Composite Amplifiers Part Four: Using Composite Amplifiers to Tame DC Offset and Noise

The fourth and final part of our series to explain the nature, characteristics, benefits, and caveats of composite amplifiers.

This is the fourth and final part in our series explaining the nature, characteristics, benefits, and caveats of composite amplifiers. The four parts of the series include:

DC Offset and the Composite Amplifier

In the case of a cascaded amplifier, the desired gain can be achieved while maintaining a large bandwidth. The DC offset of the first stage and the second stage contribute to the system output DC offset. The first stage tends to dominate. The composite amplifier nulls out the DC offset produced by the second stage. This is because that DC offset is contained within the feedback loop. This may not be obvious. So, we shall analyze this in a step-by-step fashion by referring to Figure 26.

Figure 26. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 26. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

1.   For U1A to have a differential input voltage of zero, the feedback of the system must work to make the voltage across R1 equal to VOS1.


2.  
Since we know the voltage across R1, we can apply Ohm’s law to determine the current IR1 through it. This current is assumed to flow through R2 because we also assume no current flows in or out of the op amp’s input terminals.


3. 
  The voltage across R2 is provided by Ohm’s law.


4. 
  The output voltage vOUT is determined by Kirchhoff’s voltage law.


5. 
  The voltage gain of op amp U1B is

We solve for vIN2.


6.
   The output voltage of op amp U1A (vOUT1) is determined by Kirchhoff’s voltage law.

The Multisim simulation of the circuit (see Figure 27) permits these results to be verified.

Figure 27. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 27. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

The test circuit (0) indicates the input DC offset for the AD712 is 0.7 mV. The measurement and the calculation results (1) are in close agreement. The current measurement is not included since the voltage across resistor R2 (3) is related directly and the calculation is “spot on.” The output voltage (4) measurement and calculation are nearly identical. Since an equivalent source for VOS2 is not required here, vOUT1 and vIN2 (6) are the same. This gives us confidence in our analysis.

We can calculate vOUT directly by applying Ohm’s law and Kirchhoff’s voltage law.

We recognize that 1 + R2/R1 is the voltage gain of the composite amplifier and write Equation 2.  

The output DC offset only depends on the input offset voltage of the first stage. To demonstrate this, consider Figure 28.

Figure 28. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 28. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

The “pretend” DC offset of 100 mV adds to the 0.7 mV associated with op amp U1B. The output DC offset increases from 699.2 mV to 700.6 mV. This is a negligible (only 0.2 percent) increase. Otherwise, the 100 mV DC offset would cause the second stage to saturate.

Equivalent noise voltage and voltage distortion generators can be reflected to the input of an op amp. The noise and distortion of a composite amplifier are determined by the noise and distortion produced by the first stage. The analysis proceeds in the same fashion as the analysis of the DC output offset. The noise (voltage spectral response in V/√Hz) is then determined.

Table 1 compares the single stage, cascaded, and the composite amplifiers.

Table 1

Configuration

Av

fH

VOUT(DC OFFSET)

Vout(noise)

Single Stage

1001

3.996 kHz

751 mV

15.02 µV/

Cascaded

1005

81.209 kHz

778 mV

15.10 µV/

Composite

999

136 kHz

699.3 mV

15.0 µV/

The same circuit (Figure 29) is used with a precision TL062 dual op amp. The output DC offset shows a vast improvement. The input stage sets the output DC offset. The amplitude frequency response graph is also included in Figure 29. Improved DC performance cost is achieved at the expense of bandwidth.

Figure 29. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 29. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Noise and the Composite Amplifier

The units of noise (V2/Hz and worse yet, V/√Hz) are somewhat baffling to new and experienced engineers alike. Some background is provided here to make the units less mysterious.

In general, the composite amplifier reduces the noise in a fashion like its reduction of the DC offset. However, because the noise is AC, the frequency response of the system is involved. Let’s remind ourselves of the noise produced by a resistor.

Any electrical resistance (e.g., a wire or a resistor) that operates at any temperature above absolute zero generates a random thermal noise voltage. The electrons move randomly in the resistance but they tend to collide with the atoms within that resistance.

Even though the electron movement is random, at a given instant the net movement of the electrons can be either up or down. Hence, because of the randomly-varying electron density, a random voltage will appear across that resistance. This is shown in Figure 30.

Figure 30. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 30. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Thermal noise is described as white noise because it contains all frequencies just as white light contains all colors. This description reinforces the idea the output noise depends on the frequency response of the composite amplifier.

The average (mean) value of the resistor voltage is zero. However, its root mean square (rms) value is finite and can be measured. (Recall the average value of a sine wave is also zero, but it too has a finite rms value.) The mean square value of the thermal noise voltage across a resistor is directly proportional to the absolute temperature (T), the resistance (R), and the bandwidth (Df). See Equation 3.

Recall that a bar drawn over a variable indicates it is a mean or average value (e.g. xÌ„). Boltzmann’s constant k is 1.381 x 10-23 J/K (joules per degree Kelvin) and relates absolute temperature to energy. The noise bandwidth in hertz is designated Df

If we divide both sides of Equation 3 by R, we obtain Equation 4. Equation 4 suggests the square of the average noise voltage is used because we are dealing with noise power. (Remember: P = V2/R.)

If we examine the right-hand side of Equation 4, we note that its units simplify to joules/second, which is the definition of the watt. Temperature appears in the equation because the thermal energy produces the kinetic energy associated with the randomly-moving electrons.

The rms noise voltage is obtained by taking the square root of both sides of Equation 4 which takes us to Equation 5.

A noisy resistor can be modeled as an ideal (noiseless) resistor in series with a noise voltage source as shown in Figure 31.

Figure 31. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 31. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Let’s determine the average rms noise voltage produced in a 50 Ω resistor at 17 degree C (62.6 degree F) with a noise bandwidth of 1 MHz. First, we determine the absolute temperature in kelvins.

T(K) = T(oC) + 273 = 17 + 273 = 290 K

Now we calculate the average rms noise voltage using Equation 5.

This result can be applied to form a Thevenin equivalent circuit as indicated in Figure 32.

Figure 32. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 32. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

The temperature and resistance parameters can be defined succinctly. However, the noise bandwidth appears to be dependent on a particular application. Consequently, the industry has standardized its interpretation. Let’s see how this works. We start with Equation 4.

We divide both sides by the noise bandwidth (Df). This produces Equation 6. The units on this result are V2/Hz.

If we are interested in the average rms value of the noise voltage, we can take the square root of both sides of Equation 6. 

The units associated with Equation 7 are Vrms/√Hz. The industry has standardized on a noise bandwidth (Df) of 1 Hz. Voltage noise power density has units of V2/Hz while voltage noise spectral density is in V/√Hz.

There are many mechanisms that generate noise in an op amp circuit. In op amp circuits the three most common noise sources are thermal noise, shot noise and 1/f noise.

Thermal noise and shot noise form two white-noise components and 1/f noise provides a third noise component. When combined, the total input noise voltage spectral density is formed. This is illustrated in Figure 33. Observe that a log-log graph is used.

Figure 33. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

Figure 33. Used with author’s permission from Discrete and Integrated Electronics Analysis and Design for Engineers and Engineering Technologists.

The noise generated by the second stage is contained within the overall feedback loop. It will be reduced by the bandwidth capability of the composite amplifier. The noise will peak when the frequency response peaks. The Analog Dialogue article Composite Amplifiers: High Output Drive Capability with Precision by Jino Loquinario provides the graph of the composite amplifier frequency response as affected by the first-stage bandwidth, shown in Figure 34. The graph of the noise amplitude versus frequency as controlled by the first-stage bandwidth is shown in Figure 35.

Figure 34. Composite amplifier frequency response as controlled by the first stage bandwidth. (Image courtesy of Analog Dialogue.)

Figure 34. Composite amplifier frequency response as controlled by the first stage bandwidth. (Image courtesy of Analog Dialogue.)
Figure 35. Composite amplifier noise as controlled by the first stage bandwidth. (Image courtesy of Analog Dialogue.)

Figure 35. Composite amplifier noise as controlled by the first stage bandwidth. (Image courtesy of Analog Dialogue.)

Review and Conclusions

Composite amplifiers provide the required voltage gain and preserve bandwidth requirements. A two-stage composite amplifier nulls out the DC offset associated with the second stage. The output DC offset depends on the input stage. If a low output DC offset is required, the first stage should provide an ultra-low DC offset.

Electrical noise is generated by resistance and depends on the size of the resistor, the absolute temperature, and the noise bandwidth. The output noise is determined by the input stage primarily. Second stage noise is rejected as made possible by the system bandwidth. There are three distinct noise mechanisms associated with op amp circuits: thermal noise, shot noise, and 1/f noise. The output noise is a strong function of the first stage.

Composite amplifiers also provide a reduction in signal distortion and an improvement in slew rate. The amount of frequency response peaking and the avoidance of oscillation must be understood and controlled.