Markovian modeling of classical thermal noise in two inductively coupled wire loops

Continuous Markov process theory is used to model classical thermal noise in two wire loops of resistances R1 and R2 , self-inductances L1 and L2 , and absolute temperature T, which are coupled through their mutual inductance M. It is shown that even though the currents I1 (t) and I2 (t) in the two loops become progressively noisier as M increases from 0 toward its upper bound (L1 L2 )1/2 , the fluctuation-dissipation, Nyquist, and conductance formulas all remain unchanged. But changes do occur in the spectral density functions of the currents Ii (t). Exact formulas for those functions are developed, and two special cases are examined in detail. (i) In the identical loop case (R1 =R2 =R and L1 =L2 =L), the M=0 'knee' at frequency R/2πL in the spectral density function of Ii (t), below which that function has slope 0 and above which it has slope -2, is found to split when M>0 into two knees at frequencies R/[2π(L±M)]. The noise remains white, but surprisingly slightly suppressed, at frequencies below R/[2π(L+M)], and it remains 1/f2 at frequencies above R/[2π(L-M)]. In between the two knee frequencies a rough '1/f-type' noise behavior is exhibited. The sum and difference currents I± (t)≡I1 (t)±I2 (t) are found to behave like thermal currents in two uncoupled loops with resistances R, self-inductances (L±M), and temperatures 2T. In the limit M→L, I+ (t) approaches the thermal current in a loop of resistance R and self-inductance L at temperature T, while I− (t) approaches (4kT/R)1/2 times Gaussian white noise. (ii) In the weakly coupled highly dissimilar loop case (R1 ≪R2 , L1 =L2 =L, and M≪L), I2 (t) is found, to a first approximation, not to be affected by the presence of loop 1. But the spectral density function of I1 (t) is found to be enhanced for frequencies ν≪R2 /2πL by the approximate factor (1+αν2 ), where α=(2πM)2 /R1 R2 . A concomitant enhancement, by an approximate factor of (1+2M2 R2 /L2 R1 )1/2 , is found in the high-frequency amplitude noise of I1 (t). An algorithm for numerically simulating I1 (t) and I2 (t) that is exact for all parameter values is presented, and simulation results that clarify and corroborate the theoretical findings are exhibited.