Similarly, we have for the current I b in the NA output:
I b ¼ I b0 U a
ð Þexp j2πΔft þ φ u
ð
Þ
¼ I b0 U a0 Á 1 þ m U þ jψ U
ð
Þ exp j2πΔft þ φ u
ð
Þ
½
Á exp j2πΔft þ φ u
ð
Þ
: ð6:40Þ
With account of the last notation for I b , we can write: I b ¼ I b0 (1 + σ U Á m U + jψ U ).
Now we substitute expressions after linearization into the abbreviated fluctuation
equation and take into consideration the steady-state equation, then we write the
complex fluctuation equation in the form:
Y a p þ jΔf
ð
Þ U a m U þ jψ U
ð
Þ¼I 0b σ U Á m U þ jψ U
ð
Þ
ÀK a I 0a σ aU m U þ jψ U
ð
Þ þ I nb :
ð6:41Þ
The expression for the noise current I nb with account that I na ¼ (I anCOS +
jI anASIN ) exp (Àj2πΔft + φ U ), I nb ¼ (I bnCOS + jI bnASIN ) exp (Àj2πΔft + φ U ) can be
written as:
I nb ¼ I bnCOS þ jI bnASIN
ð
Þ À K a Á I anCOS þ jI anASIN
ð
Þ þ I nb þ ξ YYn :
ð6:42Þ
The expression (6.42) for the abbreviated fluctuation equation is complicate to
obtain of analytical functions. For simplicity in the future, we shall be limited by a
case, when we can neglect by the effect of the input voltage upon the input current. In
other words, the complex (in the general case) coefficient K a can be equated to zero.
6.5.5 The Coefficient of the Local Slope σ U
Values of the coefficient σ U can be accepted for the single-stage nonlinear amplifiers
constructed, for instance, in the common emitter circuit, from 0.1 to 0.99. The
coefficient σ U is related to the coefficient of the limit circle strength of OEO as
S σ ¼ 1 À σ U . The value S σ ¼ 0 or σ U ¼ 1 corresponds to the stability boundary of the
oscillator on the inertia-less AE and with the single oscillating circuit at absence of
the retarded feedback or the delay line. Then, in the steady-state point, the coefficient
σ U < 0. If there is the delay in the feedback loop, then the value of the coefficient σ U
may take both negative and positive values. The domain of permissible values of the
coefficient σ U is necessary defined from the oscillation stability condition in the
steady-state mode. The OEO stability condition in the steady-state point, as it
follows from the differential equation (6.29), in the case of OEO with the RF filter
with the single circuit and with FOS formed by the single optical fiber, is determined
as: σ U Á cos [fT FODL ] < 0. Coefficient σ U can be also expressed through the AE
average slope in the first harmonic, which can be determined as the ratio of AE
characteristic slope to the amplitude of the first harmonic voltage.
6.5 Differential Fluctuation Equations of OEO MZ
317
I b ¼ I b0 U a
ð Þexp j2πΔft þ φ u
ð
Þ
¼ I b0 U a0 Á 1 þ m U þ jψ U
ð
Þ exp j2πΔft þ φ u
ð
Þ
½
Á exp j2πΔft þ φ u
ð
Þ
: ð6:40Þ
With account of the last notation for I b , we can write: I b ¼ I b0 (1 + σ U Á m U + jψ U ).
Now we substitute expressions after linearization into the abbreviated fluctuation
equation and take into consideration the steady-state equation, then we write the
complex fluctuation equation in the form:
Y a p þ jΔf
ð
Þ U a m U þ jψ U
ð
Þ¼I 0b σ U Á m U þ jψ U
ð
Þ
ÀK a I 0a σ aU m U þ jψ U
ð
Þ þ I nb :
ð6:41Þ
The expression for the noise current I nb with account that I na ¼ (I anCOS +
jI anASIN ) exp (Àj2πΔft + φ U ), I nb ¼ (I bnCOS + jI bnASIN ) exp (Àj2πΔft + φ U ) can be
written as:
I nb ¼ I bnCOS þ jI bnASIN
ð
Þ À K a Á I anCOS þ jI anASIN
ð
Þ þ I nb þ ξ YYn :
ð6:42Þ
The expression (6.42) for the abbreviated fluctuation equation is complicate to
obtain of analytical functions. For simplicity in the future, we shall be limited by a
case, when we can neglect by the effect of the input voltage upon the input current. In
other words, the complex (in the general case) coefficient K a can be equated to zero.
6.5.5 The Coefficient of the Local Slope σ U
Values of the coefficient σ U can be accepted for the single-stage nonlinear amplifiers
constructed, for instance, in the common emitter circuit, from 0.1 to 0.99. The
coefficient σ U is related to the coefficient of the limit circle strength of OEO as
S σ ¼ 1 À σ U . The value S σ ¼ 0 or σ U ¼ 1 corresponds to the stability boundary of the
oscillator on the inertia-less AE and with the single oscillating circuit at absence of
the retarded feedback or the delay line. Then, in the steady-state point, the coefficient
σ U < 0. If there is the delay in the feedback loop, then the value of the coefficient σ U
may take both negative and positive values. The domain of permissible values of the
coefficient σ U is necessary defined from the oscillation stability condition in the
steady-state mode. The OEO stability condition in the steady-state point, as it
follows from the differential equation (6.29), in the case of OEO with the RF filter
with the single circuit and with FOS formed by the single optical fiber, is determined
as: σ U Á cos [fT FODL ] < 0. Coefficient σ U can be also expressed through the AE
average slope in the first harmonic, which can be determined as the ratio of AE
characteristic slope to the amplitude of the first harmonic voltage.
6.5 Differential Fluctuation Equations of OEO MZ
317