Programmable Gain Transimpedance Amplifier Maximizes Dynamic Range of Spectroscopy Systems (Part 3)

Calculating TIA Noise

　　There are three main noise sources for transimpedance amplifiers: the op amp’s input voltage noise, input current noise, and the Johnson noise of the feedback resistor. All of these noise sources are typically expressed as noise density. To convert the units to V rms, find the noise power (voltage noise density squared) and then integrate over frequency. An accurate but much simpler method is to multiply the noise density by the square root of the equivalent noise bandwidth (ENBW). The closed-loop bandwidth of the amplifier can be modeled as a first-order response dominated by the feedback resistor, Rf, and the compensation capacitor, Cf. Using the specifications from the stability example, the closed-loop bandwidth is found to be:

　　(3)

　　To convert the 3 dB bandwidth to ENBW in a single-pole system, multiply by π/2:

　　(4)

　　Once ENBW is known, the rms noise caused by the feedback resistor and the current noise of the op amp can be calculated. The Johnson noise of the resistor appears directly at the output, and the current noise of the op amp appears as the output voltage after passing through the feedback resistor.

　　(5)

　　(6)

　　Here, k is the Boltzmann constant and T is the temperature (in K).

　　The final source is the voltage noise of the op amp. Output noise is equal to input noise multiplied by noise gain. The best way to think about the noise gain of a transimpedance amplifier is to start with the inverting amplifier shown in Figure 7.

　　Figure 7. Inverting amplifier noise gain.

　　The noise gain of this circuit is:

　　(7a)

　　Using the photodiode amplifier model shown in Figure 4a, the noise gain is:

　　(7b)

　　Where Zf is the parallel combination of the feedback resistor and capacitor, and Zin is the parallel combination of the operational amplifier input capacitance and the photodiode shunt capacitance and shunt resistor.

　　This transfer function contains many poles and zeros, and it would be very tedious to calculate by hand. However, using the values ​​in the example above, we can make a rough approximation. At frequencies close to DC, the resistors dominate and the gain is close to 0 dB because the shunt resistance of the diode is two orders of magnitude larger than the feedback resistor. As the frequency increases, the impedance of the capacitor decreases and begins to dominate the gain. Since the total capacitance from the inverting pin of the op amp to ground is much larger than the feedback capacitor, Cf, the gain begins to increase with frequency. Fortunately, the gain does not increase indefinitely because the pole formed by the feedback capacitor and resistor prevents the gain from increasing, and eventually the bandwidth of the op amp takes over and the gain begins to roll off.

　　Figure 8 shows the amplifier's noise gain vs. frequency, along with the location of the poles and zeros in the transfer function.

　　Figure 8. Amplifier noise gain transfer function.

　　As with the resistor noise density, the most accurate way to convert the output noise density of Figure 8 to voltage noise Vrms is to square the noise density, integrate over the entire spectrum, and then calculate the square root. However, inspection of the response reveals that a much simpler method produces only a small error. For most systems, the first zero and pole occur at a relatively low frequency than the second pole. For example, using the specifications shown in Table 1 and Table 2, the circuit has the following poles and zeros:

　　(8)

　　(9)

　　(10)

　　The peak noise is:

　　(11)

　　Note that fz1 and fp1 occur at relatively low frequencies compared to fp2. Simply assuming that the output noise is equal to the plateau noise from DC to fp2 (N2 from Equation 11) will greatly simplify the math required to calculate the output noise.

　　Under this assumption, the output noise is equal to the input noise density times the plateau gain times the ENBW, or fp2 × ​​π/2:

　　(12)

　　Once the equivalent output noise of all three noise sources is known, they can be combined to find the total system output noise. The three noise sources are independent of each other and are Gaussian, so they can be root summed square (RSS) rather than added. When combining multiple terms using RSS, if one term is about three orders of magnitude larger than the other terms, the result will be dominated by that term.

　　(13)

　　The response in Figure 8 clearly shows that the op amp’s noise bandwidth is much greater than the signal bandwidth. The extra bandwidth does nothing but contribute noise, so a low-pass filter can be added to the output to attenuate noise at frequencies outside the signal bandwidth. Adding a single-pole RC filter with a 34 kHz bandwidth reduces the voltage noise from μVrms to 45 μVrms and the total noise from 256 μVrms to only 52 μVrms.

　　Noise contributed by the programmable gain stage

　　If a PGA is added after the transimpedance amplifier, the noise at the output will be the sum of the PGA noise plus the TIA noise times the additional gain. For example, if the application requires gains of 1 and 10, and a PGA with a total input noise density of 10 nV/√Hz is used, the output noise contributed by the PGA will be 10 nV/√Hz or 100 nV/√Hz.

　　To calculate the total noise of the system, the noise contribution of the TIA and the noise contribution of the PGA can again be squared, as shown in Table 3. This example assumes that the PGA includes a 34 kHz filter. It can be seen that at a gain of 10, the noise contribution of the TIA multiplied by the PGA gain appears at the output of the PGA.

　　As we would expect, the output noise of a PGA operating at a gain of 10 is slightly greater than that of a PGA operating at a gain of 1.

　　Noise Advantages of a Single Gain Stage

　　Another approach is to use a transimpedance amplifier with programmable gain and eliminate the PGA stage entirely. Figure 9 shows a theoretical circuit with two programmable transimpedance gains (1 MΩ and 10 MΩ). Each transimpedance resistor requires its own capacitance to compensate for the input capacitance of the photodiode. To keep with the previous example, the signal bandwidth remains at 34 kHz for both gain settings. This means that a 0.47 pF capacitor should be chosen in parallel with the 10 MΩ resistor. In this case, the output voltage noise when using a 1 MΩ resistor is the same as Equation 12. When using a 10 MΩ transimpedance gain, the larger resistors result in higher Johnson noise, higher current noise (this time multiplied by 10 MΩ instead of 1 MΩ), and higher noise gain. Again, the three main noise sources are:

　　(14)

　　(15)

　　(16)

　　(17)

　　(18)

　　The total output noise is:

　　(19)

　　Adding a single-pole RC filter with a bandwidth of 34 kHz at the output reduces the noise, and the total system noise is 460 μVrms. Due to the higher gain, fp2 is closer to the signal bandwidth, so the noise reduction is not as significant as using a gain of 1 MΩ.

　　A summary of the noise performance of the two amplifier architectures is shown in Table 4. For a transimpedance gain of 10 MΩ, the total noise is approximately 12% lower than the two-stage circuit.