How does op-amp feedback work?

Fig. 1 shows an ideal linear time invariant feedback amplifier where the basic amplifier A (an amplifier with intrinsic or basic gain A) is in a circuit with an ideal negative feedback block B = beta. The circuit is ideal with no time delays and the feedback signal received at the summing node is exactly180 degrees out of phase with the input signal Vin. The feedback therefore subtracts from the input signal Vin --instead of in-phase with the input which would add to the signal strength and be an unstable condition (if A is unity or greater.) In-phase feedback is called positive feedback and is generally not found in op-amp or amplifier circuits but is commonly employed in comparator circuits to provide hysteresis. Vs is the voltage sum of the input signal and the signal A* B = A(beta).

We can now do some simple algebra to find the overall amplifier gain including the feedback.

Vout = Vs *A

Vs = Vin – B*Vout

Vin = Vs + B*Vout

The gain G can be written as

G = Vout / Vin = A*Vs/(Vs + B*Vs*A)

Canceling out Vs from the numerator and denominator (Vs not = 0) we have

G = A / (1 + B *A)

The last equation is the classical ideal feedback amplifier gain equation.

We can use the same equation for op-amps when we equate B with beta. Recall that beta for a simple op-amp circuit is

beta = Rin / (Rin + Rf)

So then we can write from the above that


G = A/ (1+A*(Rin / (Rin+Rf))

Dividing the numerator and denominator be A (A not = 0), we have


G =1/ ((1/A)+Rin/(Rin+Rf))

Now since A is usually a very large value we have an amplifier for which gain is not dependent on A but the external components Rin and Rf:

G = (Rf + Rin) / Rin = 1 / beta

where beta is the quantity for op-amps we discussed in the previous article.

What Affects Bandwidth of an Op-amp Cicuit?

How do we relate the various quantities we talked about in the previous article? Fig. 1 above relates the quantities on a gain vs. frequency plot – sometimes called a Bode plot.
The graph shows an example of what a typical plot would look like for any given voltage op-amp. As you note, the maximum gain vs. frequency for the op-amp intrinsic gain is a strong function of frequency, usually falling off at a rate of 20DB per decade with increase in frequency, starting at approximately 10 to 100 HZ. We can extend the bandwidth of our op-amp circuit by actually attenuating the maximum amplifier intrinsic gain using the beta factor to multiply it by a number less than 1. For example if we reduced the gain by the factor beta (A *beta or 60DB reduction), we achieve the target gain of 20DB out to approximately 5KHZ. So larger values of beta reduce gain but increase the flat bandwidth.

What is Op-amp Noise Gain?

The attenuation factor of the output signal at the summing point of the amplifier (the negative input terminal of the op-amp) is called beta (a Greek letter) and is given by the ordinary divider equation as

beta = Rin / (Rin + Rfb)

The so-called “loop gain” (more precisely the open loop gain) is the gain around the loop as the product of the intrinsic open-loop gain of the op-amp itself and the attenuation factor beta, as

Go = A * beta = A * Rin / (Rin + Rf)

For the inverting circuit, the signal gain is given more exactly with a correction factor for the finite gain of the op-amp itself and we can write the corrected signal gain as

Gs = - (Rfb / Rin ) * (1/ (1 + (1/Go))

Looking at the equation above the last one we can see that the quantity Go takes into account the finite gain of the op-amp and the beta factor.

Let the gain correction factor be written as gcf, then we can write Gs as

Gs = - (Rfb / Rin) * gcf

where gcf is approximately 1.0 or a little less than 1.0.
For the non-inverting amplifier, we still have the same value for beta or the attenuation factor from the output to the summing point as

beta = Rin / (Rfb + Rin)

and we will have the same correction factor for signal gain as we had above. Therefore the non-inverting signal gain is given by


Gs_ni = ((Rin + Rfb) / Rin) * gcf


So what is noise gain? Noise gain is defined as the inverse of the beta factor or

NG = 1 / beta

In the case of unity gain amplifier circuits, if we consider the non-inverting unit gain circuit, the noise gain is

NGni = (Rin + Rfb) / Rin = 1 / beta

and Rfb is zero or Rin is infinite, so

NGni = 1 /1 = 1

So the noise gain for the non-inverting amp is 1.0.

For the unity gain inverting amplifier we have

NGi = (Rin + Rfb) / Rin

and Rin = Rfb , so

NGi = (Rin + Rin ) / Rin = 2.0

Therefore the noise gain of the non-inverting amplifier is lower than the inverting amplifier in the unity gain configuration. Why is this important? Because the noise gain will act on the voltage and current off-set errors of the op-amp as well as other op-amp noise sources. For a sensitive low level amplifier circuit we should therefore prefer the non-inverting configuration.

The op-amp summing circuit

The summer circuit of Fig. 1 has a number of applications. We can sum DC signals, AC signals, logic signals, to any arbitrary waveform that is within the dynamic range of the output of the operational amplifier. We can write a simple formula for the action of the summer circuit:

Vo = - RF * ( (V1 / R1 ) + (V2 / R2) + (V3 / R3) + …)

where the Vi’s can be functions of time as well as Vo. We could also generalize the Ri’s into general impedances Zi’s which could be inductors, capacitors, resistors, or combinations thereof. Discussing all of these possibilities is beyond the scope of our discussion.

We should mention however, that the circuit has a number of possible applications such as mixing different signal frequencies, adding or subtracting bits to a logic sequence, reconstruction of time multiplexed frequencies (frequencies must be within the bandwidth of the op-amp), and other applications that may be imagined.

One interesting application is the mixing of two or more frequencies to form a composite signal. For example if we added 1000 HZ, 2000 HZ, and 3000 HZ sine waves we would get a composite signal whose shape would depend not only on the frequencies but also the amplitudes and phases of the individual signals. If the composite signal is analyzed with a fast Fourier transform (FFT), we would find spectral lines for the different frequencies with the spectral amplitudes relative to each signal strength.
The following equation

Vo = -RF * ( (A1*sin(w1*t+p1) / R1) + (A2*sin(w2*t+p2) / R2 + …)

where the w1, w2, … are the radian frequencies and p1, p2, … are the phases.

One interesting case is when all the frequencies to be added are the same but the phases are different. The sum will be a sine wave with a single frequency equal to the original input frequency, but its phase will be the average of the phases of the original input frequencies if the phase differences are small. But the output signal will of course be inverted. Note however, if two signals of the same frequency but have 180 degrees difference in phase, we will have a cancellation effect for the two signals. Relative signal amplitudes of course will have an effect on the sum.

The Integrator and how you can use it

The op-amp integrator is a handy circuit that can be used to produce a linear voltage ramp or measure a capacitance value.

In the circuit of Fig. 1, if we put in a pulse of known time and peak value we can compute some things. First the input current is given by

Iin = Vp / R

If we also know the width of our input pulse, say Tc seconds, then we can compute the value of the capacitance by measuring Vo (if Tc is short enough to avoid pushing the op-amp output to its maximum output.) C is determined from the equation:

C = - (Vp / R) * Tc / (-Vo)

An Ocilloscope could be used to observe the input pulse height and time and the output pulse voltage output to determine the value of C.

In some unusual instances we may wish to measure an unknown resistor value such as a very high value. R can be determined from the equation

R = -(Vp / C) * Tc / (-Vo)

Notice that the output ramp is inverted so we have to use the negative signs to have R and/ or C be positive real values. The time to integrate, Tintg in Fig. 1, is determined by the energy contained in the input pulse and the time constant of the circuit RC.

The differential amplifier

The differential amplifier is a very useful circuit where it is necessary to amplify either an AC or audio signal that is a differential input-- that is two input lines, neither one of which is ground or common. It can also be used to amplify small DC signals that are of a differential nature such as a strain gage or bridge measurement. Looking at Fig. 1, a circuit is shown that would be suitable for AC signals in the range of 20 HZ to 200KHZ with the component values listed below.
In Fig. 1, assume that R1 and R3 are matched in value and R2 and R4 are matched. Then the gain of the amplifier is

Vo = Vdiff * R2 / R1

where Vdiff is the differential voltage of the signal source. In terms of the two input lines from the differential source, with voltage values of V1 (high side input) and V2 (low side input) relative to ground, the output voltage will be

Vo = (V2 – V1) * R2 /R1

If the resistors are not matched the output will be in error according to the amount of miss-match. The output voltage due to the non-inverting input will be

Vo+ = V2 * ( R3 / ( R3 + R4 )) * (R1 + R2) / R1

and the output due to the inverting input will be

Vo- = - V1 * R4 /R3

The net output will then be

Vo = (Vo+) + (Vo-)

The error in gain can now be computed in terms of the individual resistor values. The action of the differential amplifier is to combine the non-inverting signal with the inverting signal inputs and produce an output that reflects the difference (or sum) of the two inputs multiplied by the effective gain.
If the signal is DC only then the capacitors in Fig. 1 should be removed. Otherwise, the capacitors can be used to couple an AC signal with isolation of the amplifier from the location of the source. High voltage capacitors can be used to provide the required amount of isolation.
For you brave souls out there, below is a list of component values needed to build an experimental amplifier with a gain of ten. The V+ and V- voltages for the CMOS type amplifier should be + or – 5 volts DC maximum. The amplifier circuit should have a pass band from 20HZ to 200KHZ with harmonic distortion of -70DB or lower, according to SPICE simulations. Traces should be kept very short and the power supply rails should be very well filtered to block power supply noise.

List of components:

Op-amp IC LTC6241HV
C1, C2 0.22 MFD
C3 10 PF
R1, R3 10K
R3, R4 100K


The load resistance should be greater than 100 ohms to avoid over heating of the amplifier or excessive output current and likely distortion. Also, although the amplifier is a “rail-to-rail” amplifier, it is wise not to overload the input with large input voltages to avoid “clipping” and distortion.

What is a Voltage Follower?

The voltage follower is another handy type of amplifier that has a gain of 1.0 (or very close to 1.0 .) This is also called a unity gain amplifier. Why do we need an amplifier that does not increase or decrease the signal level? The reason is that we can buffer a weak current source signal to obtain the same signal level out, but with a strong source that reproduces the original source voltage signal within the bandwidth capability of the amplifier with very good accuracy.
How does it work? Remember the non-inverting amplifier discussed in the previous two articles posted here? If you look back at the previous article and at fig. 1 in this article, you will note that R1 has been removed from the circuit and R2 has been replaced with a short circuit (or sometimes with a relatively low value such as 1000 ohms.) The gain of the non-inverting amplifier was given as

G = ( R1 + R2 ) / R1

In the case of our voltage follower amplifier you can think of R1 as being a very high value and R2 as a very small value. So the gain of the voltage follower is, for example with R2 = 1000:

Gvf = (N + 1000 ) /N

where N is a very large value. In the limit as N approaches an infinite value, the value of R2 at even 1000 ohms is not very significant, so Gvf approaches 1.0 as N becomes larger and larger. Or if we just set R2 = 0 , then we have Gvf = N / N = 1.0.

Note also that we don't have to worry about common mode noise with this circuit unless the external traces are long.

So now we know about the voltage follower. The supply voltage connections are not shown but will be discussed in later articles.