5.1.2 Closed PFB Circuit
Let us consider the circuit of positive feedback in OEO DM presented in Figs. 3.1a
and 3.2b. The laser pumping current can be written as the instantaneous value of the
pumping current i L ¼ J 1L (the first harmonic) in the electric laser input (or in the
output of the RF filter F). Now we present the instantaneous voltage values (mentioned on the OEO DM diagram (Fig. 5.1a) u PD (t), u F (t), u A (t), u C (t), u L (t) in the time
domain: for the photodetector (PD) u PD ¼ U 10PD Re [exp( j2πft À Φ PD )]; for the RF
filter (F) u F ¼ U 10F Re [exp( j2πft À Φ F )]; for the RF amplifier
(A) u A ¼ U 10A Re [exp( j2πft À Φ A )]; for the RF coupler u C ¼ U 10C Re [exp
( j2πft À Φ C )]; for the laser u L ¼ Z L i L ¼ U 10L Re [exp( j2πft À Φ L )]; here U 10MZ ;
U 10PD ; U 10F ; U 10A ; U 10C ; U 10L are amplitudes of first harmonics, Φ MZ ; Φ PD ; Φ F ;
Φ A ; Φ C ; Φ L are appropriate phase incursions), Z L is the input resistance of the laser,
and i 1L is the AC of the current in the QWLD input.
We introduce designation U PD , U F , U A , U C , U L , E L as the complex quantities of
the appropriate variables u PD (t), u F (t), u A (t), u C (t), u L (t). We introduce the complex
transfer function of RF FOLD K FOLD : K FOLD ¼ U PD /U L ¼ |
K FODL | exp (Àj2πfT FOS À jΦ FS ) where T FOS is the delay time in the fiber-optical
system (FOS), Φ FS in the phase incursion in RF FODL without the account of the
phase incursion in FOS, |K FODL | is the module of the RF FODL transfer function.
Let us introduce the complex transfer functions for blocks F, A, С, relatively, K F ,
K A , K C : K F ¼ U F /U PD ; K A ¼ U A /U F ; K C ¼ U L /U A . At that, for RF filter, we have:
K F ¼ U F =U PD ¼
jω
ð Þ 1=T EF
ð
Þ
jω
ð Þ
2 þ 1=T EF
ð
Þjω ð Þþ 2π f 0e
ð
Þ
2 , where the current oscillation frequency
ω ¼ 2πf, δ e is the losses in the filter F, δ e ¼ 1/T EF ¼ f 0e /( f 0e T EF ) ¼ f 0e /Q EF , T EF is
the time constant of the RF filter, Q EF is the Q-factor of the RF filter, f F0 ¼ f 0e is the
natural frequency of RF filter (or the resonance circuit), relatively. As the RF
amplifier (A), we take the only “ideal” (i.e., without account the output voltage of
the stage on the current in the stage input) non-inertia amplifying stage (Fig. 5.1a).
At that, the tube triode connected on the common anode circuit (or the stage on the
field-effect transistor (FET) connected on the common source circuit) can be such an
ideal amplifying stage. For simplicity, we neglect by the electron inertia and the
reaction of the grid current (input current) upon the RF filter and the anode reaction.
Then the equation of OEO (Fig. 3.2a), which couples the instantaneous values of the
amplifier input voltage and the output current i A (u PD ), can be written in the form:
y jω
ð Þu PD ¼ i A u PD
ð Þ,
ð5:3Þ
where y( jω) is the symbolic control conductance, which is (for typical RF oscillator)
a ratio of the symbolic load conductance y l to the FB coefficient k fb ; ω ¼ 2πf, jω is
the symbolic operator of differentiation [1].
Introducing the time constant T EF and the natural frequency f 0e for the RF filter
(Fig. 5.1a), we can write for the symbolic load conductance of the amplifier the
following equation (Eq. 3.48): y l ( jω) ¼ ( jω)
2 + (1/T EF )( jω) + (2πf 0e )
2 /[( jω)(1/T EF )].
5.1 Symbolic and Abbreviated Equations of OEO
207
Let us consider the circuit of positive feedback in OEO DM presented in Figs. 3.1a
and 3.2b. The laser pumping current can be written as the instantaneous value of the
pumping current i L ¼ J 1L (the first harmonic) in the electric laser input (or in the
output of the RF filter F). Now we present the instantaneous voltage values (mentioned on the OEO DM diagram (Fig. 5.1a) u PD (t), u F (t), u A (t), u C (t), u L (t) in the time
domain: for the photodetector (PD) u PD ¼ U 10PD Re [exp( j2πft À Φ PD )]; for the RF
filter (F) u F ¼ U 10F Re [exp( j2πft À Φ F )]; for the RF amplifier
(A) u A ¼ U 10A Re [exp( j2πft À Φ A )]; for the RF coupler u C ¼ U 10C Re [exp
( j2πft À Φ C )]; for the laser u L ¼ Z L i L ¼ U 10L Re [exp( j2πft À Φ L )]; here U 10MZ ;
U 10PD ; U 10F ; U 10A ; U 10C ; U 10L are amplitudes of first harmonics, Φ MZ ; Φ PD ; Φ F ;
Φ A ; Φ C ; Φ L are appropriate phase incursions), Z L is the input resistance of the laser,
and i 1L is the AC of the current in the QWLD input.
We introduce designation U PD , U F , U A , U C , U L , E L as the complex quantities of
the appropriate variables u PD (t), u F (t), u A (t), u C (t), u L (t). We introduce the complex
transfer function of RF FOLD K FOLD : K FOLD ¼ U PD /U L ¼ |
K FODL | exp (Àj2πfT FOS À jΦ FS ) where T FOS is the delay time in the fiber-optical
system (FOS), Φ FS in the phase incursion in RF FODL without the account of the
phase incursion in FOS, |K FODL | is the module of the RF FODL transfer function.
Let us introduce the complex transfer functions for blocks F, A, С, relatively, K F ,
K A , K C : K F ¼ U F /U PD ; K A ¼ U A /U F ; K C ¼ U L /U A . At that, for RF filter, we have:
K F ¼ U F =U PD ¼
jω
ð Þ 1=T EF
ð
Þ
jω
ð Þ
2 þ 1=T EF
ð
Þjω ð Þþ 2π f 0e
ð
Þ
2 , where the current oscillation frequency
ω ¼ 2πf, δ e is the losses in the filter F, δ e ¼ 1/T EF ¼ f 0e /( f 0e T EF ) ¼ f 0e /Q EF , T EF is
the time constant of the RF filter, Q EF is the Q-factor of the RF filter, f F0 ¼ f 0e is the
natural frequency of RF filter (or the resonance circuit), relatively. As the RF
amplifier (A), we take the only “ideal” (i.e., without account the output voltage of
the stage on the current in the stage input) non-inertia amplifying stage (Fig. 5.1a).
At that, the tube triode connected on the common anode circuit (or the stage on the
field-effect transistor (FET) connected on the common source circuit) can be such an
ideal amplifying stage. For simplicity, we neglect by the electron inertia and the
reaction of the grid current (input current) upon the RF filter and the anode reaction.
Then the equation of OEO (Fig. 3.2a), which couples the instantaneous values of the
amplifier input voltage and the output current i A (u PD ), can be written in the form:
y jω
ð Þu PD ¼ i A u PD
ð Þ,
ð5:3Þ
where y( jω) is the symbolic control conductance, which is (for typical RF oscillator)
a ratio of the symbolic load conductance y l to the FB coefficient k fb ; ω ¼ 2πf, jω is
the symbolic operator of differentiation [1].
Introducing the time constant T EF and the natural frequency f 0e for the RF filter
(Fig. 5.1a), we can write for the symbolic load conductance of the amplifier the
following equation (Eq. 3.48): y l ( jω) ¼ ( jω)
2 + (1/T EF )( jω) + (2πf 0e )
2 /[( jω)(1/T EF )].
5.1 Symbolic and Abbreviated Equations of OEO
207