Bias Card low noise bias lines noise analysis

The following discussion is valid for BC Rev. F0, F1, F2, F3, F4 where LN_BIAS DACs are used in bipolar configuration and U16 and U17 are populated with AD797A.

The low noise detector bias lines in Rev F bias cards are driven by a bipolar DAC (MAX5444AEUB+) whose output is buffered by an opamp (AD797) in a non-inverting configuration (gain G=2), and inverted by a second opamp (AD797). These two signals are then each fed through a series resistance before going to the backplane and MDM connectors. The noise performance of these bias lines is determined by summing the noise contributions of:

• the input resistance to the op amp: R_i = ~6.3kΩ, mostly due to the output resistance of the DAC
• the matched feedback resistances of the buffer, which are internal to the DAC (RFB and INV pins): not listed in the datasheet, but measured as R_f1 = R_f2 = ~12kΩ
• the voltage noise of the AD797: e_n = 0.9 nV/√Hz at 1kHz
• the voltage noise of the DAC 2.5V reference voltage, ADR441: e_n,ref = ~50 nV/√Hz
• current noise of the AD797: i_n = 2.0 pA/√Hz, which multiplies both the input resistance to the non-inverting input (~6.3kΩ) and the parallel combination of the feedback resistors at the inverting input (~12kΩ each)

Using these numbers, the total noise at 1 kHz and T = 300K can be calculated as:

<math> \begin{align} v_{nt}(1\;\mathrm{kHz},300\;\mathrm{K}) &= \sqrt{ \quad 4kTR_i \quad + 4kT\left(R_{f_1}^{-1} + R_{f_2}^{-1}\right)^{-1} + \qquad e_n^2 \qquad + \quad\; e_{n,ref}^2 \quad\; + i_n^2 \left[ R_i^2 + \left(R_{f_1}^{-1} + R_{f_2}^{-1}\right)^{-2}\right]} \\ &\approx \sqrt{\left(10.2\;\mathrm{\frac{nV}\sqrt{Hz}}\right)^2 + \left(10.0\;\mathrm{\frac{nV}\sqrt{Hz}}\right)^2 + \left(0.9\;\mathrm{\frac{nV}\sqrt{Hz}}\right)^2 + \left(50\;\mathrm{\frac{nV}\sqrt{Hz}}\right)^2 + \;\qquad\left(17.4\;\mathrm{\frac{nV}\sqrt{Hz}}\right)^2\;\qquad}\\ &\approx 55\;\mathrm{\frac{nV}\sqrt{Hz}} \end{align} </math>

This characterizes the flat high frequency noise spectrum of the positive polarity bias line before the series resistance. The dominant source of noise is the voltage reference (e_n,ref). The ~233Ω of series resistance has a thermal noise of about 2 nV/√Hz, which summed in quadrature does not contribute to the overall noise. The noise of the negative bias line is small compared to the output noise of the first stage buffer in the positive line. The negative line doubles the total noise by inverting the output noise of the positive line and adding this noise to the output, but offsets this by also doubling the resistance in the bias line. The total current noise through a detector bias element winds up roughly the same as if it were a single ended bias line.

Also, the detector band of interest is up to only 10 Hz, where 1/f noise is important. The 1/f characteristics of the op amp and voltage reference are not well documented in the respective datasheets.