*Last month, Part 1 of this article introduced novel self-injection- locked compact-coupled-planar-resonator (CCPR) oscillators. Part 2 concludes this article with more details on CCPR technology and some product examples. *

Edward^{5} proposed a novel, compact, high-Q multilayer integrable printed helical resonator that offers an optimum ratio of loaded quality factor to unloaded quality factor (Q_{L}/Q_{0} ) for minimum phase noise for a given VCO topology. *Figure 5* shows an integrable planar helical resonator coupled to coplanar waveguide (CPW) for VCO applications.^{6} But such high-Q helical resonators are limited in tuning range for given phase-noise, size, and cost requirements.^{5,6}

A recent publication^{11} described the design of an extended resonance oscillator (ERO) in which the resonator group delay is maximized for low phase noise. From ref. 6, the oscillator's loaded Q factor, Q_{L}, is

where

φ(ω) = the phase of the oscillator's open loop transfer function at a steady state and

τ_{d} = the group delay.

*Figure 6* shows a typical ERO circuit using an N-way power divider and combiner where the condition for coherent power combining can be obtained by making phase difference between successive device output ports (θ_{dn}) equal to the phase delay between the corresponding device input ports (θ_{gn}).

From ref. 7, Q_{L} is proportional to the absolute value of the group delay; therefore, the main design objective for the ERO is to maximize group delay by incorporating (N >2) multiple devices. From ref. 11, the group delay τ_{d} of the N-device ERO depicted in *Fig. 3* can be described by

where

I_{i} = the input current,

I_{0} = the output current, and

V_{0N} = the voltage at the output of the Nth device.

From ref. 11, the figure of merit (FOM), F, can be given by

where

τ_{d} = the group delay and

L = the insertion loss.

The relative noise contribution of the N-device ERO circuit with respect to a two-device ERO can be given by^{11}:

where

τ_{Dn} = the group delay of the N-device ERO and

L_{N} = the insertion loss of the N-device ERO.

From ref. 11, an eight-device ERO will yield about a 13-dB improvement in phase noise in comparison to a twodevice ERO, but there is a limitation in the number of devices for a given tuning range, noise factor, and power dissipation. The typical ERO circuit shown in *Fig. 6* is limited to narrow/fixed frequency applications, sensitive to temperature variations, and requires larger real estate and power, therefore, not a promising alternative to DROs.

The new approach presented here simplifies the limitation of the ERO by incorporating a stub-based tuning mechanism to maximize the group delay for a given operating mode. The present work describes a novel topology that improves the Q factor in compact size, and also suited for MMIC process. *Figures 7 and 8* show typical stub-tuned planar-coupled resonators (STPCRs). They use open and shorted stubs depending upon the injection strength for a given mode, operating frequency, and tuning range. *Figure 7 *shows open stubs of lengths l_{1} and l_{2} (l_{1,2} = λ_{0}/4Δl), which form the self-coupling mechanism (without using a coupling capacitor). The two unequal planar open stubs exhibit resonant frequencies below and above f0, in which the lengths of the resonators are symmetrically offset by the amount Δl (Δl0). This approach provides a selfcoupling mechanism without a lumped capacitor as a coupling element. The input admittance Y_{i}(ω) for this configuration is given by Eqs. 12-17.

where

Y_{0} = the characteristic admittance,

Z_{0} = the characteristic impedance,

v_{p }= the phase velocity,

φ = the phase shift,

γ(ω) = the propagation constant,

G_{i}(ω) = input conductance, and

B_{i}(ω) = input susceptance.

From ref. 15, R_{p} can be found by Eq. 18.

From refs. 14 and 16, C_{p} and L_{p} can be given by Eqs. 19 and 20 (*Fig. 4* ):

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From refs. 18, 19, and 20, and Q factor can be given by Eqs. 21-23. From refs. 18-23, R_{p}, C_{p}, L_{p}, , and Q_{L} are dependent on the value of the offset length Δl of the open-stub-tuned resonators. Similarly, the resonant characteristic of shorted stubs (l1,2=λ0/2Δl) is given by Eq. 24.

The expression of phase noise, according to the discussion appearing on p. 332 in ref. 9, can be given by Eq. 25, where:

m = the ratio of the loaded and unloaded Q.

From eqs. 23 and 24, m can be given in terms of the coupling coefficient by using Eq. 26:

From refs. 25 and 27, the minimum phase noise can be found by differentiating Eq. 11 with respect to m, and equating to zero for minimum value of phase noise as Eqs. 27 and 28:

where

(f_{m}) = the ratio of the sideband power in a 1-Hz bandwidth at f_{m} to the total power (in dB),

f_{m} = the offset frequency,

f_{0} = the oscillator center frequency,

f_{c} = the flicker corner frequency,

Q_{L} = the loaded Q,

Q_{0} = the unloaded Q,

F = the noise factor,

k = Boltzman's constant,

T = temperature (in degrees K),

P_{o} = average output power,

R = tuning diode equivalent noise resistance, and

K_{0} = voltage gain.

From ref. 28, for low-phase-noise applications, m_{opt} and _{opt} should be dynamically controlled and must lie in the vicinity of 0.5 and 1, respectively, for best phase noise.

*Figure 9* shows the typical layout of the DCO473542-5 (4.73 to 5.42 GHz), DXO810900-10 (8.1 to 10.9 GHz), and DXO10351090-5 (10.350 to 10.900 GHz) MCSTPR VCOs for the validation of the new approach by dynamically controlling m_{opt} and _{opt} for minimum noise figure. *Figure 10* shows simulated phase noise for a 10-GHz MCSTPR VCO (*Fig. 9*) offset 1 MHz from the carrier with respect to m_{opt} and _{opt}.

From Eqs. 25 and 28, for different values of noise figure F (F_{3} > F_{2} > F_{1}), the phase noise can be lowest for corresponding m_{opt} = 0.5 and _{opt} = 1, the plot is typically like a bathtub curve, shifted symmetrically about m_{opt}.

*Figure 11 *shows a measured phase noise plot of the discrete version of the MCSTPR VCO, typically 132 dBc/Hz offset 1 MHz from the carrier, within 3 dB of the simulated results.

For validation of the novel MCSTPR approach, example tunable DCO/DXO VCOs using a SiGe heterojunction- bipolar-transistor (HBT) active device (a model BFP 740 from Infineon) were fabricated on Rogers substrate material with a dielectric constant of 3.38 and thickness of 30 mils (microstripline). *Figure 12* shows a block diagram of the DCO/ DXO VCO series sources used for validating the the approach of achieving minimum phase noise performances over the operating band. The oscillators measure 0.35 x 0.35 x 0.16 in.

* Figure 13* shows the layout of the tunable DXO10351090-5 VCO, where the MCSTPR sets up optimum standing waves (within the resonator) and the noise impedance transfer function over the tuning range (10.350 to 10.900 GHz) by controlling mopt (by optimizing injection-locking) and opt (by optimizing mode-tuning using open and shorted stubs).^{23-28}

As depicted in *Fig. 14*, the dynamic mode-coupling mechanism exhibits 10 to 12 dB improvement in phase noise performance offset 10 kHz from 10-GHz carrier frequencies.

The measured RF power at 10 GHz is +1.2 dBm after compensating losses from the connectors and bias tee. The DXO circuit (*Fig. 13*) offers promising phase noise (better than 104 dBc/Hz offset 100 kHz) for broadband operation (10.350-10.900 GHz). *Figures 15-31* , on the online version of this article on www.mwrf.com, show measured performance for VCOs based on stub-tuned planar resonators, including the DCO473542-5 (4730-5420 MHz), DXO810900-10 (8100 -10900 MHz), and DXO10351090-5 (10350-10900 MHz).

This new approach to designing tunable oscillators with planar resonators yields compact VCOs with excellent phase-noise performance and in configurations that can be readily adapted to modern RF integrated circuit (RFIC) and MMIC semiconductor manufacturing processes.

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