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

2 Oscillators
is typically used. The reactive part of Z L is chosen to resonance the circuit,
X L = −X in . (2.24)
When oscillation occurs between the load network and the transistor, oscillation simultaneously
occurs at the output port, which we can show as follows. For steady-stage
oscillation at the input port, we must have Γ L Γ in = 1, as derived in (2.17). Then from
the definition of the S parameters, we have
1
Γ L
= Γ in = S 11 + S 12S 21 Γ T
1 − S 22 Γ T
= S 11 − ∆Γ T
1 − S 22 − ∆Γ T
(2.25)
where ∆ = S 11 S 22 − S 12 S 21 . Solving for Γ T gives
Then,
Γ T = 1 − S 11Γ L
S 22 − ∆Γ L
. (2.26)
Γ out = S 22 + S 12S 21 Γ L
1 − S 11 Γ L
= S 22 − ∆Γ L
1 − S 11 Γ L
, (2.27)
which shows that Γ T Γ out = 1, and hence Z T = −Z out . Thus the condition for oscillation
of the terminating network is satisfied. Note that the appropriate S parameters to use in
the above development are generally the large-signal parameters of the transistor.
Negative
resistance
device
Load
network
(tuning)
Transistor
[S]
Terminating
network
ΓL
ZL
Γin
Zin
Γout
(Zout)
ΓT
(ZT)
Fig. 2.5: Circuit for a two-port transistor oscillator.
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