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I'm using an ac coupled noise matching transformer to measure the voltage noise of a low ohmic sample as a function of bias current. The complete circuit is in an enshielded enclosure. When the transformer primary is fully floating (without R2), the circuit is quite susceptible to external noise. The circuit is shown below

schematic

simulate this circuit – Schematic created using CircuitLab

However when a high value resistor to ground (R2) is connected to the primary, the pickup drops significantly.

Why is that?

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The transformer inevitably has capacitance between primary and secondary windings and, any common mode signal fluctuations from the "battery powered floating bias current source", will cause a differential signal to be delivered to the single-ended pre-amp via the inter-winding capacitance.

The 1 Mohm resistor provides an alternative shunt path for these common mode signals and thus you see noise reduce. If your pre-amp were differential and balanced you might find that things improve significantly.

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Due to C1, or rather the lack thereof in the other input, the two input impedances are not truly symmetrical (relative to ground), reducing the benefits of a truly differential input.

Install a 2uF capacitor in the place of C1 and also on the other input line.

The improved symmetry is likely to reduce the noise.

Everything @Andy said is still perfectly valid, so combining both methods may provide the least noise.

A second 1M resistor to ground on the other transformer pin may increase symmetry even further!

Finally, ensure that the two low voltage wires/traces are as close to each other as possible, and, if possible, twisted - even if your enclosure is electrically shielded, any external magnetic field (e.g. from nearby mains wiring in a wall) can still penetrate your enclosure and induce a magnetic field (and therefore a voltage) in the loop you have drawn on the left.

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