# Achieving Faster Composite Op-Amp Dynamics by Expanding the Frequency Bandwidth

February 23, 2020 by Dr. Sergio Franco## This is Part 3 of the series of articles about composite amplifiers. In this section, we are going to show how to achieve faster op-amp dynamics by expanding the frequency bandwidth.

In Parts 1 and 2 of a three-article series on composite amplifiers, we have investigated how to boost the output current drive capability of an op-amp and simulate our example voltage buffer in PSpice.

Now, we are going to show how to achieve faster op-amp dynamics by expanding the frequency bandwidth.

### Expanding the Frequency Bandwidth

The open-loop gain of most op-amps exhibits a constant gain-bandwidth product (constant GBP). The most salient consequence of this constancy is the fact that the higher the noise gain of an op-amp circuit is, the lower the closed-loop bandwidth. For instance, if we configure the op-amp as a noninverting amplifier, in which case the noise gain coincides with the closed-loop gain *A*, then the closed-loop bandwidth is

\[f_B = \frac {GBP}{A}\]

**Equation 1**

**Equation 1**

So, if we use an op-amp with GBP = 1 MHz and configure it for a noninverting gain of *A* = 10 V/V, then we get *f _{B}* = 10

^{6}/10 = 100 kHz. For

*A*= 100 V/V we get

*f*= 10 kHz, and for A = 1,000 V/V we get

_{B}*f*= 1 kHz.

_{B}What if we wanted to use this op-amp as an *audio preamplifier* with a gain of 1,000 V/V and a bandwidth of *f _{B}* = 20 kHz, which represents the upper limit of the audio range?

Clearly, a single 1-MHz op-amp won’t do it, so let’s see if we can enlist the help of a second, similar op-amp to raise *f _{B}* from 1 kHz to 20 kHz. Figure 1 shows a popular realization of this concept.

**Figure 1. **(a) Composite amplifier to achieve a wider bandwidth. (b) Straight-line Bode plots

**Figure 1.**(a) Composite amplifier to achieve a wider bandwidth. (b) Straight-line Bode plots

In the figure, you can see (a) a composite amplifier to achieve a *wider bandwidth*. and (b) straight-line Bode plots where:

- |
*a*| is the open-loop gain of each op-amp, and*f*is the transition frequency (_{t}*f*= GBP in the present rendition)_{t} - |
*a*| is the composite amplifier’s open-loop gain_{c} - |
*A*| is the closed-loop gain of_{2}*OA*, and_{2}*f*is its –3-dB frequency_{2} - |
*A*| is the composite amplifier’s closed-loop gain, and_{c}*f*is its –3-dB frequency_{c} - |β| is the feedback factor around the composite amplifier
- a
_{0}, A_{c0}, and A_{20}identify the DC values of the above gains

Here, *OA _{1}* is the primary op-amp and

*OA*is the secondary op-amp, both having an open-loop gain of

_{2}*a*.

*OA*is configured as a noninverting amplifier with a closed-loop gain of

_{2}*A*with a DC value of

_{2}

\[A_{20} = 1 + \frac {R_4}{R_3}\]

**Equation 2**

**Equation 2**

By Equation 1, with GBP replaced by *f _{t}*, the closed-loop bandwidth of

*OA*is

_{2}

\[f_{2} = \frac {f_t}{A_{20}}\]

*Equation 3*

*Equation 3*

Together, *OA _{1}* and

*OA*form a composite amplifier with an open-loop gain of

_{2}

\[a_{c} = a \times A_2\]

*Equation 4*

*Equation 4*

The presence of *OA _{2}* inside

*OA*’s feedback loop has two effects:

_{1}- It expands the open-loop gain from
*a*to*a*. Due to the logarithmic nature of decibels (the log of a_{c}*product*equals the*sum*of the logs), the DC values*a*and_{0}*A*_{20}*add up*in the manner shown. - It establishes a pole frequency at
*f*, which causes the slope of the |_{2}*a*| curve to change from –20 dB/dec to –40 dB/dec, as shown. This pole frequency will erode the phase margin of the loop around_{c}*OA*, so we must be vigilant that the overall circuit does not get destabilized._{1}

The composite amplifier of Figure 1(a) is in turn configured as a noninverting amplifier with a feedback factor of *β = R _{1}/(R_{1 }+ R_{2})*. The reciprocal 1/β is called the

*noise gain*, and

\[\frac {1}{\beta} = 1 + \frac {R_2}{R_1}\]

**Equation 5**

**Equation 5**

(Recall that for a noninverting op-amp the noise gain and the closed-loop gain coincide, so A_{c0} = 1/β). Were *OA _{1}* operating alone, its closed-loop bandwidth would be

*f*(see Figure 1(b)).

_{1}However, the presence of *OA _{2}* expands the closed-loop bandwidth from

*f*to

_{1}*f*, where

_{c}*f*is the

_{c}*crossover frequency*of the |

*a*| and |1/β| curves. It is precisely this bandwidth expansion that we wish to exploit.

_{c}To gain better insight, consider the PSpice circuit of Figure 2, simulating a composite amplifier with a closed-loop gain of 1,000 V/V, or 60 dB.

**Figure 2**. PSpice circuit for a composite amplifier using Laplace blocks to simulate 1-MHz op-amps.

**Figure 2**. PSpice circuit for a composite amplifier using Laplace blocks to simulate 1-MHz op-amps.

Figure 3(a) shows the effect of stepping EOA2’s closed-loop gain |A2| in 10-dB increments via R4. For |A2| = 0 dB things go as if EOA1 were operating alone, giving a closed-loop DC gain of 1,000 V/V (= 60 dB) with a closed-loop bandwidth of 1 kHz.

**Figure 3. **Visualizing the effect of 10-dB increments in **EOA2**’s closed-loop gain |A2| in the circuit of Figure 2. Effect on the composite amplifier’s (a) open-loop gain |ac|, and (b) closed-loop gain |Ac|.

**Figure 3.**Visualizing the effect of 10-dB increments in

**EOA2**’s closed-loop gain |A2| in the circuit of Figure 2. Effect on the composite amplifier’s (a) open-loop gain |ac|, and (b) closed-loop gain |Ac|.

Increasing |A2| *expands* the composite amplifier’s open-loop gain |ac| along both the vertical and the horizontal axes, while at the same time *reducing* EOA2’s closed-loop bandwidth *f _{2}*, as per Equation 3.

Figure 3(b) shows the effect on the composite amplifier’s closed-loop gain *A _{c}*: all curves exhibit the same DC value of 60 dB; however, the bandwidth increases with |A2|.

It is interesting to observe, in Figure 4(a), how **OA1** and **OA2** cooperate, in complementary fashion, to maintain a constant DC value of 60 dB.

**Figure 4.** Visualizing the effect of 10-dB increments in **EOA2**’s closed-loop gain |A2| upon EOA1’s closed-loop gain |A1| in the circuit of Figure 2 (b). The composite amplifier’s closed-loop gain |Ac| for phase margins of 45° and 65°.

**Figure 4.**Visualizing the effect of 10-dB increments in

**EOA2**’s closed-loop gain |A2| upon EOA1’s closed-loop gain |A1| in the circuit of Figure 2 (b). The composite amplifier’s closed-loop gain |Ac| for phase margins of 45° and 65°.

As |A2| rises, |A1| drops in such a way that their DC values keep adding up to 60 dB as 0 + 60, or 10 + 50, or 20 + 40, or 30 + 30. However, also **OA2**’s pole frequency *f _{2}* drops, and in so doing it gradually erodes

**OA1**’s phase margin. How far can we raise |A2| before

*f*destabilizes the composite amplifier? This depends on the phase margin we are willing to accept.

_{2}In the absence of *OA _{2}*, the circuit would conform to the situation corresponding to the 1/β

_{1}curve of Figure 2(a) of Part 1, indicating a phase margin of

*ɸ*= 90°. With

_{m}*OA*present,

_{2}*ɸ*gets eroded according to

_{m}

\[\phi_m = 90^\circ - tan ^{-1}\frac {f_c}{f_2}\]

**Equation 6**

**Equation 6**

Now, exploiting the constancy of the GBP on the |*a*| curve of Figure 1(b), we write

\[A_{c0} \times f_c = f_t \times A_{20}\]

**Equation 7**

**Equation 7**

Combining with Equations 3 and 7 and solving for the *f _{c}/f_{2}* ratio gives

*ɸ*in terms of the DC gains A

_{m}_{20}and A

_{c0}

\[\phi_m = 90^\circ - tan ^{-1}\frac {{A_{20}^2}}{A_{c0}}\]

**Equation 8**

**Equation 8**

Turning around Equation 8, we can find how far we can increase *A _{20}* for a given

*ɸ*and A

_{m}_{c0}

\[A_{20} = \sqrt{A_{c0} \times tan(90^\circ-\phi_m})\]

**Equation 9**

**Equation 9**

A popular strategy is to impose *f _{2}* =

*f*, a situation corresponding to the 1/β

_{c}_{2 }curve of Figure 2(a) of Part 1, for which

*ɸ*= 45°. This is achieved by making

_{m}*A*=

_{20}*(A*. So, for the PSpice circuit of Figure 2, we need

_{c0})^{1/2}*A*= (1,000)

_{20}^{1/2}= 31.6 V/V, which we implement with

*R*= 30.6 kΩ. As shown in Figure 4(b), the ensuing closed-loop gain exhibits some peaking around 22 kHz, and a –3-dB frequency of about 40 kHz.

_{4}If the application calls for the absence of peaking, then we shoot for *ɸ _{m}* = 65°, which marks the onset of peaking. Using Equation 9 we find

*A*= 21.6 V/V, which we implement with

_{20}*R*= 20.6 kΩ in our PSpice circuit of Figure 2. The ensuing response has a –3-dB frequency of about 30 kHz. This is considerably higher than the bandwidth of 1 kHz that OA1 would yield if acting alone.

_{4}It is worth pointing out that besides expanding the bandwidth, the presence of *OA _{2}* also raises the DC loop gain by

*A*. In our circuit example of Figure 2, without

_{20}*OA*we would have

_{2}*f*= 1 kHz and a DC loop gain of

_{B}*T*= βa

_{0}_{0}= 10

^{–3}×10

^{5}= 100. With

*OA*present and configured for

_{2}*A*= 21.6 V/V to give

_{20}*ɸ*= 65°,

_{m}*f*gets raised from 1 kHz to 30 kHz, and

_{B}*T*gets raised from 200 to 200×

_{0}*A*= 200×21.6 > 4,000, thus improving the DC precision considerably.

_{20}You can readily implement the composite amplifier under discussion using a dual op-amp package.

In the next article, we'll talk about another method of achieving faster op-amp dynamics: raising the slew-rate.

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