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When talking amplifiers, I sometimes get the impression that whole groups of electronics hobbyists and engineering students have been brainwashed by endlessly hearing and repeating the mantra “Gain-bandwidth product is a constant.” Hearing this statement so often, they actually start to believe this is a universal truth.

I was reminded of this once again some time ago, when I saw a video on the otherwise excellent YouTube channel “w2aew.” At around minute 2:23, the host says “...for a single-pole response, the product of the DC-gain and the bandwidth is constant":

However, this statement isn’t universally true—sometimes it’s true, sometimes it’s approximately true, and sometimes it’s completely wrong.

You might have thought I was going to tell you about the case of a voltage amplifier with a current feedback op amp—a statement that you probably already knew isn’t true—but I am not. Instead I’ll focus on what’s probably the simplest op-amp voltage amplifier configuration you can imagine, the mother of all op-amp circuits: the inverting single-pole op-amp amplifier shown in *Figure 1*.

*1. Standard inverting op-amp amplifier.*

To avoid confusion, let’s first define all symbols that will be involved *(see table)*.

To deduce the connection between *A _{OL,}*

_{0}

*, BW*

_{OL}, A_{CL,}_{0},

_{}and

*BW*all we need to do is apply Kirchhoff’s law to the inverting node of the op amp. We assume our op amp is or can be approximated by a single-pole model, which means we can write:

_{CL},_{}** **** **** **

Applying Kirchhoff’s law gives:

We also have:

which gives us:

Substituting Equation 1 in Equation 2 gives:

Equation 3 might look a bit daunting at first sight, but it was intentionally written that way to allow identification with a single-pole representation of the system:

Inspecting Equations 3 and 4, we get:

Multiplying the respective left and right sides in Equation 5 gives us the following exact formula:

Equation 6 is seldom used because in most practical cases:

which implies:

and therefore the equation can be rewritten as:

A closer look at Equation 7 shows the cause of the problem with the mantra: Only when *A _{CL,}*

_{0 }is substantially greater than 1 can that equation be approximated by

*A*

_{CL,}_{0 }·

*BW*=

_{CL }*A*

_{OL,}_{0 }·

*BW*for the standard inverting op-amp amplifier. If this isn’t the case, though, we can get a substantial error.

_{OL }Take, for example, the case of a phase inverter (i.e., *R*1 = *R*2 or *A _{CL,}*

_{0 }= 1) with a 741 op amp, for which

*A*

_{OL,}_{0 }= 200000 and

*BW*= 5 Hz. Using the mantra, we would find a closed-loop bandwidth of 1 MHz. However, using Equation 7, we find the correct bandwidth to be 500 kHz, which means the mantra gives us an overestimation of the bandwidth with a factor 2! More generally, the procentual error on the predicted closed-loop bandwidth using the mantra-formula is given by:

_{OL }So, if you want your prediction to be less than 5% off, your closed-loop DC gain should be at least 20, i.e. 26 dB.

*2. Amplitude characteristics for a standard inverting op-amp amplifier.*

The graphs in *Figure 2*, which were made using PSpice and a first order op-amp model, also show *A*_{CL,0 }• *BW _{CL}* ≠

*A*

_{OL,0 }•

*BW*,

_{OL}*since the oblique asymptotes of the closed-loop system don’t coincide with the one of the open-loop system as is often erroneously drawn in textbooks.*

^{1}To clarify this point, consider:

For *f *→ ∞, this becomes:

Considering the vertical axis in *Figure 2* is in dB and the horizontal axis is logarithmic, we get: 20log(*A _{CL}*) = 20log(

*A*

_{CL,}_{0}

*BW*) − 20log

_{CL}*f.*The term 20log(

*A*

_{CL,}_{0}

*BW*) determines the vertical position of the oblique asymptotes. Because of Equation 7, the latter term is equal to:

_{CL}and obviously a function of *A _{CL,}*

_{0 }itself. Only if

*A*

_{CL,}_{0 }

*>>*1, the oblique asymptote of 20log

*A*will coincide with the one of 20log

_{CL }*A*.

_{OL}As an afterthought, note that for a standard *non*-inverting op-amp amplifier, the mantra-formula is *correct*, but for the standard inverting configuration you’d better memorize (1 + *A**CL,*0)*BW**CL *= *A**OL,*0 *BW**OL *from now on.

When actually measuring the bandwidth of the above-mentioned unity-gain inverter, don’t be surprised if you notice some peaking in the amplitude characteristic. This is caused by a complex pole in the real 741 and the presence of parasitic capacitance between the inverting and non-inverting input of the op amp, but that’s another story.

*Hugo Coolens received his engineering degree in electronics in 1983 from KIHO Ghent. He is currently head lecturer at the Faculty of Engineering Technology, KU Leuven TC Ghent.*

**Reference:**

1. Willy M. C. Sansen, *Analog Design Essentials*, p.155.