Applying jumps at the two impasse points QA, and QB we obtain a closed dynamic route. This means that the solution waveforms become periodic after a short transient time (starting from any initial capacitance voltage) and the op-amp circuit functions as an oscillator. Note that the oscillation is not sinusoidal. Such oscillators are usually called relaxation oscillators. To figure out what kind of waveforms will be generated note that the closed dynamic route operates always in the saturation regions (except for the short transient time at the very beginning). These are the segments Q1-QA, and Q4-QB. It means that the output voltage (vout) will alternate between the two saturation levels, +Esat and –Esat. The output will be a square wave with a duty cycle of 50 % if the saturation levels are symmetrical. 1 Any circuit which exhibits an impasse point is the result of poor modeling. This impasse point can be resolved, for this circuit, by inserting a very small linear inductor (representing the inductance of the connecting wires) in series with the
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