High Order Harmonic Oscillators in Microwave and Millimeter-wave ...

2 Oscillators
resistance implies energy dissipation, a negative resistance implies an energy source. The
condition of (2.16) controls the frequency of oscillation. The condition in (2.14), that
Z L = −Z in for steady-state oscillation, implies that the reflection coefficients Γ L and Γ in
are related as
Γ L = Z L − Z 0
Z L + Z 0
= −Z in − Z 0
−Z in + Z 0
= Z in + Z 0
Z in − Z 0
= 1
Γ in
. (2.17)
The process of oscillation depends on the nonlinear behavior of Z in , as follows. Initially,
it is necessary for the overall circuit to be unstable at a certain frequency, that is,
R in (I,jω) + R L < 0. Then any transient excitation or noise will cause an oscillation to
build up at the frequency, ω. As I increases, R in (I,jω) must become less negative until the
current I 0 is reached such that R in (I 0 ,jω 0 )+R L = 0, and X in (I 0 ,jω 0 )+X L (jω 0 ) = 0. Then
the oscillator is running in a steady-state. The final frequency, ω 0 generally differs from the
start up frequency because X in is current dependent, so that X in (I,jω) ≠ X in (I 0 ,jω 0 ).
Thus we see that the conditions of (2.15) and (2.16) are not enough to guarantee
a steady-state of oscillation. In particular, stability requires that any perturbation in
current or frequency will be damped out, allowing the oscillator to return to its original
state. This condition can be quantified by considering the effect of a small change, δI,
in the current and a small change, δs in the complex frequency s = α + jω. If we let
Z T (I,s) = Z in (I,s) + Z L (s), then we can write a Taylor series for Z T (I,s) about the
operating point I 0 ,ω 0 as
Z T (I,s) = Z T (I 0 ,s 0 ) + ∂Z T
∂s
∣ δs + ∂Z t
∣ δI = 0, (2.18)
s0 ,I 0 ∂I s0 ,I 0
since Z T (I,s) must still equal zero if oscillation is occurring. In (2.18), s 0 = jω 0 is the
complex frequency at the original operating point. Now use the fact that Z T (I 0 ,S 0 ) = 0,
and that ∂Z T
∂s
= −j ∂Z T
, to solve (2.18) for δs = δα + jδω:
∂ω
δs = δα + jδω = −∂Z T/∂I
∂Z T /∂s
∣
s0 ,I 0
δI = −j(∂Z T/∂I)(∂Z ∗ T/∂ω)
|∂Z T /∂ω| 2 δI. (2.19)
Now if the transient caused by δI and δω is to decay, we must have δα < 0 when δI > 0.
Equation (2.19) then implies that
I m
{ ∂ZT
∂I
∂Z ∗ T
∂ω
}
< 0, (2.20)
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