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:
 Part One: When Cascaded Amplifiers are Not an Option
 Part Two: The Composite Amplifier Choice
 Part Three: Composite Amplifier ClosedLoop Frequency Response
 Part Four: Using the Composite Amplifier to Tame DC Offset and Noise
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 stepbystep fashion by referring to Figure 26.
1. For U1A to have a differential input voltage of zero, the feedback of the system must work to make the voltage across R_{1} equal to V_{OS1}.
2. Since we know the voltage across R_{1}, we can apply Ohm’s law to determine the current I_{R1} through it. This current is assumed to flow through R_{2} because we also assume no current flows in or out of the op amp’s input terminals.
3. The voltage across R_{2} is provided by Ohm’s law.
4. The output voltage v_{OUT} is determined by Kirchhoff’s voltage law.
5. The voltage gain of op amp U1B is
We solve for v_{IN2}.
6. The output voltage of op amp U1A (v_{OUT1}) is determined by Kirchhoff’s voltage law.
The Multisim simulation of the circuit (see Figure 27) permits these results to be verified.
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 R_{2} (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 V_{OS2} is not required here, v_{OUT1} and v_{IN2} (6) are the same. This gives us confidence in our analysis.
We can calculate v_{OUT} directly by applying Ohm’s law and Kirchhoff’s voltage law.
We recognize that 1 + R_{2}/R_{1} 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.
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 
A_{v} 
f_{H} 
V_{OUT(DC OFFSET)} 
V_{out(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.
Noise and the Composite Amplifier
The units of noise (V^{2}/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 randomlyvarying electron density, a random voltage will appear across that resistance. This is shown in Figure 30.
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 = V^{2}/R.)
If we examine the righthand 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 randomlymoving 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.
Let’s determine the average rms noise voltage produced in a 50 â„¦ 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(^{o}C) + 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.
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 V^{2}/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 V^{2}/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 whitenoise 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 loglog graph is used.
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 firststage bandwidth, shown in Figure 34. The graph of the noise amplitude versus frequency as controlled by the firststage bandwidth is shown in Figure 35.
Review and Conclusions
Composite amplifiers provide the required voltage gain and preserve bandwidth requirements. A twostage 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 ultralow 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.