Thermal Noise Behavior of the Bridge Circuit

Abstract

This paper considers a connection between the deterministic and noisy behavior of nonlinear networks. Specifically, a particular bridge circuit is examined which has two possibly nonlinear energy storage elements. By proper choice of the constitutive relations for the network elements, the deterministic terminal behavior reduces to that of a single linear resistor. This reduction of the deterministic terminal behavior, in which a natural frequency of a linear circuit does not appear in the driving-point impedance, has been shown in classical circuit theory books (e.g. [1,2]). The paper shows that, in addition to the reduction of the deterministic behavior, the thermal noise at the terminals of the network, arising from the usual Nyquist-Johnson noise model associated with each resistor in the network, is also exactly that of a single linear resistor. While this result for the linear time-invariant (LTI) case is a direct consequence of a well-known result for RLC circuits, the nonlinear result is novel. We show that the terminal noise current is precisely that predicted by the Nyquist-Johnson model for R if the driving voltage is zero or constant, but not if the driving voltage is time-dependent or the inductor and capacitor are time-varying.

Limits to the fluctuation-dissipation theorem for nonlinear circuits

“…All derivations are exact, involving no approximations, except for the last. Further details, including a stochastic calculus derivation for the LTI bridge, treatment of a dual circuit, and more explicit calculations in some proofs have been omitted here for brevity but can be found in [5]. Other mathematically oriented studies of noise in nonlinear circuits include [8]- [11].…”

“…2) is associated with each resistor, then the spectrum of the shortcircuit terminal current in a matched bridge circuit is also that of a Nyquist-Johnson noise model for a single resistor of value . The verification can be done by standard frequency-domain techniques or by stochastic calculus [5]. The high-pass filtering of the branch is precisely balanced by the low-pass filtering of the branch, so that the terminal noise spectrum is flat.…”