A Novel Integrated Power Module with Solid-State …
15
G D = −2
I dc
V r. f
.
I 1 (u)
I 0 (u)
.
(1 − cos θ)
θ
(17)
B D = ω(C d + C a ) ≈ ωC d
(18)
where I dc is the DC bias current flowing through the diode from a constant current
source and for thin avalanche zone practically C a C d .
Let us now apply the threshold approximation, i.e. near oscillation threshold v r.f
→ 0 and so u is very small and we can expand the modified Bessel’s functions I 0 (u)
and I 1 (u) in power series which yields:
G D | S D R = −3
α
θ 2 (1 − cos θ)I dc
1 −
9
8
α
2
θ 2 V
2
r. f
(19a)
G D | D D R = −
3
2
α
θ 2 (1 − cos θ)I dc
1 −
9
32
α
2
θ 2 V
2
r. f
(19b)
where the terms proportional to V
4
r. f and higher powers are neglected. Here: for SDR
diode, u =
3α
θ
V r. f as
ωW
v s
= θ ; while for DDR diode, u =
3α
2θ
V r. f as
ωW
v s
= 2θ (vide
Fig. 10) and note that
I 1 (u)
I 0 (u)
=
u
2
+
u
3
16
+ · · · .
Now, when a negative resistance device with an admittance: Y D = −G D + j B D
is connected to a passive circuit whose admittance is: Y C = G C + j B C , then the
condition for stable oscillation due to device-circuit interaction is given by:
G D (V, ω) − G C (ω) = 0
(20a)
B D (V, ω) + B C (ω) = 0
(20b)
The net zero conductance of the circuit-diode combination ensures the stability
of the oscillator and the frequency of oscillation is being determined by the resonant
frequency of the system, when the total susceptance is zero.
In general, an equivalent lumped circuit representation of IMPATT diode
(including its series resistance) and the passive RF circuit may be given by Fig. 11.
Referring to this circuit, the device and circuit admittances can be written as:
Y D = −(−G D − B
2
D R s ) + j B D
(21a)
Y C = g L +
−
j
ω 0 L c
(21b)
where B D and G D are given by Eqs. (18) and (19a, 19b), respectively, R s is the diode’s
parasitic series resistance, g L is the load conductance referred to the diode plane, and
L C is the circuit inductance.
15
G D = −2
I dc
V r. f
.
I 1 (u)
I 0 (u)
.
(1 − cos θ)
θ
(17)
B D = ω(C d + C a ) ≈ ωC d
(18)
where I dc is the DC bias current flowing through the diode from a constant current
source and for thin avalanche zone practically C a C d .
Let us now apply the threshold approximation, i.e. near oscillation threshold v r.f
→ 0 and so u is very small and we can expand the modified Bessel’s functions I 0 (u)
and I 1 (u) in power series which yields:
G D | S D R = −3
α
θ 2 (1 − cos θ)I dc
1 −
9
8
α
2
θ 2 V
2
r. f
(19a)
G D | D D R = −
3
2
α
θ 2 (1 − cos θ)I dc
1 −
9
32
α
2
θ 2 V
2
r. f
(19b)
where the terms proportional to V
4
r. f and higher powers are neglected. Here: for SDR
diode, u =
3α
θ
V r. f as
ωW
v s
= θ ; while for DDR diode, u =
3α
2θ
V r. f as
ωW
v s
= 2θ (vide
Fig. 10) and note that
I 1 (u)
I 0 (u)
=
u
2
+
u
3
16
+ · · · .
Now, when a negative resistance device with an admittance: Y D = −G D + j B D
is connected to a passive circuit whose admittance is: Y C = G C + j B C , then the
condition for stable oscillation due to device-circuit interaction is given by:
G D (V, ω) − G C (ω) = 0
(20a)
B D (V, ω) + B C (ω) = 0
(20b)
The net zero conductance of the circuit-diode combination ensures the stability
of the oscillator and the frequency of oscillation is being determined by the resonant
frequency of the system, when the total susceptance is zero.
In general, an equivalent lumped circuit representation of IMPATT diode
(including its series resistance) and the passive RF circuit may be given by Fig. 11.
Referring to this circuit, the device and circuit admittances can be written as:
Y D = −(−G D − B
2
D R s ) + j B D
(21a)
Y C = g L +
−
j
ω 0 L c
(21b)
where B D and G D are given by Eqs. (18) and (19a, 19b), respectively, R s is the diode’s
parasitic series resistance, g L is the load conductance referred to the diode plane, and
L C is the circuit inductance.