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**Editor’s note: ***This article was originally published in EDN on August 1, 1997 and has been rebuilt by Michael Steffes for clarity and added information. It will be referenced many times in Steffes’ future articles.—Steve Taranovich*

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 transconductance, g_{m}, which produces the ultra-high 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 response. Passive post-filtering 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 amplifiers’ performance 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, C_{S} and C_{F}, to the standard inverting op-amp configuration (**Figure 1**). Previous discussions of this circuit focused on using C_{F} to compensate for a parasitic C_{S}. The following analysis shows you how to set both C_{S} and C_{F} to get a well‑controlled, closed-loop, second-order lowpass frequency response at any signal gain for even the most decompensated op amp.

**Figure 1 **A simple, but previously unexplored, compensation circuit consist of C_{F} and C_{S}. The technique allows you to use a decompensated voltage feedback op amp at low gains with the high frequency benefits of a current feedback op amp but the DC accuracy of a voltage feedback device.

You can easily analyze this circuit using a single-pole, open-loop model for the op amp. Without C_{S} and C_{F}, 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 signal swing at the V+ input, the inverting configuration for most op amps achieves higher slew rates, higher full-power bandwidth, and lower distortion. The trade-offs to getting these inverting-mode benefits are an input impedance set by R_{G} 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:

The significant components of this transfer function are:

- ZF/RG, which would be the signal gain if the op amp were ideal (had infinite open-loop gain and bandwidth),
- 1+ZF/ZG, which is the noise-gain portion of the loop gain (and also equal to the gain from the noninverting input to the output), and
- A(s), which is the open-loop gain over frequency for the op amp.

At DC, the denominator of **Equation 1 **is approximately 1, whereas the numerator is equal to -R_{F}/R_{G}, 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 1V/V (0dB) 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.

**Figure 2** For the compensation network design, Bode analysis points out the unity gain intersection of the sloping portion of the noise gain curve (Z_{o}), the pole set by the feedback compensation network (P_{1}), the low frequency noise gain (G_{1}), and the noise gain at loop gain crossover (G_{2})

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 ω_{0} 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 C_{S} and C_{F}. 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, Z_{F} and Z_{G}, and the op amp’s open-loop-gain expression A(s) into **Equation 1** yields

**(5)**

Re-arranging 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 A_{OL} and a dominant pole at ω_{A}. The noise gain has a DC gain of 1+R_{F}/R_{G}, a low-frequency zero, and a high-frequency pole to flatten the noise gain to 1+C_{S}/C_{F} 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, Z_{0}, and P_{1}. GBP is simply the gain-bandwidth product of the selected op amp (GBP=A_{OL} ω_{A} /2π Hz). Z_{0,} which equals 1/(2πR_{F}(C_{S}+C_{F})), 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 G_{1}Z_{0}=Z_{1}. G_{1} and G_{2} are the low-frequency and high-frequency noise gains, respectively.

P_{1}, the feedback-network pole, is equal to 1/(2πR_{F}C_{F}). This pole and Z_{0} are the two things you can adjust to control the closed-loop frequency response. P_{1} is also equal to Z_{0}G_{2}, which is simply Z_{0} 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 Z_{0} and GBP. This point turns out to be the characteristic frequency, F_{0}, of the closed-loop second-order response (see **sidebar, “**Second-order lowpass-response characteristics**“**). When you set P_{1} to less than this geometric mean, the noise gain crosses the open-loop response at a gain equal to G_{2}. The noise gain crosses the open-loop response at F_{C}, 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 G_{2}.

One of the key assumptions in this analysis is that you control G_{2} 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 -R_{F}/R_{G}. 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, G_{2}, 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 (C_{F} and C_{S}), the following methodology uses radian frequency units. Converting those units to the hertz shown in Figure 2 simply requires a division by 2π.

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

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 ω_{0} and Q that show a clear path to a design methodology.

Specifically, you can simplify the terms inside the radical for ω_{0} by recognizing that AOL is much greater than 1+R_{F}/R_{G}. Dropping the 1+R_{F}/R_{G} of that term, recognizing that A_{OL}ω_{A}=GBP and that 1/((C_{F}+C_{S})*R_{F})=Z_{0} (in Figure 2), and simplifying the expression for Q in the denominator yields the following equations, where G_{2}=1+C_{S}/C_{F} and G_{1}=1+R_{F}/R_{G}:

Referring back to the Bode plot of Figure 2, these simple equations indicate that the closed-loop, second-order response has a characteristic frequency, ω_{0}, that is the geometric mean of Z_{0} and the amplifier’s GBP. Also, the ratio of that characteristic frequency to the sum of the high-frequency, loop-gain crossover frequency (F_{C}) and the zero frequency in the noise gain (Z_{1}) 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.

[Continue reading on EDN US: Design for maximum bandwidth]

**Author’s biography** (1997)

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

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