Memo: Op-amp’s gain factor and noise problems

First, here’s a quick cheatsheet on closed-loop gain factor for inverting and non-inverting amplifiers. The gain of inverting amplifier is given by the resistance R2 across the feedback loop divided by the resistance R1 across the forward input.

A = R2/R1

Meanwhile, the gain of non-inverting amplifier is given by

A = 1 + R2/R1

as calculated in the previous note. Kirchhoff’s current law is applied.

Gain factor and cut-off frequency relationship

The cut-off frequency is not only affected by the properties of a given component but it is also dependent on the gain factor of the amplifier circuit. Higher gain amplifiers have lower cut-off frequencies and this can be charted as an exponential decay function.

To easier express this function in linear terms, the relationship is converted to logarithmic terms:

20 log(0.5) = -6.02 [dB]

with 0.5 is the approximated V/Vref value from experimentation.

On a side note, below are two commonly seen decibel functions in electronics for power and voltage respectively.

{\displaystyle G_{\mathrm {dB} }=10\log _{10}\left({\frac {1000~\mathrm {W} }{1~\mathrm {W} }}\right)=30.}

{\displaystyle G_{\mathrm {dB} }=20\log _{10}\left({\frac {31.62~\mathrm {V} }{1~\mathrm {V} }}\right)=30.}

Naturally, the gain factor for op-amps is expressed in terms of voltages (the second formula).

Differential amplifier

A useful circuit using op-amps to amplify and de-noise a weak signal is a differential amplifier. The basic circuit uses one op-amp in the arrangement below:

V1 and V2 inputs are the same signals, one of which is inverted. This can be done using an inverting amplifier with the gain factor of 1 (inverting unity gain amplifier) but this arrangement is subjected to high-frequency quirks as stated in the previous memo, especially phase shifting property.

If the inputs are identical and not phase-inverted, the output will be zero.

The output of this particular circuit is an amplified signal with transmission line noise suppressed. Note that the transmission line noise being suppressed is the one common on both inputs after the inversion. Noises occurred before inversion cannot be suppressed this way.

Instrumentation amplifier (three op-amps)

Normally, to ensure proper signal flow, the inputs are tunneled through two additional voltage followers (two more op-amps) in order to reduce the impedance. The full circuit (called “instrumentation amplifier) would then look like the following:

with all R values are the same except for Rgain.

Rgain can be of any value. It serves as a “common ground” connector and it also tweaks the voltage gain of the full circuit. Increasing the value of Rgain decreases the gain of the differential amplifier.

In reality, the gain of the above circuit can be tweaked further, following this function

{\frac {V_{\mathrm {out} }}{V_{2}-V_{1}}}=\left(1+{2R_{1} \over R_{\mathrm {gain} }}\right){R_{3} \over R_{2}}

Instrumentation amplifier (two op-amps)

An alternative instrumentation amplifier design using two op-amps can be seen below:

Though this setup saves on component costs, it does have a few disadvantages, notably, the lack of support for unity gain (not a problem for most scenarios but can be if the instrumentation amplifier is used solely for noise canceling).

Furthermore, the circuit is unbalanced. The leftmost amplifier increases the input slightly and introduces some signal delay. This unbalance leads to reduced noise-canceling capability. The output can saturate if the common-mode noise of the input signal here is too high and race condition can lead to a much lower cut-off frequency (compared to three op-amps version).

The gain factor here is controlled by RG in the same manner as Rgain in the other circuit.

G = 1 + R2/R1 + (2*R2)/RG

Bonus: Choosing the base resistance

When calculating the resistance of an analog system, the first resistor (base resistor) is chosen as a compromise between power consumption and noise tolerance. Low resistance (or impedance for AC systems) allows more current draw. This is welcome when loads are concerned but it is a waste of energy when sensors are concerned. The higher the base resistance, the lower the power consumption will be.

On the other hand, high resistance circuits are more susceptible to noise. The relationship between resistance and noise is given by Johnson-Nyquist noise voltage function

{\overline {v_{n}^{2}}}=4k_{\text{B}}TR

where kB is Boltzmann constant 1.38 x 10^-23 [J/K], T is the absolute temperature in Kelvin [K], and R is the system’s resistance. In other words, a small increase in resistance increases in Johnson noise voltage by the power of two.


Increase R, increase noise, decrease current consumption

Decrease R, decrease noise, increase current consumption

Choose the right compromise for the application.

Bonus: Using op-amp with a unipolar power source

Unipolar or single-rail power sources that only have GND and VCC terminals need to offset the ground terminal to create a -VCC source. This offset can be done using a voltage divider circuit in conjunction with a voltage follower. Be mindful that the with only half of the voltage range, the op-amp might experience unexpected floating values.

The voltage follower (or unity gain amplifier) must be connected in series to the middle point of the voltage divider in order to create a new, offset virtual ground. This eliminates the added impedance from the voltage divider, ensuring sufficient current draw power devices using the virtual ground.

For example, such a system of a voltage divider and voltage follower can be employed to create +2.5V and -2.5V bi-polar supply from Arduino’s 5V and 0V unipolar supply. In this case, the op-amp will produce 5V (relative to the Arduino’s true ground) as HIGH signal and 0V as LOW signal. Inversely, it will see 0V from the Arduino as -2.5V (relative to the virtual ground) and 5V as +2.5V.

The previous section on choosing base resistance also applies here, and even more so with the halved voltage range doubling susceptibility to signal distortion.

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