Before continuing, please consider reviewing the information in the previous two articles on VFAs (voltage-feedback amplifiers) and CFAs (current-feedback amplifiers):

- Part I: Intro to CFAs: Current-Feedback Amplifiers vs. Voltage-Feedback Amplifiers
- Part II: Benefits of CFA Design vs. VFAs

### Configuring a CFA: The Importance of R_{f}

In general, each node in a circuit contributes a pole frequency of the type

*Equation (1)*

*Equation (1)*

where *R _{node}* is the equivalent resistance presented by that node, and

*C*is its stray capacitance towards ground.

_{node}In the last article, we considered a specific CFA schematic, shown below:

**Figure 1. **Circuit schematic of a CFA (top), and its constituent blocks (bottom).

**Figure 1.**Circuit schematic of a CFA (top), and its constituent blocks (bottom).

Here, each node other than the gain node is such that *R _{node}* <<

*R*, where

_{gn}*R*is the equivalent resistance presented by the gain node. As a result, the open-loop transimpedance gain

_{gn}*z(*

*jf*

*)*is dominated by the gain-node pole

*f*

_{p}*= 1/(2πR*, with all remaining poles typically clustered at least some 3–4 decades above this dominant pole.

_{gn}C_{gn})The optimum value of *R _{F}* specified in datasheets is the result of a

*compromise*between the desire to keep the

*1/ß*curve as low as possible to maximize the loop gain

*T*, and the need to avoid pushing its crossover frequency

*f*into the region of excessive phase lag due to the

_{x}**higher-frequency**poles.

Thus, when configuring a CFA for voltage-follower operation, we must include *R _{F}* in its feedback path, as shown in Figure 2

*(a)*.

*(a) (b)*

**Figure 2.** (a) A voltage-follower CFA, and (b) its 1/β curve. Using a wire instead of R_{F} would destabilize the circuit and cause it to oscillate.

**Figure 2.**(a) A voltage-follower CFA, and (b) its 1/β curve. Using a wire instead of R

_{F}would destabilize the circuit and cause it to oscillate.

Using a plain wire, as in the case of a VFA, would push down the *1/β* curve till it coincides with *r _{n}* and crosses the |

*z*| curve in a region of excessive phase shift, where there would be no phase margin left, and the circuit would surely oscillate.

So long as a CFA is equipped with the proper *R _{F}*, it can be used in virtually all the resistive applications that are typical of the VFA, such as

*inverting*and

*non-inverting amplifiers*,

*summing amplifiers*,

*difference amplifiers*, and

*I-V converters*.

It works well also in filter applications that utilize op amps with resistive feedback.

### Stability Issues with CFAs: Integrator Applications

Despite the versatility of a CFA with an appropriate *R _{F}*, using the CFA as the popular Miller integrator is not allowed on account of stability considerations. To see why, refer to Figure 3

*(a)*, where we note that since the feedback element is the impedance

*Z*= 1/(

_{F}*j*2

*πfC*), we now have 1/

*β*=

*Z*+

_{F}*r*.

_{n}

*(a) (b)*

**Figure 3. **(a) A Miller integrator and (b) its 1/β curve.

**Figure 3.**(a) A Miller integrator and (b) its 1/β curve.

At low frequencies, where |*Z _{F}*| >>

*r*, we have |

_{n}*1/β*| →

*1/(2πfC)*, and at high frequencies, where |

*Z*| <<

_{F}*r*, we have |

_{n}*1/β*| →

*r*. As depicted in Figure 3

_{n}*(b)*, the crossover frequency is again in a region of excessive phase shift, where no phase margin is left and the circuit is likely to oscillate.

### How to Gain Stability by Using a Dual CFA

The alternative realization of the above concept uses two CFAs to provide the integration function without destabilizing either CFA:

**Figure 4.** Integrator realization using a dual CFA.

**Figure 4.**Integrator realization using a dual CFA.

Here, *CFA _{2}* works as a unity-gain inverting amplifier, driving the capacitor’s right plate with the voltage –

*V*. The current through

_{o}*R*, (where

_{F1}= V_{o}/z_{1}*z*is the open-loop gain of

_{1}*CFA*) is vanishingly small, so we assume the non-inverting input of

_{1}**to be at zero potential.**

*CFA*_{1}This allows us to write, by KCL,

*Equation (2)*

*Equation (2)*

giving the transfer function,

*Equation (3)*

*Equation (3)*

This is the transfer function of a *non-inverting integrator* having* f _{o}* as its

*unity-gain frequency*. This circuit, easily realizable with a dual CFA IC, enjoys also the advantage of

*active frequency compensation*

**[1]**, a highly desirable feature for coping with

*Q-enhancement*issues in dual-integrator-loop filters.

### How to Counteract Phase Lag in a CFA

There are situations in which a (small) feedback capacitance is allowed, and this is when it is necessary to counteract the phase lag arising from the presence of a substantial stray capacitance at the inverting input.

A typical example is the I-V conversion of a current-output digital-to-analog converter (DAC), which appears as a current sink *I _{i}* with a parallel stray capacitance

*C*on the order of tens or even hundreds of picofarads, as depicted in Figure 5

_{s}*(a)*.

*(a) (b)*

**Figure 5.** (a) I-V converter, and (b) investigating its stability via linearized Bode plots: subscript u stands for uncompensated (C_{F} = 0) and subscript c for compensated (C_{F} in place).

**Figure 5.**(a) I-V converter, and (b) investigating its stability via linearized Bode plots: subscript u stands for uncompensated (C

_{F}= 0) and subscript c for compensated (C

_{F}in place).

Ideally, the circuit would give *V _{o} = *

*R*. We wish to assess the effect of

_{F}I_{i}*C*graphically. We observe that at low frequencies, where

_{s}*C*acts as an open circuit, we still have

_{s}*1/β*→

*R*+

_{F}*r*. The presence of

_{n}*C*starts to be felt when its impedance |

_{s}*Z*| becomes equal to the resistance seen by

_{s}*C*itself, which is

_{s}*R*||

_{F}*r*. This occurs at the frequency

_{n}

*f*such that |

_{p}*1/(j2πf*| =

_{p}C_{s})*R*||

_{F}*r*, or

_{n}

*Equation (4)*

*Equation (4)*

where the fact that *r _{n}* <<

*R*has been exploited.

_{F}Past *f _{p}*, the

*1/β*curve starts to

*rise*, indicating that

*f*is a

_{p}*zero frequency*for

*1/β*, and therefore a

*pole frequency*for the loop gain

*T = zβ*, where

*z*is the open-loop gain of the CFA. This pole erodes the phase margin of the circuit, bringing it on the verge of oscillation, so we need some form of frequency compensation.

To counteract the *phase lag* due to *C _{s}* we introduce

*phase lead*via a feedback capacitor

*C*, as shown. A good starting point is to break the

_{F}*1/β*curve right at the crossover frequency

*f*, which will result in a phase margin of about 45°.

_{x}Considering that *r _{n}* <<

*R*, we can approximate the crossover frequency between the

_{F}*R*+

_{F}*r*curve and the

_{n}*z*curve as

*f*. Then

_{t}*f*is the

_{x}*geometric mean*of

*f*and

_{p}*f*, so imposing

_{t}

*Equation (5)*

*Equation (5)*

and solving for *C _{F} *gives

*Equation (6)*

*Equation (6)*

If a greater phase margin than the estimated 45° is desired, it can be achieved by suitably increasing the value of *C _{F}*. This task is best done

*empirically*by observing the step response with an oscilloscope and raising

*C*until the overshoot is lowered to an acceptable value.

_{F}

### The Composite Amplifier: The Best of CFAs and VFAs

The fast dynamics (wide bandwidths as well as high slew rates) and low-distortion characteristics of CFAs make them suited to high-speed applications such as video systems, radar systems, IF and RF stages, DSL, and **automated test equipment** applications.

VFAs, on the other hand, offer better DC characteristics (low input offset voltage and bias currents), lower noise, and higher loop gains, so they are better suited to precision applications.

Figure 6 shows a composite amplifier that offers the best of both worlds.

**Figure 6.** A composite amplifier enjoying the best of both the VFA and the CFA worlds.

**Figure 6.**A composite amplifier enjoying the best of both the VFA and the CFA worlds.

The circuit uses a VFA with a constant gain-bandwidth product (GBP) of 10 MHz to achieve a closed-loop gain of 100 V/V with a closed-loop bandwidth of also 10 MHz. Were it to act alone, the VFA would be capable of a bandwidth of only (10 MHz)/100 = 100 kHz. However, cascading it with a much faster CFA with a gain of 100 will trick the VFA into amplifying only by 1 V/V, that is, into acting as a mere voltage follower, whose closed-loop bandwidth we know to coincide with its GBP, in turn coinciding with *f _{t}*.

**Figure 5**. Bode plots for the composite amplifier of Figure 6. Cascading the VFA with a CFA having a closed-loop gain A_{CFA} of 40 dB shifts the VFA’s open-loop gain a_{VFA} upwards also by 40 dB, thus resulting in the composite open-loop gain *a*_{comp}*, and in the composite closed-loop gain A*_{comp}. Note how the VFA is tricked into acting as a unity-gain voltage follower with closed-loop gain A_{VFA} of 0 dB.

**Figure 5**. Bode plots for the composite amplifier of Figure 6. Cascading the VFA with a CFA having a closed-loop gain A

_{CFA}of 40 dB shifts the VFA’s open-loop gain a

_{VFA}upwards also by 40 dB, thus resulting in the composite open-loop gain

_{comp}

_{comp}. Note how the VFA is tricked into acting as a unity-gain voltage follower with closed-loop gain A

_{VFA}of 0 dB.

To avoid destabilizing the VFA by introducing any substantial delay within its feedback loop, the closed-loop bandwidth of the CFA should be much higher (by, say, a decade or more) than the GBP of the VFA, an easily achievable goal with the faster CFAs.

The circuit enjoys the better input characteristics of the VFA (low input DC errors and noise) as well as the maximum achievable loop gain while providing the high slew rate and lower distortion of the CFA. Note also that any overheating by the output stage of the CFA will never reach the input stage of the VFA, thus reducing significantly the effects of input thermal drift.

In this article, we continued our investigation of the CFA circuit topology. We discussed how the CFA can be used in the resistive applications often suited to VFAs (i.e., inverting and non-inverting amplifiers, summing amplifiers, difference amplifiers, and I-V converters).

We also looked at the limits to this concept as stability prevents the CFA from being appropriate for Miller integrator applications but how integration may be achieved via the use of a dual CFA.

Finally, we explored the concept of the composite amplifier, which combines the strengths of two separate amplifiers to attain performance not possible with a single amplifier alone.

### References

**[1]** http://online.sfsu.edu/sfranco/BookOpamp/OpampsJacket.pdf

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