https://www.ti.com.cn/cn/lit/an/sloa020a/sloa020a.pdf?ts=1640863691642&ref_url=https%253A%252F%252Fwww.google.com%252Fhttps://www.ti.com.cn/cn/lit/an/sloa020a/sloa020a.pdf?ts=1640863691642&ref_url=https%253A%252F%252Fwww.google.com%252F

2 Development of the Circuit Equations 

 

 

 

 

 

 

 

A Several things must be mentioned at this point in the analysis. First, the transfer functions for the noninverting and inverting equations, 11 and 16, are different. For a common set of ZG and ZF values, the magnitude and polarity of the gains are different. Second, the loop gain of both circuits, as given by equations 13 and 17, is identical. Thus, the stability performance of both circuits is identical although their transfer equations are different. This makes the important point that stability is not dependent on the circuit inputs. Third, the A gain block shown in Figure 1 is different for each op-amp circuit. By comparison of equations 5, 11, and 16 we see that ANON–INV = a and AINV = aZF ÷ (ZG + ZF). Equation 7 shows that the error is inversely proportional to the loop gain; thus, the accuracy of identical closed-loop gain inverting and noninverting op-amp circuits is different. Equation 17 is used to compensate all op amp circuits. First, we determine what compensation method to use. Second, we derive the compensation equations. Third, we analyze the closed-loop transfer function to determine how it is modified by the compensation. The effect of the compensation on the closed-loop transfer function often determines which compensation technique will be used.

3 Internal Compensation

 Op amps are internally compensated to save external components and to enable their use by less knowledgeable people. It takes some measure of analog knowledge to compensate an analog circuit. Internally compensated op amps normally are stable when they are used in accordance with the applications instructions. Internally compensated op amps are not unconditionally stable. They are multiple pole systems, but they are internally compensated such that they appear as a single pole system over much of the frequency range. The cost of internal compensation is that it severely decreases the closed-loop bandwidth of the op amp.

Internal compensation is accomplished in several ways, but the most common method is to connect a capacitor across the collector-base junction of a voltage gain transistor (see Figure 6). The Miller effect multiplies the capacitor value by an amount approximately equal to the stage gain, thus, the Miller effect uses small value capacitors for compensation. Figure 7 shows the gain/phase diagram for an older op amp (TL03X). When the gain crosses the 0-dB axis (gain equal to one) the phase shift is about 100°, thus, the op amp must be modeled as a second order system because the phase shift is more than 90°.

 This yields a phase margin of φ = 180° – 100° = 80°, thus the circuit should be very stable (Reference 1 explains feedback analysis tools). Referring to Figure 8, the damping ratio is one and the expected overshoot is zero. Figure 7 shows approximately 10% overshoot which is unexpected, but inspecting Figure 7 further reveals that the loading capacitance for the two plots is different. The pulse response is loaded with 100 pF rather than 25 pF shown for the gain/phase plot, and this extra loading capacitance accounts for the loss of phase margin.

 Why does the loading capacitance make the op amp unstable? Look closely at the gain/phase response between 1 MHz and 9 MHz, and observe that the gain curve changes slope drastically while the rate of phase change approaches 120°/decade. The radical gain/phase slope change proves that several poles are located in this area. The loading capacitance works with the op-amp output impedance to form another pole, and the new pole reacts with the internal op-amp poles. As the loading capacitor value is increased, its pole migrates down in frequency, causing more phase shift at the 0-dB crossover frequency. The proof of this is given in the TL03X data sheet where plots of ringing and oscillation versus loading capacitance are shown.

 Figure 9 shows similar plots for the TL07X which is the newer family of op amps. Notice that the phase shift is 100 when the gain crosses the 0-dB axis. This yields a phase margin of 80, which is close to being unconditionally stable. The slope of the phase curve changes to 180/decade about one decade from the 0-dB crossover point. The radical slope change causes suspicion about the 90° phase margin. Furthermore, the gain curve must be changing radically when the phase is changing radically. The gain/phase plot may not be totally false, but it sure is overly optimist

 

The TL07X pulse response plot shows approximately 20% overshoot. There is no loading capacitance indicated on the plot to account for a seemingly unconditionally stable op amp exhibiting this large an overshoot. Something is wrong here: the analysis is wrong, the plots are wrong, or the parameters are wrong. Figure 10 shows the plots for the TL08X family of op amps which are sisters to the TL07X family. The gain/phase curve and pulse response is virtually identical, but the pulse response lists a 100-pF loading capacitor. This little exercise illustrates three valuable points: first, if the data seems wrong it probably is wrong, second, even the factory people make mistakes, and third, the loading capacitor makes op amps ring, overshoot, or oscillate.  

 

 The frequency and time-response plots for the TLV277X family of op amps is shown in Figures 11 and 12. First, notice that the information is more sophisticated because the phase response is given in degrees of phase margin; second, both gain/phase plots are done with substantial loading capacitors (600 pF), so they have some practical value; and third, the phase margin is a function of power supply voltage.

 

 At VCC = 5 V, the phase margin at the 0-dB crossover point is 60°, while it is 30° at VCC = 2.7 V. This translates into an expected overshoot of 18% at VCC = 5 V, and 28% at VCC = 2.7 V. Unfortunately, the time response plots are done with 100-pF loading capacitance, hence we can not check our figures very well. The VCC = 2.7-V overshoot is approximately 2%, and it is almost impossible to figure out what the overshoot would have been with a 600-pF loading capacitor. The small-signal pulse response is done with mV-signals, and that is a more realistic measurement than using the full-signal swing.

 Internally compensated op amps are very desirable because they are easy to use, and they do not require external compensation components. Their drawback is that the bandwidth is limited by the internal compensation scheme. The op-amp open-loop gain eventually (when it shows up in the loop gain) determines the error in an op-amp circuit. In a noninverting buffer configuration, the TL277X is limited to 1% error at 50 kHz (VCC = 2.7 V) because the op amp gain is 40 dB at that point. Circuit designers can play tricks such as bypassing the op amp with a capacitor to emphasize the high-frequency gain, but the error is still 1%. Keep equation 7 in mind because it defines the error. If the TLV277X were not internally compensated, it could be externally compensated for a lower error at 50 kHz because the gain would be much higher.

4 External Compensation, Stability, and Performance

This section is approached on a compensation type basis. Nobody compensates an op amp because it is there; they have a reason to compensate the op amp, and that reason is usually stability. They want the op amp to perform a function in a circuit where it is potentially unstable. Internally and noninternally compensated op amps are compensated externally because certain circuit configurations do cause oscillations. Several potentially unstable circuit configurations are analyzed in this section, and the reader can extend the external compensation techniques as required.

Other reasons for externally compensating op amps are noise reduction, flat amplitude response, and obtaining the highest bandwidth possible from an op amp. An op amp generates noise, and noise is generated by the system. The noise contains many frequency components, and when a high-pass filter is incorporated in the signal path, it reduces high-frequency noise. Compensation can be employed to roll off the op amp’s high-frequency, closed-loop response, thus, causing the op amp to act as a noise filter. Internally compensated op amps are modeled with a second order equation, and this means that the output voltage can overshoot in response to a step input. When this overshoot (or peaking) is undesirable, external compensation can increase the phase margin to 90° where there is no peaking. An uncompensated op amp has the highest bandwidth possible. External compensation is required to stabilize uncompensated op amps, but the compensation can be tailored to the specific circuit, thus yielding the highest possible bandwidth consistent with the pulse response requirements.

5 Dominant-Pole Compensation

We saw that capacitive loading caused potential instabilities, thus, an op amp loaded with an output capacitor is a circuit configuration that must be analyzed. This circuit is called dominant pole compensation because if the pole formed by the op amp output impedance and the loading capacitor is located close to the zero frequency axis, it becomes dominant. The op-amp circuit is shown in Figure 13, and the open-loop circuit used to calculate the loop gain (Aβ) is shown in Figure 14.

Thevenin equivalent circuit 

 

 

 

Several conclusions can be drawn from equation 24 depending on the location of the poles. If the Bode plot of equation 23, the op amp transfer function, looks like that shown in Figure 15, it only has 25° phase margin, and there is approximately 48% overshoot. When the pole introduced by ZO and CL moves towards the zero frequency axis it comes close to the τ2 pole, and it adds phase shift to the system. Increased phase shift increases peaking and decreases stability. In the real world, many loads, especially cables, are capacitive, and an op amp like the one pictured in Figure 15 would ring while driving a capacitive load. The load capacitance causes peaking and instability in internally compensated op amps when the op amps do not have enough phase margin to allow for the phase shift introd

Prior to compensation, the Bode plot of an uncompensated op amp looks like that shown in Figure 16. Notice that the break points are located close together thus accumulating about 180° of phase shift before the 0 dB crossover point; the op amp is not usable and probably unstable. Dominant pole compensation is often used to stabilize these op amps. If a dominant pole, in this case ωD, is properly placed it rolls off the gain so that τ1 introduces 45 phase at the 0 dB crossover point. After the dominant pole is introduced the op amp is stable with 45° phase margin, but the op-amp gain is drastically reduced for frequencies higher than ωD. This procedure works well for internally compensated op amps, but is seldom used for externally compensated op amps because inexpensive discrete capacitors are readily available

 6 Gain Compensation

When the closed-loop gain of an op-amp circuit is related to the loop gain, as it is in voltage feedback op amps, the gain can be used to stabilize the circuit. This type of compensation can not be used in current feedback op amps because the mathematical relationship between the loop gain and ideal closed-loop gain does not exist. The loop gain equation is repeated as equation 27. Notice that the closed-loop gain parameters ZG and ZF are contained in equation 27, hence the stability can be controlled by manipulating the closed-loop gain parameters.

The original loop-gain curve for a closed-loop gain of one is shown in Figure 17, and it is or comes very close to being unstable. If the closed-loop noninverting gain is changed to 9, then K changes from K/2 to K/10. The loop-gain intercept on the Bode plot (see Figure 17) moves down 14 dB, and the circuit is stabilized.

 

 Gain compensation works for inverting or noninverting op-amp circuits because the loop gain equation contains the closed-loop gain parameters in both cases. When the closed-loop gain is increased, the accuracy and the bandwidth decrease. As long as the application can stand the higher gain, gain compensation is the best type of compensation to use. Uncompensated versions of normally internally compensated op amps are offered for sale as stable op amps with minimum gain restrictions. As long as the gain in the circuit you design exceeds the gain specified, this is economical and a safe mode of operatio

 7 Lead Compensati

 The compensation capacitor introduces a pole and zero into the loop equation. The zero always occurs before the pole because RF >RF||RG. When the zero is properly placed it cancels out the τ2 pole along with its associated phase shift. The original transfer function is shown in Figure 19, drawn as solid lines. When the RFC zero is placed at ω = 1/τ2, it cancels out the τ2 pole causing the Bode plot to continue on a slope of –20 dB/decade. When the frequency gets to ω = 1/(RF||RG)C, this pole changes the slope to –40 dB/decade. Properly placed, the capacitor aids stability, but what does it do to the closed-loop transfer function? The equation for the inverting op amp closed-loop gain is repeated below.

 

The op-amp gain (a) the forward gain (A) and the ideal closed-loop gain are plotted in Figure 20. The op-amp gain is plotted for reference only. The forward gain for the inverting op amp is not the op-amp gain. Notice that the forward gain is reduced by the factor RF/(RG +RF), and it contains a high-frequency pole. The ideal closed-loop gain follows the ideal curve until the 1/RFC breakpoint (same location as 1/τ2 breakpoint), and then it slopes down at –20 dB/decade. Lead compensation sacrifices the bandwidth between the 1/RFC breakpoint and the forward gain curve. The location of the 1/RFC pole determines the bandwidth sacrifice, and it can be much greater than shown here. The pole caused by RF, RG, and C does not appear until the op amp’s gain has crossed the 0 dB axis, thus, it does not effect the ideal closed-loop transfer function. 

The plot of the noninverting op amp with lead compensation is shown in Figure 21. There is only one plot for both the op-amp gain (a) and the forward gain (A), because they are identical in the noninverting circuit configuration. The ideal starts out as a flat line, but it slopes down because its closed-loop gain contains a pole and a zero. The pole always occurs closer to the low-frequency axis because RF > RF||RG. The zero flattens the ideal closed-loop gain curve, but it never does any good because it can not fall on the pole. The pole causes a loss in the closed-loop bandwidth by the amount separating the closed-loop and forward-gain curves. 

Although the forward gain is different in the inverting and noninverting circuits, the closed-loop transfer functions take very similar shapes. This becomes truer as the closed-loop gain increases because the noninverting forward gain approaches the op-amp gain. This relationship can not be relied on in every situation, and each circuit must be checked to determine the closed-loop effects of the compensation scheme. 

8 Compensated Attenuator Applied to Op Amp

Stray capacitance on op-amp inputs is a problem that circuit designers are always trying to get away from because it decreases closed-loop frequency response or causes peaking. The circuit shown in Figure 22 has some stray capacitance (CG) connected from the inverting input to ground. Equation 34 is the loop-gain equation for the circuit with input capacitance.

Op amps having high input and feedback resistors are subject to instability caused by stray capacitance on the inverting input. Referring to equation 34, when the 1/(RF||RGCG) pole moves close to τ2 the stage is set for instability. Reasonable component values for a CMOS op amp are RF = 1 MΩ, RG = 1 MΩ, and CG = 10 pF. The resulting pole occurs at 318 kHz, and this frequency is lower than the breakpoint of τ2 for many op amps. There is 90 phase shift resulting from τ1, the 1/(RF||RGC) pole adds 45° phase shift at 318 kHz, and τ2 adds another 45° phase shift at about 600 kHz. This circuit is unstable because of the stray input capacitance. The circuit is compensated by adding a feedback capacitor as shown in Figure 23.

The compensated attenuator Bode plot is shown in Figure 24. Adding the correct 1/RFCF breakpoint cancels out the 1/RGCG breakpoint, the loop gain is independent of the capacitors. Now is the time to take advantage of the stray capacitance. CF can be formed by running a wide copper strip from the output of the op amp over the ground plane under RF; do not connect the other end of this copper strip. The circuit is tuned by removing some copper (a razor works well) until all peaking is eliminated. Then measure the copper, and have an identical trace put on the print-circuit board.

 The inverting and noninverting closed-loop gain equations are a function of frequency. Equation 37 is the closed-loop gain equation for the inverting op amp. When RFCF = RGCG equation 37 reduces to equation 38 which is independent of the breakpoint. This also happens to the noninverting op-amp circuit. This is one of the few occasions when the compensation does not affect the closed-loop gain frequency response.

9 Lead-Lag Compensation 

 

 Referring to Figure 26, a pole is introduced at ω = 1/RC, and this pole reduces the gain 3 dB at the breakpoint. When the zero occurs prior to the first op-amp pole it cancels out the phase shift caused by the ω = 1/RC pole. The phase shift is completely canceled before the second op-amp pole occurs, and the circuit reacts as if the pole was never introduced. Nevertheless, Aβ is reduced by 3 dB or more, so the loop gain crosses the 0-dB axis at a lower frequency. The beauty of lead-lag compensation is that the closed-loop ideal gain is not affected as is shown below. The Thevenin equivalent of the input circuit is calculated in equation 40, the circuit gain in terms of Thevenin equivalents is calculated in equation 41, and the ideal closed-loop gain is calculated in equation 42.

 

 RG Equation 42 is intuitively obvious because the RC network is placed across a virtual ground. As long as the loop gain (Aβ) is large, the feedback will null out the closed-loop effect of RC, and the circuit will function as if it weren’t there. The closed-loop log plot of the lead-lag compensated op amp is given in Figure 27. Notice that the pole and zero resulting from the compensation occur and are gone before the first amplifier poles come on the scene. This prevents interaction, but it is not required for stability.

10 Comparison of Compensation Schemes 

Internally compensated op amps can, and often do, oscillate under some circuit conditions. Internally compensated op amps need an external pole to get the oscillation or ringing started, and circuit stray capacitances often supply the phase shift required for instability. Loads, such as cables, often cause internally compensated op amps to ring severely.

Dominant pole compensation is often used in IC design because it is easy to implement. It rolls off the closed-loop gain early; thus, it is seldom used as an external form of compensation unless filtering is required. Load capacitance, depending on its pole location, usually causes the op amp to ring. Large load capacitance can stabilize the op amp because it acts as dominant pole compensation.

The simplest form of compensation is gain compensation. High, closed-loop gains are reflected in lower-loop gains, and in turn, lower-loop gains increase stability. If an op-amp circuit can be stabilized by increasing the closed-loop gain, do it.

Stray capacitance across the feedback resistor tends to stabilize the op amp because it is a form of lead compensation. This compensation scheme is useful for limiting the circuit bandwidth, but it decreases the closed-loop gain.

Stray capacitance on the inverting input works with the parallel combination of the feedback and gain setting resistors to form a pole in the Bode plot, and this pole decreases the circuit’s stability. This effect is normally observed in highimpedance circuits built with CMOS op amps. Adding a feedback capacitor forms a compensated attenuator scheme which cancels out the input pole. The cancellation occurs when the input and feedback RC time constants are equal. Under the conditions of equal time constants, the op amp functions as though the stray input capacitance was not there. An excellent method of implementing a compensated attenuator is to build a stray feedback capacitor using the ground plane and a trace off the output node.

Lead-lag compensation stabilizes the op amp, and it yields the best closed-loop frequency performance. Contrary to some published opinions, no compensation scheme will increase the bandwidth beyond that of the op amp. Lead-lag compensation just gives the best bandwidth for the compensation.

11 Conclusion

The stability criteria often is not oscillation, rather, it is circuit performance as exhibited by peaking and ringing.

The circuit bandwidth can often be increased by connecting an external capacitor in parallel with the op amp. Some op amps have hooks which enable a parallel capacitor to be connected in parallel with a portion of the input stages. This increases bandwidth because it shunts high frequencies past the low bandwidth gm stages, but this method of compensation depends on the op amp type and manufacturer.

The compensation techniques given here are adequate for the majority of applications. When the new and challenging application presents itself, use the procedure outline here to invent your own compensation technique.

Other reference

 TL03x, TL03xA ENHANCED-JFET LOW-POWER LOW-OFFSET OPERATIONAL AMPLIFIERS

https://www.ti.com/lit/ds/symlink/tl032a.pdf?ts=1640900989080&ref_url=https%253A%252F%252Fwww.google.com%252F

control systems engineering - Books

 http://freedownloads88.weebly.com/uploads/1/3/6/2/13624622/control_systems.pdf

CONTROL SYSTEM ENGINEERING-I

https://www.vssut.ac.in/lecture_notes/lecture1423904331.pdf

1.0 Introduction to Control system 1.1 Scope of Control System Engineer 1.2 Classification of Control System 1.3 Historical development of Control system 1.4 Analogues systems 1.5 Transfer function of Systems 1.6 Block diagram representation 1.7 Signal Flow Graph(SFG)

2.0 Feedback Characteristics of Control systems and sensitivity measures 2.1 The Concept of Feedback and Closed loop control 2.2 Merits of using Feedback control system 2.3 Regenerative Feedback

3.0Control System Components 3.1 Potentiometers 3.2 DC and AC Servomotors 3.3 Tachometers 3.4 Amplidyne 3.5 Hydralulic systems 3.6 Pneumatic systems 3.7 Stepper Motors

4.0 Time Domain Performance Analysis of Linear Control Systems 4.1 Standard Test Signals 4.2 Time response of 1st order Systems 4.3 Unit step response of a prototype 2nd order system 4.4 Unit Ramp response of a second order system 4.4 Performance Specification of Linear System in Time domain 4.5 The Steady State Errors and Error Constants 4.6 Effect of P, PI, PD and PID Controller 4.7 Effect of Adding a zero to a system 4.8 Performance Indices(ISE,ITSE,IAE, ITAE) 4.9 Approximations of Higher order Systems by Lower order Problems

5.0 The Stability of Linear Control Systems 5.1 The Concept of Stability 5.2 The Routh Hurwitz Stability Criterion 5.3 Relative stability analysis

6.0 Root Locus Technique 6.1 Angle and Magnitude Criterion 6.2 Properties of Root Loci 6.3 Step by Step Procedure to Draw Root Locus Diagram 6.4 Closed Loop Transfer Function and Time Domain response 6.5 Determination of Damping ratio, Gain Margin and Phase Margin from Root Locus 6.6 Root Locus for System with transportation Lag. 6.7 Sensitivity of Roots of the Characteristic Equation.

7.0 Frequency Domain Analysis. 7.1 Correlation between Time and frequency response 7.2 Frequency Domain Specifications 7.3 Polar Plots and inverse Polar plots 7.4 Bode Diagrams 7.4.1 Principal factors of Transfer function 7.4.2 Procedure for manual plotting of Bode Diagram 7.4.3 Relative stability Analysis 7.4.4 Minimum Phase, Non-minimum phase and All pass systems 7.5 Log Magnitude vs Phase plots. 7.6 Nyquist Criterion 7.6.1 Mapping Contour and Principle of Argument 7.6.2 Nyquist path and Nyquist Plot 7.6.3 Nyquist stability criterion 7.6.4 Relative Stability: Gain Margin, and Phase Margin 7.7 Closed Loop Frequency Response 7.7.1 Gain Phase Plot 7.7.1.1 Constant Gain(M)-circles 7.7.1.2 Constant Phase (N) Circles 7.7.1.3 Nichols Chart 7.8 Sensitivity Analysis in Frequency Domain

Application Note AN-1162 Compensator Design Procedure for Buck Converter with Voltage-Mode Error-Amplifier

https://www.infineon.com/dgdl/an-1162.pdf?fileId=5546d462533600a40153559a8e17111a

 

 

 

 

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来源： https://blog.csdn.net/liugaoxingliushi/article/details/122253728