Unique compensation technique tames high-bandwidth voltage-feedback op amps
You may think that there
Michael Steffes, Burr-Brown Corp -- EDN, August 1, 1997
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Designers seeking high slew rates and low noise for dc-coupled pulse amplifiers often must turn to extremely high-gain-bandwidth, nonunity-gain-stable, voltage-feedback op amps. The lower internal compensation capacitance, which gives these op amps the nickname "decompensated," increases slew rate, and the higher input-stage trans-conductance, gm, which produces the ultrahigh gain bandwidth, decreases input-voltage noise. Unfortunately, many designers have been burned trying to apply these touchy decompensated devices to low gains. Much of the popularity for the current-feedback topology comes from its superior slew rate and stability at low gains compared with high-gain-bandwidth voltage-feedback designs. However, the high-frequency performance of a current-feedback op amp also comes with poor dc accuracy and higher output noise. Op-amp designers suggest various forms of external compensation to take advantage of the dc accuracy, low noise, and high slew rate of a decompensated voltage-feedback op amp at low signal gains. Unfortunately, previously suggested compensation schemes have many shortcomings. For example, some op amps provide access to the internal compensation node, but adding this dominant-pole compensation directly reduces the slew rate. Common lead-lag compensation techniques produce pole-zero pairs in the closed-loop response, yielding deplorable pulse response and settling characteristics. A new external compensation method provides complete control over a simple, second-order lowpass response at low signal gains. This technique allows you to achieve a well-controlled frequency response at any inverting gain for any internally decompensated op amp. The full slew rate of the decompensated op amp is available at the output, along with an output-noise voltage density that increases with frequency. This increased output noise stems from the necessary peaking in the noise gain to achieve a flat, closed-loop frequency re-sponse. Passive postfiltering can significantly reduce the effect of this noise. Using this external technique with a high-quality, decompensated voltage-feedback op amp provides significantly better absolute dc accuracy than high-speed current-feedback alternatives. Comparable noise and slew rate and considerably lower harmonic distortion than equivalent current-feedback options are also possible. With some extra effort, you can also use this compensation to emulate the gain-bandwidth independence of a current-feedback op amp. Gain-bandwidth independence using a voltage-feedback op amp can be useful in inverting-summing applications for which you might need to adjust the summing weights during the design process or as part of the application. Once you understand the topology and derive the basic transfer function, you can predict the amplifier's perform-ance based on the desired signal gain and the amplifier's characteristics. Three design examples show how the compensation technique works to maximize the achievable flat bandwidth, implement a filter, or produce a gain-bandwidth-independent design (and why you would want that). Analyze the compensation circuit The compensation technique simply consists of adding two compensation elements, CS and CF, to the standard inverting op-amp configuration (Figure 1). Previous discussions of this circuit focused on using CF to compensate for a parasitic CS. The following analysis shows you how to set both CS and CF to get a well-controlled, closed-loop, second-order lowpass frequency response at any signal gain for even the most unstable op amp. You can easily analyze this circuit using a single-pole, open-loop model for the op amp. Without CS and CF, a single-pole op-amp model would be inadequate because the higher order poles of a decompensated op amp wholly determine the closed-loop response at low gains. However, you'll see that the design methodology justifies this single-pole simplification with the compensation elements in place. Besides being the only way this compensation will work, the inverting configuration offers several other benefits. With no common-mode voltage at the input, the inverting configuration for most op amps achieves high er slew rates, higher full-power bandwidth, and lower distortion. The trade-offs to getting these inverting-mode benefits are an input impedance set by RG and a slightly higher dc noise gain for the noninverting input-voltage noise of the op amp. You can write the Laplace transfer function for the circuit of Figure 1 in Bode-analysis form as follows:  (1)where  (2) (3)and the single-pole op amp's open-loop gain is  (4)The significant components of this transfer function are
At dc, the denominator of Equation 1 is approximately 1, whereas the numerator is equal to RF/RG, which is the desired low-frequency signal gain. For stability analysis, it is common to look at the corresponding Bode plot (Figure 2). The magnitude portion of the Bode plot compares the magnitude of the noise gain with the magnitude of the open-loop gain, which are the top and bottom of the fraction in the denominator of Equation 1, respectively. At the frequency at which these two curves cross, which is loop-gain crossover, the loop gain drops to 1 and, in a simple op-amp application, the closed-loop bandwidth rolls off. Because a pole also exists in the numerator of Equation 1, this simple analysis is not sufficient to determine the closed-loop response. Normally, you would also need to consider the phase of the loop-gain terms. However, Equation 1 ultimately reduces to a simple, second-order lowpass transfer function, and you proceed with the design by controlling the v0 and Q of that transfer function. The magnitude portion of the Bode analysis provides insight into what is happening in the design, but you don't use the magnitude information to set CS and CF. You can disregard the phase plot for now with the assumption that loop-gain crossover will occur at a noise gain high enough for you to safely ignore the higher order poles of A(s). Substituting the two impedances, ZF and ZG, and the op amp's open-loop-gain expression A(s) into Equation 1 yields  (5)Rearranging this equation to produce a pole-zero expression for the noise-gain terms in the denominator yields  (6)The terms in the denominator make up the loop-gain portion of this transfer function. The op amp's open-loop gain has a high dc value of AOL and a dominant pole at vA. The noise gain has a dc gain of 1+RF/RG, a low-frequency zero, and a high-frequency pole to flatten the noise gain to 1+CS/CF at higher frequencies. The complete Bode plot (Figure 2) shows the gain-magnitude portion for this loop gain along with a number of key frequencies that are critical to the design. The key frequencies (in hertz) are GBP, Z0, and P1. GBP is simply the gain-bandwidth product of the selected op amp (GBP=AOLvA/2p Hz). Z0, which equals 1/(2pRF(CS+CF)), is the unity-gain (0-dB) intersection of the sloping portion of the noise-gain curve. The actual zero in the noise gain occurs at G1Z0=Z1. G1 and G2 are the low-frequency and high-frequency noise gains, respectively. P1, the feedback-network pole, is equal to 1/(2pRFCF). This pole and Z0 are the two things you can adjust to control the closed-loop frequency response. P1 is also equal to Z0G2, which is simply Z0 times the high-frequency noise gain set by the capacitor ratios. Another point of interest from Figure 2 is where the projection of the sloping portion of the noise-gain curve intersects the open-loop-gain curve at the geometric mean of Z0 and GBP. This point turns out to be the characteristic frequency, F0, of the closed-loop second-order response (see sidebar, "Second-order lowpass-response characteristics"). When you set P1 to less than this geometric mean, the noise gain crosses the open-loop response at a gain equal to G2. The noise gain crosses the open-loop response at FC, which would equal the closed-loop bandwidth for a unity-gain-stable op amp of the same GBP operating at a noninverting noise gain of G2. One of the key assumptions in this analysis is that you control G2 so that it's greater than the specified minimum stable gain for the op amp. Crossover at this high noise gain is the reason you can use a nonunity-gain-stable op amp at a low signal gain of RF/RG. There is, however, little consistency among op-amp manufacturers on the definition of minimum stable gain. Some manufacturers use a typical phase-margin target, others target a maximum peaking, and still others actually specify a gain that causes oscillation in the closed-loop response. Generally, most data sheets show a recommended minimum gain that does not cause oscillation. The goal in this design is for the noise gain to cross over the open-loop response at a noise gain, G2, high enough for you to safely ignore the higher order poles of A(s). If the minimum stable gain on the data sheet is really a minimum operating suggestion, it should be safe to target crossover at 1.5 times that gain. This guard band is, however, an estimate and varies from part to part and from manufacturer to manufacturer. Using the macromodels that most manufacturers provide allows you to fine-tune this target. You can extensively use the frequencies and gains in the Bode plot to gain insight into the algebraic solution for the closed-loop, second-order transfer function. Because the design seeks values for the compensation elements (CF and CS), the following methodology uses radian frequency units. Converting those units to the hertz shown in Figure 2 simply requires a division by 2p. Expanding the transfer function of Equation 6 into normal monic form (writing a polynomial from highest order to lowest order with a coefficient of 1 for the highest order term) yields  (7)where  (8)and  (9)Although seeing that this full transfer function ends up as a second-order lowpass response is encouraging, the individual terms still look a little intractable. With a bit of manipulation and judicious simplifications, you can develop simple expressions for v0 and Q that show a clear path to a design methodology. Specifically, you can simplify the terms inside the radical for v0 by recognizing that AOL is much greater than 1+RF/RG. Dropping the 1+RF/RG of that term, recognizing that AOLvA=GBP and that 1/(CF+CS)RF=Z0 (in Figure 2), and simplifying the expression for Q in the denominator yields the following equations, where G2=1+CS/CF and G1=1+RF/RG:  (10)and  (11)Referring back to the Bode plot of Figure 2, these simple equations indicate that the closed-loop, second-order response has a characteristic frequency, v0, that is the geometric mean of Z0 and the amplifier's GBP. Also, the ratio of that characteristic frequency to the sum of the high-frequency, loop-gain crossover fr equency (FC) and the zero frequency in the noise gain (Z1) sets the value of Q. If you've already selected the amplifier and the required signal gain (G1=|SIGNAL GAIN|+1), you need only set Z0 and P1 (or, equivalently, G2), to implement the compensation. Design for maximum bandwidth Virtually all the elements that determine the Q of the closed-loop response in Equation 11 are known. The system designer determines the amplifier's GBP and the desired low-frequency noise gain. Once you select a tar get Q, you need only set Z0 and G2. The key simplification to this analysis is to judiciously target a G2 that is greater than the specified minimum stable gain for the selected op amp so that you can continue to neglect the added phase shift that the high-frequency, open-loop poles introduce. To get as much bandwidth as possible, set the target G2 very close to the minimum stable gain. As previously suggested, the following design examples use a factor of 1.5 times the minimum stable gain. With G2 somewhat arbitrarily set, you can then use Equation 11 to solve for Z0. The following equation shows the solution as a quadratic equation that you must solve to set Z0:  (12)An exact solution for Z0 is  (13)However, when (G2/G1)>6Q2, a good approximation is  (14)After selecting G1 and G2 and determining Z0, you can implicitly determine P1: P1=G2Z0. Then, you can combine the equations for Z0 and G2 in Figure 2 to solve for CF and CS:  (15)and  (16)Note that with a target Q of 0.707, you can substitute Equation 13 into Equation 10 to show the maximum achievable F0, which is approximately equal to F3 dB given the following factors: a given op amp's GBP, the G1 that corresponds to the desired signal gain, and the high-frequency gain, G2, necessary for stability. The resulting equation shows the maximum achievable flat bandwidth using this compensation technique:  (17)Maximize the bandwidth of a real design One of the greatest attractions of this compensation technique is that it allows you to successfully apply a nonunity-gain-stable, voltage-feedback op amp at low signal gains and simultaneously retain the full slew rate and dc accuracy of the part. Table 1 summarizes the key specifications for a pair of good voltage-feedback op amps. The OPA627 from Burr-Brown Corp is unity-gain-stable; the OPA637 is the company's decompensated version and has a recommended minimum gain of 5. In this case, the input-voltage noise of the decompensated OPA637 is no lower than that of the OPA627, but the slew rate (and high-frequency open-loop gain) of the OPA637 is markedly higher than that of the OPA627. You can compare the OPA627 performance to the performance of the OPA637 in a complete design that uses the compensation technique. Table 2 summarizes a gain of 2 (G1=3) design target for the OPA637, the resulting key frequencies in the Bode analysis of Figure 2, and the component values necessary to set up this compensation. The selected feedback-resistor value is the result of a compromise between high input impedance (RG=RF/(G11)) and keeping the compensation capacitors greater than the parasitic values on those nodes. To implement this circuit for testing, you must also consider the components' parasitic capacitances and test-interface requirements. To include the effect of parasitics, the actual test-circuit design (Figure 3) reduces the value of CF by 0.2 pF and that of CS by the 15-pF parasitic at the input of the OPA637. The test circuit also includes 50V impedance-matching resistors at the input and output to match the assumed test-equipment source and load impedances of 50V. Adding the input-matching resistor slightly changes G1 from 3.0 to 2.95. This change has no effect on F0 and very little effect on Q because the G1Z0 portion of Equation 11 is small relative to GBP/G2. The test circuit also shows a bias-current-cancellation resistor from the noninverting input to ground. This resistor is equal to the parallel value of RF and RG to improve the output dc offset that results from bias currents. With this resistor match in place, the output dc error that results from input bias currents is simply the input-offset current times the feedback-resistor value. A large capacitor shunts this noninverting-input resistor to roll off the noise terms that might arise from the resistor's Johnson noise and bias-current noise. These two components on the noninverting input are unnecessary for the FET-input OPA637 because its bias, offset, and noise-current terms are infinitesimal relative to the voltage offset and noise terms. The test circuit in Figure 3 includes these components for general application. The test circuit's frequency response (Figure 4a) is remarkably flat for a gain-of-5-stable op amp operating at a noise gain of 3 (a signal gain of 2). The frequency response does show slight peaking, which indicates that the Q of the actual circuit is slightly greater than 0.707 instead of the target value of 0.64. This difference in Q slightly extends the bandwidth from a target of 7.7 to 9.8 MHz and introduces some overshoot and ringing into the pulse response (Figure 4b). Apparently, either the target G2 was too close to the minimum stable gain to exclude the effects of the higher order poles or the parasitic capacitances are different from the estimate. You can obtain a closer match between predicted and measured results at higher values of G2. You can compare this inverting compensation for the OPA637 to a maximum-bandwidth design using the unity-gain-stable OPA627 at a gain of 2. The inverting compensation with the OPA637 produces a slightly higher bandwidth of 9.8 MHz vs the 8 MHz of the OPA627. However, because of the slew-rate difference, the OPA627 is slew-limited for output steps greater than 2.4V, whereas the OPA637 can support nonslew-limited steps as high as 4.2V at the output. If this 4V were the input range of an ADC, the nonslew-limited pulse response that the OPA637 provides would settle to a final value more quickly than that of the OPA627. For example, this compensation of the OPA637 provides a low-gain ADC buffer with excellent settling time for large output steps. When driving a 4Vp-p input-range, 10-bit ADC, the circuit has an absolute dc accuracy (with no trims) and peak-to-peak output noise that doesn't exceed 1/4LSB. The worst-case output dc error is 0.75 mV, and the worst-case output peak-to-peak noise is 0.9 mV. The settling time to 1/2LSB is 33 nsec. The improvement in performance between unity-gain-stable and decompensated versions of the same op amp is even more significant if you use parts that have a wider difference in their minimum stable gains. Predict the output noise Any compensation technique that shapes the noise gain of a nonunity-gain-stable op amp produces higher output noise as the frequency increases. This compensation technique increases the gain for the noninverting input-voltage noise of the op amp, as the noise-gain portion of Figure 2's Bode plot shows. In most cases, the op amp's noninverting input-voltage noise dominates the total output noise for the circuit of Figure 1. Referring to the Bode plot, this input-voltage noise has a gain that starts at G1, has a zero at Z1 equal to G1Z0, and finally has second-order poles that are identical to those you set in the inverting-compensation design. The transfer function for either the noise or a signal applied to the noninverting input (V+) of Figure 1 is  (18)(Refer to Equations 8 and 9 for the terms that you can place in this equation.) One approach to describing the output noise is to compute an equivalent noise-power bandwidth (NPB) that, when you multiply it by a constant output-noise-power value, gives the same total integrated noise power as the actual frequency response. If you arbitrarily use the output noise due to the noninverting input-voltage noise amplified by G1, which is the low-frequency output noise due to the noninverting input-voltage noise, as the constant noise value, you can calculate an equivalent NPB as  (19)This noise is the square of Equation 18's gain magnitude integrated from a frequency of 0 to infinity, then divi ded by the low-frequency noise gain squared (G12). This integral simplifies considerably, and you can solve it in closed form when you target a Q of 0.707. The middle term in the denominator of Equation 19 drops out, which allows you to use integral-table solutions for forms including 1/(x4+c4). Using the terms defined in Figure 2 and assuming a Q of approximately 0.707 gives an equivalent NPB of  (20)The last term in this equation is generally much less than 1, and you can ignore it. This equation states that the NPB is approximately equal to the single-pole bandwidth that results if you simply operate the amplifier at G1 (a bandwidth equal to GBP/G1) times the ratio of that bandwidth to the characteristic frequency (F0=ˆZ0GBP) of the actual second-order closed-loop response. To use this calculated NPB, multiply the op amp's noninverting input-voltage noise by G1 to compute the low-frequency spot noise at the output. Then, multiply that result by the square root of Equation 20 to get the integrated noise (EO(RMS)). Performing these computations for the design example of Figure 3, which has a Q'0.707, gives  (21)This analysis shows the significant increase in output-voltage noise due to the increased noise gain to G2 by assuming a constant output noise and computing the required NPB to get the same integrated noise power as the actual output noise over frequency. Evaluating this integral for the NPB is based on the assumption that the op amp's frequency response of Equation 18 self-limits the output noise. Another way to look at this noise is to compute the equivalent input-voltage noise that integrates to the same power over a simple lowpass Butterworth bandwidth. This approach allows an easy comparison between this technique and other approaches for getting a desired frequency response. The NPB of a simple, sec ond-order Butterworth response equals 1.11F0=1.11F3 dB when Q=0.707 (see sidebar). You can set up an equality to define the equivalent input-spot noise (EM) that will integrate over an NPB set by 1.11F0 to the same total output-noise power as the actual response as follows:  (22)where EN is the input-voltage noise of the op amp. The solution for EM is  (23)This equation states that the increase in the equivalent input-referred spot-noise voltage is proportional to the square root of the ratio of GBP/G1 to Z1. Evaluating this equation for the design example in Figure 3 yields an equivalent input-noise voltage of 17.1 nV/ˆHz. Multiplying this result by the low-frequency noise gain, G1, and then by the square root of 1.11F0 (Table 2) gives the integrated noise. This calculation gives the same 158 mV of integrated noise as Equation 21 gives. Equation 23 is useful because it clearly shows the noise penalty you pay by using this compensation. Postfiltering can significantly reduce this effect. For instance, if you totally filter the output noise after F0, the equivalent input noise calculated using Equation 23 decreases by half (the 1.5 inside the radical changes to 0.378, which is one-fourth of 1.5). With this postfiltering at F0, using the compensation scheme of Figure 3 with the OPA637 at a noise gain of 3 (signal gain of 2) has almost twice the equivalent noninverting input-referred spot-noise voltage (8.55 nV/ˆHz) as an OPA627 implementation. However, this design exhibits more than twice the slew rate of the OPA627 (Table 1) and also has considerably higher loop gain, and therefore lower harmonic distortion, for frequencies below Z1. Implement a second-order lowpass filter Because the design target to this point has been the control of a second-order lowpass response--principally for pulse-response control--you can also use this topology and the corresponding performance equations to implement an arbitrarily selected second-order response. You can adapt Equations 10 and 11 for this purpose. First, assume that you've selected the amplifier, its corresponding GBP, and design targets for v0 and Q. Now, you can select a value for either G1 or G2 and then solve for the other. To control high-frequency noise, first solve for G1 in terms of G2 using Equations 10 and 11, and then compute the constraint on G2 to get a solution. For example, first solve Equation 10 for Z0. Then, substitute this Z0 into Equation 11 and solve for G1 as follows:  (24)and  (25)G1 has a solution only if G2>(GBP3Q/v0). Thus, for low-noise filter design, this topology is most appropriate for lower Qs and when v0 is not orders of magnitude less than the GBP. Once you recognize this fact, you can flip Equation 25 around to solve for G2 when you want to target a desired dc signal gain of 1G1:  (26)For good results, G2 should be at least greater than two times the minimum stable gain for the amplifier. You should recognize that this analysis applies to a unity-gain-stable op amp as well. Also, the denominator of Equation 26 should be greater than zero. Thus, GBP/Q must be greater than G1v0. Table 3 shows a design example and the resulting key frequencies and gains corresponding to the Bode plot of Figure 2. The Bode plot of this design shows that P1 occurs after the noise gain intersects the open-loop curve at an F0 of 5 MHz. Most simplified stability discussions strongly discourage a greater-than-40-dB/decade closure rate in the loop gain at crossover. However, this design uses this higher closure rate to achieve complex closed-loop poles. Although using this approach to design a second-order lowpass filter has a limited range of applications and appears to have a relatively high output noise, the approach offers a low-sensitivity design. The most variable portion of the design is the amplifier's GBP. A 10% increase in GBP increases v0 by 5% and decreases Q by 1.3%. This result indicates a root loci vs GBP that is principally a radial movement from the origin in the s-plane. The first example using this compensation technique squeezes as much bandwidth as possible out of a given nonunity-gain-stable op amp operated at a noise gain G1 less than the op amp's specified minimum. A filter-design application then shows that you can get a very stable and moderate Q design by intentionally putting P1 above the intersection between the open-loop response and the noise gain. A third way to apply the compensation technique also exists. The Bode plot of Figure 2, along with Equations 10 and 11 for v0 and Q, shows a way to achieve gain-bandwidth independence. Note that F0 is unaffected if you vary G1 while staying below G2 in Figure 2. Holding all other components constant and adjusting RG in Figure 1 simply changes the noise gain at the low-frequency end of the plot. However, the noise-gain curve eventually hits and moves along the same 20-dB/decade curve that intersects 0 dB at Z0 and the open-loop gain curve at F0. Changes in the low-frequency noise gain, G1, and, therefore, changes in the inverting signal gain do not affect the key Z0 and F0 frequencies in Figure 2. All that happens as G1 changes is that the Q of the second-order response changes. Considering Equation 11 along with the Bode plot for a graphical interpretation, increasing G1 increases Z1 and therefore slightly decreases the Q set by the ratio F0/(FC+Z1). If you set the initial design point for Z1 well below FC, you can make significant changes in G1 with only a slight change in Q and with no change in F0. Thus, you can make the second-order frequency response relatively constant vs signal gain and hold all other components constant. The root loci of the second-order response to changes in RG is a constant v0 circle with a very slight Q sensitivity to RG and therefore to G1. Making the initial Z1 low relative to FC is equivalent to making G2 much greater than G1. This setting implies that the design will target a relatively low F0 to circumvent GBP limits and will move even closer to emulating the benefits of current-feedback op amps. You can also interpret this setting in another way. Initially setting G2 high (with P10) sets the closed-loop bandwidth relatively low. Then, starting at G1=G2, the closed-loop bandwidth should be close to FC. As you decrease G1, the compensation inhibits the bandwidth extension that normally occurs in a voltage-feedback amplifier as the gain decreases. Instead, the compensation shapes the noise gain to move the closed-loop response closer to that of a simple resistive noise gain of G2. One way to approach this design is to initially target a Q of 0.707 (maximally flat Butterworth) and to recognize that F3 dB equals F0 in Figure 2 when Q=0.707. Then, with a target F3 dB that is significantly less than the maximum-bandwidth design, you can solve Equation 10 for the required Z0 as in Equation 24. Then, you can select a midrange value for G1 and set G2 using Equation 26. Also, you can then select an RF and use Equations 15 and 16 to set the capacitor values. Once you set these values, you can vary RG over a range of gains with no effect on v0 and only a slight effect on Q. Approaching current-feedback performance To illustrate a design with frequency-response independence for which you can change the signal gain without strongly impacting the frequency response (Figure 5a), set up the OPA637 for a flat second-order response with a nominal F0=F3 dB=5 MHz with Q=0.707 at a midrange gain (Table 4). As in the maximum-bandwidth design, the measured small-signal frequency response (Figure 5b) has a Q that is a little higher than expected. In both designs, the higher order poles of the open-loop response appear to cause a slight increase in the closed-loop Q. The scale on Figure 5b's plot is 1 dB/div to show fine detail. As the signal gain changes from 1 to 7--corresponding to a change in the dc noise gain, G1, from 2 to 8--the 3-dB bandwidth changed only from 6.1 to 4.8 MHz. In other words, a four-times increase in the gain of this voltage-feedback op amp produces only a 21% decrease in bandwidth. Decreasing the target bandwidth further by increasing G2 produces results with a better match to theory and an even more insensitive design. Moving this design target down in frequency to get closer to gain-bandwidth independence increases the output noise. Using a high-gain-bandwidth voltage-feedback op amp in this fashion approaches the gain-bandwidth independence and slew rate of current-feedback designs and even brings along with it the higher noise you normally see with current-feedback parts. The inverting-summing configuration (Figure 6) is another good application for this compensation technique when set up for gain-bandwidth independence. This configuration sums multiple signals to the output by combining the signals at the inverting node. Inverting susumming is one common application for current-feedback op amps because of their approximate gain-bandwidth independence. Now, you can achieve some of those same benefits using this external compensation and still get the exceptional dc accuracy of a part such as the OPA637. For this circuit, each channel sees a signal gain equal to RF/RG. Without compensation, the noise gain increases as you add each channel. When you use a voltage-feedback op amp, this increased noise gain always leads to a decreased bandwidth for all inputs. However, using the compensation technique, you can add, remove, and adjust the gain of individual channels with a relatively minor impact on their frequency response for any input to the output. Applying this compensation technique to a nonunity-gain-stable op amp can give good wideband performance. Acknowledgment The author would like to acknowledge Dr Aram Budak, retired professor of electrical engineering, Colorado State University (Fort Collins), for his enlightening emphasis on wringing the intuition out of the equations.
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summing is one common application for current-feedback op amps because of their approximate gain-bandwidth independence. Now, you can achieve some of those same benefits using this external compensation and still get the exceptional dc accuracy of a part such as the OPA637.
For this circuit, each channel sees a signal gain equal to RF/RG. Without compensation, the noise gain increases as you add each channel. When you use a voltage-feedback op amp, this increased noise gain always leads to a decreased bandwidth for all inputs. However, using the compensation technique, you can add, remove, and adjust the gain of individual channels with a relatively minor impact on their frequency response for any input to the output. Applying this compensation technique to a nonunity-gain-stable op amp can give good wideband performance.
Acknowledgment
The author would like to acknowledge Dr Aram Budak, retired professor of electrical engineering, Colorado State University (Fort Collins), for his enlightening emphasis on wringing the intuition out of the equations.
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Second-order lowpass-response characteristics |
A good understanding of the second-order lowpass transfer function is important to understanding this compensation technique. Equation A shows the gener al form of the Laplace transfer function of a second-order lowpass response: (A)The characteristic frequency is the radial distance in the s-plane from the origin to the poles when they are complex-conjugate pairs. The units in Equation A normally give this frequency in radians; you can convert values to hertz by dividing by 2p. The Q indicates how complex the poles are. The angle that the vector makes with the negative-real axis in the s-plane from the origin to the complex poles is given by cos1(1/2Q). Some key values for Q are
Another interesting point for Q is the geometric mean of the Bessel and Butterworth Qs. This point gives a Q=0.639, which is attractive because it gives the highest bandwidth (for a given v0) with less than 2% overshoot in the pulse response. You can describe the 3-dB bandwidth as a function of v0 and Q as follows:  (B)Evaluating the radical portion of Equation B at se veral of the possible values of Q suggested above gives the ratio of the 3-dB bandwidth to the characteristic frequency. For Q=0.577, F3 dB=0.79F0. For Q=0.639, F3 dB=0.90F0. For Q=0.707, F3 dB=F0. Some other useful second-order lowpass-response equations include  (C)which is the peak frequency when Q>0.707;  (D)which is the amount of peaking when Q>0.707;  (E)which is the percent overshoot in the pulse response when Q>0.50; and  (F)which is the noise-power bandwidth when Q>0.5. |
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Table 1 --Op-amp specifications |
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| Typical specifications | OPA627 | OPA637 |
| GBP (MHz) | 16 | 80 |
| Minimum stable gain | 1 | 5 |
| Slew rate (V/µsec) | 55 | 135 |
| Input-voltage noise (nV/squareroot(Hz)) | 4.5 | 4.5 |
| Input capacitance (Cdifferential+ Ccommon mode) (pF) | 15 | 15 |
| Input-offset voltage (mV) | 0.1 (low grade) |
0.28 (low grade) |
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Table 2 --OPA637 gain of 2 example |
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| Design features | Values | Notes |
| Target Q | 0.64 | See box |
| Target G2 | 7.5 | 1.5 times the minimum stable gain |
| Computed Z0 | 0.93 MHz | From Equation 13 |
| Resulting Z1 | 2.77 MHz | G1•Z0 |
| Resulting P1 | 6.93 MHz | G2•Z0 |
| Resulting F0 | 8.6 MHz | =Z0•GBP |
| Resulting F-3 dB | 7.74 MHz | If Q=0.64 |
| Selected RF | 2 kilohms | |
| Required RG | 1 kilohms | |
| Required CF | 11.5 pF | From Equation 15 |
| Required CS | 75 pF | From Equation 16 |
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Table 3 --Lowpass-filter design |
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| Design targets | Resulting design |
Component values (RF=2 kilohms) |
| lowercaseomega0=5 MHz | Z0=312 kHz | RG=1 kilohms |
| Q=2 | G2=51.2 | CF=5 pF |
| G1=3 | P1=16 MHz | CS=249 pF |
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Table 4 --Gain-bandwidth-independent design |
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Design targets |
Resulting design |
Predicted Q and F-3 d-vs gain | |||
| G 1 | -R F /R G | Q | F -3 dB (MHz) | ||
| F0=5 MHz | Z0=312 kHz | 2 | -1 | 0.776 | 5.59 |
| Nominal G1=4 | G2=13.74 | 3 | -2 | 0.74 | 5.31 |
| Nominal Q=0.707 | P1=4.3 MHz | 4 | -3 | 0.707 | 5 |
| 5 | -4 | 0.677 | 4.64 | ||
| 6 | -5 | 0.65 | 4.24 | ||
| 7 | -6 | 0.624 | 3.75 | ||
| 8 | -7 | 0.601 | 3.16 | ||
Second-order lowpass-response characteristics
A good understanding of the second-order lowpass transfer function is important to understanding this compensation technique. Equation A shows the gener al form of the Laplace transfer function of a second-order lowpass response:
(
A
)
The characteristic frequency is the radial distance in the s-plane from the origin to the poles when they are complex-conjugate pairs. The units in Equation A normally give this frequency in radians; you can convert values to hertz by dividing by 2p. The Q indicates how complex the poles are. The angle that the vector makes with the negative-real axis in the s-plane from the origin to the complex poles is given by cos1(1/2Q). Some key values for Q are
-
when Q&0.5, the poles are both real;
-
when Q=0.5, two repeated real poles occur at v0;
-
when Q=0.577, the frequency response is a second-order Bessel with the best phase linearity; and
-
when Q=0.707, the frequency response is a second-order Butterworth with maximum flatness.
At Q=0.707, the poles are at ±458 to the negative-real axis in the s-plane. The pulse response for a Butterworth lowpass response shows about 4.3% overshoot. For Q>0.707, the frequency response begins to peak, extending the 3-dB bandwidth but also introducing additional overshoot in the step response.
Another interesting point for Q is the geometric mean of the Bessel and Butterworth Qs. This point gives a Q=0.639, which is attractive because it gives the highest bandwidth (for a given v0) with less than 2% overshoot in the pulse response.
You can describe the 3-dB bandwidth as a function of v0 and Q as follows:
(
B
)
Evaluating the radical portion of Equation B at se veral of the possible values of Q suggested above gives the ratio of the 3-dB bandwidth to the characteristic frequency.
For Q=0.577, F3 dB=0.79F0. For Q=0.639, F3 dB=0.90F0. For Q=0.707, F3 dB=F0.
Some other useful second-order lowpass-response equations include
(
C
)
which is the peak frequency when Q>0.707;
(
D
)
which is the amount of peaking when Q>0.707;
(
E
)
which is the percent overshoot in the pulse response when Q>0.50; and
(
F
)
which is the noise-power bandwidth when Q>0.5.
Table 1
--Op-amp specifications
Typical specifications
OPA627
OPA637
GBP (MHz)
16
80
Minimum stable gain
1
5
Slew rate (V/µsec)
55
135
Input-voltage noise (nV/squareroot(Hz))
4.5
4.5
Input capacitance (Cdifferential+ Ccommon mode) (pF)
15
15
Input-offset voltage (mV)
0.1
(low grade)
0.28
(low grade)
Table 2
--OPA637 gain of 2 example
Design features
Values
Notes
Target Q
0.64
See box
Target G2
7.5
1.5 times the minimum stable gain
Computed Z0
0.93 MHz
From Equation 13
Resulting Z1
2.77 MHz
G1•Z0
Resulting P1
6.93 MHz
G2•Z0
Resulting F0
8.6 MHz
=Z0•GBP
Resulting F-3 dB
7.74 MHz
If Q=0.64
Selected RF
2 kilohms
Required RG
1 kilohms
Required CF
11.5 pF
From Equation 15
Required CS
75 pF
From Equation 16
Table 3
--Lowpass-filter design
Design targets
Resulting design
Component values
(RF=2 kilohms)
lowercaseomega0=5 MHz
Z0=312 kHz
RG=1 kilohms
Q=2
G2=51.2
CF=5 pF
G1=3
P1=16 MHz
CS=249 pF
Table 4
--Gain-bandwidth-independent design
Design
targets
Resulting
design
Predicted Q and F-3 d-vs gain
G 1
-R F /R G
Q
F -3 dB (MHz)
F0=5 MHz
Z0=312 kHz
2
-1
0.776
5.59
Nominal G1=4
G2=13.74
3
-2
0.74
5.31
Nominal Q=0.707
P1=4.3 MHz
4
-3
0.707
5
5
-4
0.677
4.64
6
-5
0.65
4.24
7
-6
0.624
3.75
8
-7
0.601
3.16
|
Author's biography Michael Steffes is a strategic marketer for high-speed signal-processing components at Burr-Brown Corp (Tucson, AZ). He has a BSEE from the University of Kansas (Lawrence) and an MBA from Colorado State University (Fort Collins), and he has helped develop numerous amplifier ICs. His spare-time interests include history, classic literature, travel, and running. You can reach Steffes at steffes_michael@burr-brown.com. |
Author's biography
Michael Steffes is a strategic marketer for high-speed signal-processing components at Burr-Brown Corp (Tucson, AZ). He has a BSEE from the University of Kansas (Lawrence) and an MBA from Colorado State University (Fort Collins), and he has helped develop numerous amplifier ICs. His spare-time interests include history, classic literature, travel, and running. You can reach Steffes at steffes_michael@burr-brown.com.
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