Markovian modeling of classical thermal noise in two inductively coupled wire loops
Abstract
Continuous Markov process theory is used to model classical thermal noise in two wire loops of resistances R_{1} and R_{2} , selfinductances L_{1} and L_{2} , and absolute temperature T, which are coupled through their mutual inductance M. It is shown that even though the currents I_{1} (t) and I_{2} (t) in the two loops become progressively noisier as M increases from 0 toward its upper bound (L_{1} L_{2} )^{1/2} , the fluctuationdissipation, Nyquist, and conductance formulas all remain unchanged. But changes do occur in the spectral density functions of the currents I_{i} (t). Exact formulas for those functions are developed, and two special cases are examined in detail. (i) In the identical loop case (R_{1} =R_{2} =R and L_{1} =L_{2} =L), the M=0 'knee' at frequency R/2πL in the spectral density function of I_{i} (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/f^{2} at frequencies above R/[2π(LM)]. In between the two knee frequencies a rough '1/ftype' noise behavior is exhibited. The sum and difference currents I_{+/} (t)≡I_{1} (t)+/I_{2} (t) are found to behave like thermal currents in two uncoupled loops with resistances R, selfinductances (L+/M), and temperatures 2T. In the limit M>L, I_{+} (t) approaches the thermal current in a loop of resistance R and selfinductance 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 (R_{1} <<R_{2} , L_{1} =L_{2} =L, and M<<L), I_{2} (t) is found, to a first approximation, not to be affected by the presence of loop 1. But the spectral density function of I_{1} (t) is found to be enhanced for frequencies ν<<R_{2} /2πL by the approximate factor (1+αν^{2} ), where α=(2πM)^{2} /R_{1} R_{2} . A concomitant enhancement, by an approximate factor of (1+2M^{2} R_{2} /L^{2} R_{1} )^{1/2} , is found in the highfrequency amplitude noise of I_{1} (t). An algorithm for numerically simulating I_{1} (t) and I_{2} (t) that is exact for all parameter values is presented, and simulation results that clarify and corroborate the theoretical findings are exhibited.
 Publication:

Physical Review E
 Pub Date:
 March 1997
 DOI:
 10.1103/PhysRevE.55.2588
 Bibcode:
 1997PhRvE..55.2588G