Design of the CMOS Voltage-Controlled Oscillator
4.2 Microwave Circuit Design of VCO
The traditional circuit design of microwave oscillator has separated two ways with one-port network and two-port network[1.8].
4.2.1 One-Port Negative Resistance Microwave Oscillator
Here we present some of the basic principles of operation for one-port negative oscillators; much of this material will also apply to two-port (transistor) oscillators.
Fig. 4.3 Circuit for a one-port negative-resistance oscillator
Fig. 4.3 shows the canonical RF circuit for a one-port negative-resistance oscillator, where Zin = Rin + jXin is the input impedance of the active device[1]. In general, this impedance is current (or voltage) dependent, as well as frequency dependent, which we can indicate by writing Zin(I,jw) = Rin(I,jw)+ jXin(I,jw). The device is terminated with a passive load impedance. ZL = RL + jXL. Applying Kirchhoff's voltage law gives (ZL+Zin)I=0 (4.2) If oscillation is occurring, such that the RF current I is nonzero, conditions must be satisfied:
RL + Rin =0, (4.3) XL + Xin =0. (4.4)
Since the load is passive. RL > 0 and equation (4.3) indicates that Rin < 0. Thus, while a positive resistance implies energy dissipation, a negative resistance implies an
energy source. The condition of equation (4.3) controls the frequency of oscillation.
The condition in equation (4.2), that ZL = -Zin for steady-state oscillation, implies that the reflection coefficients ΓL and Γin are related as
in The process of oscillation depends on the nonlinear behavior of Zin, as follows.
Initially, it is necessary for the overall circuit to be unstable at a certain frequency, that is, Rin(I,jw)+RL < o. Then any transient excitation or noise will cause an oscillation to build up at the frequency, w. As I increases, Rin(I,jw) must become less negative until the current I0 is reached such that Rin(I0,jw0)+RL = 0, and X in(I0,jw0)+XL(jw0) =0. Then the oscillator is running in a stable state. The final frequency, w0, generally differs from the startup frequency because Xin is current dependent, so that Xin(I,jw) ≠X
in(I0,jw0).
Thus we see that the conditions of equation (4.3) and (4.4) are not enough to guarantee a stable 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 ZT(I,s) = Zin(I,s) + ZL(s), then we can write a Taylor series for since ZT(I,s) must still equal zero if oscillation is occurring. In equation (4.6), s0 = jω0
is the complex frequency at the original operating point. Now use the fact that
Equation (4.7) then implies that ,
or >0. maximum oscillator stability. Cavity and dielectric resonators are often used for this purpose.
Effective oscillator design requires the consideration of several other issues, such as the selection of an operating point for stable operation and maximum power output, frequency-pulling, large-signal effects, and noise characteristics.
Fig. 4.4 Linear variation of the negative resistance as a function of the current amplitude
In a given oscillator design, the input impedance of the active device is known for small-signal conditions. A practical way of designing RL is to select the value of RL for maximum oscillator power. If the magnitude of the negative resistance is a linearly decreasing function of A which shows in Fig. 4.4, we can express Rin(A)in the form
Hence, the value of A that maximizes the oscillation power is found from 3 0
2 2
1 2
0 =
−
=
AM
A A dA R
dP
which gives the desired value of A, denoted by Ao,max, that maximizes the power. That is,
M
o A
A 3
2
max
, =
At Ao,max, the value of RIN(Ao,max) is
) 3 ( ,max R0
A
RIN o =−
Hence a convenient value of RL, which maximizes the oscillator power, is
3 R0
RL = (4.11) Observe that equation (4.11) is valid when the negative input resistance varies linearly with amplitude. In practice, the selection of RL according to equation (4.11) produces good results.
4.2.2 Two-Port Network of Microwave Oscillator
Fig. 4.5 Two-port oscillator model
The general block diagrams for two-port negative-resistance oscillators are shown in Figs. 4.5(a) and 4.5(b)[1.8].The transistor network is characterized by its S parameters, ZT is the terminating network impedance, and ZL is the load impedance.
Observe the notation used in Figs. 4.5(a) and 4.5(b), which shows that in an oscillator either port of the transistor can be used as the terminating port. Once the terminating port is selected, the other port is referred to as the input port. The load-matching network is connected to the input port, in agreement with the one-port notation used in Fig. 4.3.
When the two-port is potentially unstable, an appropriate ZT permits the two-port to be represented as a one-port negative-resistance device with input impedance ZIN, as shown in Fig. 4.3. The conditions for a stable oscillation are given by equation (4.3), (4.4), and (4.9). To start the oscillation, the value of RL, is selected according to equation (4.11) (i.e., RL = R0/3 or, in general, RL = |RIN(0,ω)|/3).
When the input port is made to oscillate, the terminating port also oscillates. The fact that both ports are oscillating can be proved as follows.
The input port is oscillating when
=1 and from equation (4.13) and (4.14) it follows that
=1 Γ ΓOUT T
which shows that the terminating port is also oscillating.
A design procedure for a two-port oscillator is as follows[8]:
1. Use a potentially unstable transistor at the frequency of oscillation ωo.
2. Design the terminating network to make |ΓIN| > 1. Series or shunt feedback can be used to increase |ΓIN|.
3. Design the load network to resonate ZIN, and to satisfy the start of oscillation condition in equation (11).That is, let
( )
ω0 IN( )
ω0L X
X =− (4.15) and
Frequency determination of LC tank
Device size determination of cross-coupled pair (Wp/Wn)
excess noise minimization and I tail current trimming
Low phase noise or symmetry of output waveform
Yes NO
3 R0
RL = or, in general,
( )
3 , 0ω
IN L
R = R (4.16)
This design procedure is popular due to its high rate of success. However, the frequency of oscillation will shift somewhat from its designed value at ω0. This occurs because the oscillation power increases until the negative resistance is equal to the load resistance and Xin varies as a function of A (i.e., as a function of the oscillation power). Also, there is no assurance that the oscillator is providing optimum power.