the total, so-called, oscillating characteristic
4 is defined by existence of blocking
interstage connections, which extracts the DC components of signals. In other
words, if the first transistor “falls” into the nonlinear mode, we shall have the
dependence of the DC component of the output voltage of the amplitude of the
input signal, which will lead to the operating point displacement on the transfer
characteristic. In the future, for simplicity sake, we consider the single amplifier
stage with the transfer characteristic, i C ¼ αu B À βu
3
B . The transfer slope of the
nonlinear amplifier can be defined as S NA ¼ i C =u B ¼ α À βu
2
B . Let the harmonic
signal in the form u B ¼ U B cos(ωt À ϕ 0 ), where U B is the amplitude, ω, φ are
frequency and phase, acts in the stage input. At that, the oscillating characteristic
defined as the function of the first harmonic amplitude of the collector current I C ¼ f
(U B ), is obtained as the average value or as the ratio of the integral on ξ ¼ ωt À ϕ 0
parameter to π, and it is equal:
I K ¼
1
π
Z π
Àπ
αu B cos ξ À β u B cos ξ
ð
Þ
3
h
i
cos ξdξ ¼ αU B À 3=4
ð ÞβU
3
B :
ð3:40Þ
The average (during the period) slope on the first harmonic amplitude can be
defined as: S NA1 ¼ I C =U B ¼ α À 3=4
ð
ÞβU
2
B . In the general case, the active element
(AE) can be inertial. The expression for the instantaneous value of the u PD voltage in
the PD output or in the NA input can be written in the form: u PD ¼
K PD E
2
0L K FODL U 0M exp j2π f 0 t
ð
Þ , where the transfer function K FOLD is defined as
K FODL ¼ K PD M
2
z B 1M exp j2π f 0 T FOS þ T 1M þ ΔT M =2
ð
Þ
½
ð 3:41Þ
where T 2M , T 1M are delays in the optical channels OC2 and OC1, relatively, in the
MZ modulator T 2M , ΔT M ¼ T 1M À T 2M .
The expression for the instantaneous value of NA output voltage u NA or in the
input of RF filter can be written as: u NA ¼ S NA E
2
0L K BLZ u PD t À T FOS
ð
Þ
Â
Ã
.
3.3.7 The Feedback Network in OEO
3.3.7.1 The Positive Feedback Network in OEO (Spanned a Laser)
The series-connected fiber optical system (FOS) including in general case the optical
filters, the optical amplifiers and the optical modulators, the photodetector, the
nonlinear RF amplifier, and the RF filter with the natural frequency and the
4 Under “oscillating characteristic” of the RF amplifier we understand the function of the first
harmonic of the output (collector) current versus the amplitude of the first harmonic of input
(base) voltage.
3.3 Mathematical Description of Transfer Functions of OEO Components
113
4 is defined by existence of blocking
interstage connections, which extracts the DC components of signals. In other
words, if the first transistor “falls” into the nonlinear mode, we shall have the
dependence of the DC component of the output voltage of the amplitude of the
input signal, which will lead to the operating point displacement on the transfer
characteristic. In the future, for simplicity sake, we consider the single amplifier
stage with the transfer characteristic, i C ¼ αu B À βu
3
B . The transfer slope of the
nonlinear amplifier can be defined as S NA ¼ i C =u B ¼ α À βu
2
B . Let the harmonic
signal in the form u B ¼ U B cos(ωt À ϕ 0 ), where U B is the amplitude, ω, φ are
frequency and phase, acts in the stage input. At that, the oscillating characteristic
defined as the function of the first harmonic amplitude of the collector current I C ¼ f
(U B ), is obtained as the average value or as the ratio of the integral on ξ ¼ ωt À ϕ 0
parameter to π, and it is equal:
I K ¼
1
π
Z π
Àπ
αu B cos ξ À β u B cos ξ
ð
Þ
3
h
i
cos ξdξ ¼ αU B À 3=4
ð ÞβU
3
B :
ð3:40Þ
The average (during the period) slope on the first harmonic amplitude can be
defined as: S NA1 ¼ I C =U B ¼ α À 3=4
ð
ÞβU
2
B . In the general case, the active element
(AE) can be inertial. The expression for the instantaneous value of the u PD voltage in
the PD output or in the NA input can be written in the form: u PD ¼
K PD E
2
0L K FODL U 0M exp j2π f 0 t
ð
Þ , where the transfer function K FOLD is defined as
K FODL ¼ K PD M
2
z B 1M exp j2π f 0 T FOS þ T 1M þ ΔT M =2
ð
Þ
½
ð 3:41Þ
where T 2M , T 1M are delays in the optical channels OC2 and OC1, relatively, in the
MZ modulator T 2M , ΔT M ¼ T 1M À T 2M .
The expression for the instantaneous value of NA output voltage u NA or in the
input of RF filter can be written as: u NA ¼ S NA E
2
0L K BLZ u PD t À T FOS
ð
Þ
Â
Ã
.
3.3.7 The Feedback Network in OEO
3.3.7.1 The Positive Feedback Network in OEO (Spanned a Laser)
The series-connected fiber optical system (FOS) including in general case the optical
filters, the optical amplifiers and the optical modulators, the photodetector, the
nonlinear RF amplifier, and the RF filter with the natural frequency and the
4 Under “oscillating characteristic” of the RF amplifier we understand the function of the first
harmonic of the output (collector) current versus the amplitude of the first harmonic of input
(base) voltage.
3.3 Mathematical Description of Transfer Functions of OEO Components
113