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Analog Electronics Lecture 6 Op amp Stability Analysis and Opamp Circuits Electronic Devices, 9th edition Thomas L. Floyd Muhammad Amir Yousaf © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Lecture: Stability analysis and compensation of op-amps Op-amp Circuits Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Stability analysis and compensation of op-amps Op-amp’s gain is so high that even a slightest input signal would saturate the output. Negative feedback is used to control the gain. A classic form of feedback equation Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Loop Gain Feedback equation: The term AolB( or AB for simplicity) is called ‘Loop Gain’ When loop gain is higher i.e. AB >> 1 o The system gain is determined by only feedback factor B. o Feedback factor is implemented by stable passive components o Thus in ideal conditions the closed loop gain is predictable and stable because B is predictable and stable. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Stability analysis and compensation of op-amps System output heads to infinity as fast as it can when 1+ AB approaches to zero. Or |AB| =1 and ∠AB = 180o If the output were not energy limited the system would explode the world. System is called unstable under these conditions Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Feedback equation for Op-amp feedback systems Non-inverting amplifier + Vout – Vin Rf Vf Feedback circuit Ri Non-inverting amplifier AB A Electronic Devices, 9th edition Thomas L. Floyd R i R R i f © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Feedback equation for Op-amp feedback systems Inverting amplifier Rf Ri – Vout Vin + AB A R i R R i f Replace ZF with Rf and ZG with Ri Loop Gain for both inverting and non inverting amplifier circuits are identical, hence the stability analysis is identical. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Bode plot analysis of Feedback circuits Bode plots of loop gain are key to understanding Stability: Stability is determined by the loop gain, when AB = -1 = |1| ∠180o instability or oscillation occurs Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Loop gain plots are key to understanding Stability: Notice that a one pole can only accumulate 90° phase shift, so when a transfer function passes through 0 dB with a one pole, it cannot oscillate. A two-slope system can accumulate 180° phase shift, therefore a transfer function with a two or greater poles is capable of oscillation. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Op-amp transfer function The transfer function of even the simplest operational amplifiers will have at least two poles. At some critical frequency, the phase of the amplifier's output = -180° compared to the phase of its input signal. The amplifier will oscillate if it has a gain of one or more at this critical frequency. This is because: (a) the feedback is implemented through the use of an inverting input that adds an additional -180° to the output phase making the total phase shift -360° (b) the gain is sufficient to induce oscillation. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Phase Margin, Gain Margin Phase Margin = ΦM Phase margin is a measure of the difference in the actual phase shift and the theoretical 180° at gain 1 or 0dB crossover point. Gain Margin = AM The gain margin is a measure of the difference of actual gain (dB) and 0dB at the 180° phase crossover point. For Stable operation of system: ΦM > 45o or AM > 2 (6dB) Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Phase Margin, Gain Margin The phase margin is very small, 20o So the system is nearly stable A designer probably doesn’t want a 20° phase margin because the system overshoots and rings badly. Increasing the loop gain to (K+C) shifts the magnitude plot up. If the pole locations are kept constant, the phase margin reduces to zero and the circuit will oscillate. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Compensation Techniques: Dominant Pole Compensation (Frequency Compensation) Gain Compensation Lead Compensation Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Dominant Pole Compensation (Frequency Compensation) Dominant Pole Compensation is implemented by modifying the gain and phase characteristics of the amplifier's open loop output or of its feedback network, or both, in such a way as to avoid the conditions leading to oscillation. This is usually done by the internal or external use of resistance-capacitance networks. A pole placed at an appropriate low frequency in the open-loop response reduces the gain of the amplifier to one (0 dB) for a frequency at or just below the location of the next highest frequency pole. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Dominant Pole Compensation (Frequency Compensation) The lowest frequency pole is called the dominant pole because it dominates the effect of all of the higher frequency poles. Dominant-pole compensation can be implemented for general purpose operational amplifiers by adding an integrating capacitance. The result is a phase margin of ≈ 45°, depending on the proximity of still higher poles. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Gain Compensation The closed-loop gain of an op-amp circuit is related to the loop gain. So the gain can be used to stabilize the circuit. AB A R i R R i f Gain compensation works for both inverting and non-inverting op-amp circuits because the loop gain equation contains the closed-loop gain parameters in both cases. As long as the application can stand the higher gain, gain compensation is the best type of compensation to use. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Gain Compensation Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Lead Compensation It consists of putting a zero in the loop transfer function to cancel out one of the poles. The best place to locate the zero is on top of the second pole, since this cancels the negative phase shift caused by the second pole. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Lead Compensation Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Lecture: Stability analysis and compensation of op-amps Op-amp Circuits Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Comparators A comparator is a specialized nonlinear op-amp circuit that compares two input voltages and produces an output state that indicates which one is greater. Comparators are designed to be fast and frequently have other capabilities to optimize the comparison function. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Comparator with Hysteresis Noise contaminated signal may cause an unstable output. To avoid this, hysteresis can be used. Hysteresis is incorporated by adding regenerative (positive) feedback, which creates two switching points: The upper trigger point (UTP) and the lower trigger point (LTP). After one trigger point is crossed, it becomes inactive and the other one becomes active. VUTP Vin 0 t VLTP +Vout (max) –Vout(max) Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Schmitt Trigger A comparator with hysteresis is also called a Schmitt trigger. The trigger points are found by applying the voltage-divider rule: VUTP R2 R1 R 2 V out ( m ax ) V LT P and R2 R1 R 2 What are the trigger points for the circuit if the maximum output is ±13 V? Vin V out ( m ax ) – Vout VUTP R2 R1 R 2 V out ( m ax ) 10 k 47 k + 10 k + 13 V + R1 47 k R2 10 k = 2.28 V By symmetry, the lower trigger point = 2.28 V. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Output Bounding Some applications require a limit to the output of the comparator (such as a digital circuit). The output can be limited by using one or two Zener diodes in the feedback circuit. The circuit shown here is bounded as a positive value equal to the zener breakdown voltage. Vin Ri 0V +VZ – 0 + Electronic Devices, 9th edition Thomas L. Floyd –0.7 V © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Comparator Applications Simultaneous or flash analog-to-digital converters use 2n-1 comparators to convert an analog input to a digital value for processing. Flash ADCs are a series of comparators, each with a slightly different reference voltage. The priority encoder produces an output equal to the highest value input. VREF R Vin (analog) Op-amp comparators + – R + – R R R – (7) (6) + (5) (4) – (3) (2) + (1) (0) – In IC flash converters, the priority encoder usually includes a latch that holds the converter data constant for a period of time after the conversion. Electronic Devices, 9th edition Thomas L. Floyd R + – R Priority encoder + D2 D1 D0 Binary output Enable input + – R © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Comparator Applications Over temperature sensing circuit: o R1 is temperature sensing resistor with a negative temperature coefficient. o R2 value is set equal to the resistance of R1 at critical temperature oAt normal conditions R1 > R2 driving op-amp to low At R1=R2, balance bridge creates high op-amp output, energizes relay, activates alarm. An over temperature sensing circuit Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Summing Amplifier A summing amplifier has two or more inputs; normally all inputs have unity gain. The output is proportional to the negative of the algebraic sum of the inputs. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Example Summing Amplifier What is VOUT if the input voltages are +5.0 V, 3.5 V and +4.2 V and all resistors = 10 k? Rf R1 VOUT = (VIN1 + VIN2 + VIN3) = (+5.0 V 3.5 V + 4.2 V) = 5.7 V Electronic Devices, 9th edition Thomas L. Floyd VIN1 R2 VIN2 10 k – R3 VIN3 VOUT + © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Averaging Amplifier An averaging amplifier is basically a summing amplifier with the gain set to Rf /R = 1/n (n is the number of inputs). The output is the negative average of the inputs. What is VOUT if the input voltages are +5.0 V, 3.5 V and +4.2 V? Assume R1 = R2 = R3 = 10 k and Rf = 3.3 k? Rf R1 VIN1 R2 VOUT = ⅓(VIN1 + VIN2 + VIN3) = ⅓(+5.0 V 3.5 V + 4.2 V) = 1.9 V Electronic Devices, 9th edition Thomas L. Floyd VIN2 3.3 k – R3 VIN3 VOUT + © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Scaling Adder A scaling adder has two or more inputs with each input having a different gain. It is useful when one input has higher weight than the other. The output represents the negative scaled sum of the inputs. Assume you need to sum the inputs from three microphones. The first two microphones require a gain of 2, but the third microphone requires a gain of 3. What are the values of the Rf input R’s if Rf = 10 k? R1 VIN1 R2 R1 R 2 R3 Rf Av 3 Electronic Devices, 9th edition Thomas L. Floyd Rf 10 k Av 1 10 k 3 2 VIN2 5.0 k 10 k – R3 VIN3 VOUT + 3.3 k © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Scaling Adder: D/A Converter An application of a scaling adder is the D/A converter circuit shown here. The resistors are inversely proportional to the binary column weights. Because of the precision required of resistors, the method is useful only for small DACs. +V 8R 20 Rf 4R 21 2R – VOUT 2 2 + R 23 Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. R/2R Ladder DAC A more widely used method for D/A conversion is the R/2R ladder. The gain for D3 is 1. Each successive input has a gain that is half of previous one. The output represents a weighted sum of all of the inputs (similar to the scaling adder). Inputs An advantage of the R/2R ladder is that only two values of resistors are required to implement the circuit. Electronic Devices, 9th edition Thomas L. Floyd D0 D1 D2 D3 R3 2R R6 R5 2R R7 2R R2 R1 2R R4 2R R R R8 Rf = 2 R – R Vout + © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. The Integrator An op-amp integrator simulates mathematical integration, a summing process that determines total area under curve. The ideal integrator is an inverting amplifier that has a capacitor in the feedback path. The output voltage is proportional to the negative integral (running sum) of the input voltage. Electronic Devices, 9th edition Thomas L. Floyd IC C R – Vin Ii Vout + Ideal Integrator © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. The Integrator Capacitor in the ideal integrator’s feedback is open to dc. Rf C This implies open loop gain with dc offset. That would lead to saturation. The practical integrator overcomes these issues– the simplest method is to add a relatively large feedback resistor. R Vin – Vout + Practical Integrator Rf should be large enough Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Example The Integrator If a constant level is the input, the current is constant. The capacitor charges from a constant current and produces a ramp. The slope of the V in output is given by the equation: V out t Ri C Sketch the output wave: Rf +2.0 V Vin V out t 220 k t (ms) 0V 0.0 2.0 V 0.5 V in RiC 1.0 1.5 2.0 2 V 10 k 0.1 μF C 2 V /m s Vin Ri 0.1mF – 10 k Vout +1.0 V + Vout Electronic Devices, 9th edition Thomas L. Floyd 0V 0.0 1.0 V t (ms) 0.5 1.0 1.5 2.0 © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. The Differentiator R C An op-amp differentiator simulates mathematical differentiation, a process to determine instantaneous rate of change of a function. Vin – Vout + Ideal Differentiator The ideal differentiator is an inverting amplifier that has a capacitor in the input path. The output voltage is proportional to the negative rate of change of the input voltage. Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. Example The Differentiator The output voltage is given by V V out C R f C t +1.0 V Sketch the output wave: Vin 0V 0.0 1.0 V t (ms) 0.5 1.0 1.5 2.0 V V out C R f C t Rf 1 V 10 k 0.1 μF 2 V 0.5 m s 10 k Vin Rin C – 220 0.1mF Vout + +2.0 V Vout Electronic Devices, 9th edition Thomas L. Floyd Rc 0V 0.0 2.0 V t (ms) 0.5 1.0 1.5 10 k 2.0 © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved. References Slides by ‘Pearson Education’ for Electronic Devices by Floyd ‘Op.amp for every one’ by Ron Mancini ’Stability Analysis for volatge feedback op-amps’, Application Notes byTexas Instruments (TI) ’Feedback amplifiers analysis tool’ by TI ‘Feedback, Op Amps and Compensation’ Application Note 9415 by Intersil Modified by Muhammad Amir Yousaf Electronic Devices, 9th edition Thomas L. Floyd © 2012 Pearson Education. Upper Saddle River, NJ, 07458. All rights reserved.