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.

Edward⁵ 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/Q₀ ) 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 publication¹¹ 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,
I₀ = 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 by¹¹:

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 l₁ and l₂ (l_1,2 = λ₀/4±Δl), 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 (Δl<<λ₀). 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
Y₀ = the characteristic admittance,
Z₀ = 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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