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A low voltage and wide tuning range voltage controlled oscillator (VCO) at 2.4 GHz is developed for ISM applications. A design procedure with the parasitics of packaged devices considered is offered. A VCO with a 200 MHz tuning range for less than 2 V using a packaged varactor is demonstrated.
YaoHuang Kao
and ChunChieh Chien
Institute of Communication Engineering, National ChiaoTung University
HsinChu, Taiwan, ROC
Designing VCOs with low voltage and wide tuning range is an interesting topic for wireless applications.^{1,2} For low cost reasons, surfacemount devices are still widely used. In practice, these devices possess parasitics resulting in a self resonant frequency (SRF), compatible with the operating frequency, and causing some complexity in the frequency control. A design methodology, focusing on frequency control and device selection, is presented here. The goal is to build a VCO with a tuning range from 2.3 to 2.5 GHz with a tuning sensitivity larger than 100 MHz/V under low voltage operation (3 V).
Wideband VCOs have been studied by many researchers.^{38} Also, a number of commercial design software packages are available. In the case of the oneport negative resistance approach of Kurokawa^{7} , the onset of oscillation must satisfy the conditions where the negative resistance equals the positive external loss and the total reactance is zero. The configuration of a common emitter transistor with a series feedback capacitor is considered here. The required negative resistance is produced from the positive feedback of a capacitor in the emitter terminal. However, the suitable choice of the crucial elements, such as the feedback capacitor and varactor, is not clearly indicated. The problem is that the polarity of the reactance in the feedback arm may be changed and can lead to the disappearance of the negative resistance as the operating frequency exceeds its SRF. Besides, the typical range of capacitance of commercially available varactors using a silicon process varies from 3 to 50 pF at the test voltage (generally 2 V).^{9} The right choice of varactor is very crucial for design success, especially at low operating voltage. An analytic approach is presented here to obtain the optimum device choice and avoid the parasitics, especially for the varactor and the feedback capacitor. Once the key device properties are determined, the exact frequency of the VCO can be predicted by computeraided analysis.
FREQUENCY CONTROL
The schematic of the VCO with a series feedback capacitor at the emitter terminal is shown in Figure 1. It belongs to the class of Clapp oscillators. Three issues should be considered simultaneously during the design. They are the appropriate region of negative resistance, the tuning ratio (0.2/2.3 GHz = 8.7 percent) and the operating frequency (2.3 to 2.5 GHz). Except for the junction capacitance of the transistor, four elements in the series tank circuit are to be determined, which are the emitter series capacitor C_{e} , the DC blocking capacitor C_{b} , the varactor capacitance C_{v} and the external inductor L_{ex} . Each element has its own parasitic and SRF. In general, the smaller the capacitance, the higher the SRF. L_{e} , L_{b} and L_{v} are the equivalent parasitic inductors for C_{e} , C_{b} and C_{v} , respectively, as shown in Figure 2. For simplicity, the small parallel package capacitor( 0.1 pF) in the varactor model is neglected.

Negative Resistance
Note that the SRF of the capacitor C_{e} plays an important role in the behavior of the negative resistance. At the frequency of interest, a BJT transistor is represented by an equivalent model with only the junction capacitor C_{π } 2C_{je0 } + g_{m} _{F} and the base resistance r_{π} taken into account in the input port, where C_{je0 } is the junction capacitance between base and emitter at zero bias, g_{m} is the transconductance and _{F} is the forward base transit time.^{10} The input negative impedance is given as^{8}
when r_{π} >> 1/ωC_{π} . The real part of Z_{in} is negative at low frequency and increases monotonically with the zerocrossing point in the frequency domain roughly equal to the SRF of C_{e} . Above the SRF, the negative resistance disappears. Hence, the SRF of capacitor C_{e} acts as the limit for the negative resistance. It can be easily measured with a network analyzer (HP8753). The dependence of SRF as a function of the capacitance for a typical series of 0603 (= 0.06 x 0.03 in^{2} ) capacitors (Philips) is shown in Figure 3. The fitted curve satisfies the relation

For 2.5 GHz operation, the capacitor C_{e} should be less than 5 pF.
Tuning Range
It is obvious that the oscillation frequency in the series tank circuit is
where
The equivalent parasitic inductors L_{b} and L_{e} in C_{b} and C_{e} , respectively, satisfy Equation 2 derived from
The tuning ratio is given as
where
ΔC_{eff } = variation of the capacitance C_{eff} as a function of tuning voltage
Because of the series connection, the tuning capability depends on the ratios of C_{v} , C_{π} and C_{b} to C_{e} , which is now less than 5 pF. Fortunately, although under low operating voltage, the variation of a varactor capacitance from one to three volts is quite significant. For comparison, the parameters of the equivalent circuit for two typical varactors (Philips BB883 and BB142) are extracted from the oneport scattering parameter S_{11} and are listed in Table 1. The BB883 device has larger C_{v} than the BB142 varactor. The ratio of (C_{v} (1)C_{v} (3))/C_{v} (1) is nearly 50 percent in both cases. The notation C_{v} (V) means the capacitance of varactor at the tuning voltage V. Hence, the tuning ratio is roughly equal to 16.6 percent if C_{v} (1) = C_{e} and without the C_{π} and C_{b} effects. The ratio becomes 12.5 percent if C_{v} (1) = 2C_{e} . When C_{π} and C_{b} are taken into account, the ratio is reduced. In view of the availability of varactors with small capacitance values (3 pF), the range for the C_{e} selection is very narrow. In addition, a transistor with larger capacitance C_{¼} is preferred. Hence, a transistor with medium f_{t} is chosen. Figure 4 shows the dependences of the tuning ratio as a function of the emitter capacitor C_{e} for the BB142 and BB833 varactors, respectively, with C_{π} = 4.58 pF and two different C_{b} (10 and 100 pF). C_{π} = 4.58 pF is taken from a typical UHF transistor at low current operation (5 mA), such as BFS520 with C_{jeo } = 1.245 pF and _{F } = 8.61 ps. The figure shows clearly for the varactor BB142 that the capacitor C_{e} has a large degree of freedom from approximately 2 pF (mark P or Q) to 5 pF. However, for BB833, the useful range is small from 4 pF (mark X) to 5 pF. Even the case with C_{b } = 10 pF is impractical because the crossing point (mark Y) is over 5 pF.

Operating Frequency
The operating frequency is achieved by trimming the external inductor L_{ex} . However, the equivalent parasitic inductors from C_{b} and C_{e} and solder pads may lead to an impractical value of L_{ex} . These inductors are absorbed as parts of the external inductor. Because L_{b} is slightly proportional to C_{b} , L_{ex} is given from Equations 2 and 3 as
where
K_{0 } = constant 13.6π from Equation 2
For the BB142 varactor the useful range of L_{ex} is under 3 nH (mark R or S), while the required inductance for BB833 is nearly equal to zero (mark Z). Zero inductance is an impractical value for circuit realization. It implies that a varactor with large ratio Cv(1)/Ce (2 here), which would lead to the disappearance of the negative resistance, is not applicable in this case.
DESIGN PROCEDURE AND RESULTS
Based on the previous study, a procedure for the design of a wide tuning VCO is offered. First, C_{e} is chosen with its SRF slightly higher than the required upper limit (2.5 GHz). A transistor with medium ft is chosen to give the proper junction capacitance C_{¼} . As for C_{b} , a tradeoff between quality factor and tuning ratio is made. The larger the C_{b} , the wider the tuning. It leads to a small L_{ex} and thus low quality factor. To obtain a higher Q factor, a choice of C_{b} >10 C_{e} is suggested. For the desired tuning ratio, Cv(1) C_{e} is chosen (BB142). Once all the capacitors are chosen, L_{ex} is trimmed according to Equation 5.
The VCO is designed with the BFS520 transistor and BB142 varactor. C_{e } = 2 pF and C_{b } = 40 pF are chosen. The supply voltage is 3 V with a current consumption of 5.5 mA. The f_{T} is approximately 5.5 GHz. The external inductor is obtained from the pad. The tuning capability of the VCO is shown in Figure 5. The phase noise is shown in Figure 6. The output performance is listed in Table 2. It was observed that the grounding point of C_{e} is indeed crucial for the upper limit. Multiple vias are employed.

CONCLUSION
In this article, a low voltage, wide tuning VCO with applications to the 2.4 GHz ISM band is presented. The frequency control is clarified by special emphasis on the inherent SRF of the devices. The design procedure starts from selecting the feedback capacitor C_{e} at the emitter terminal whose SRF is taken to be larger than the highest operating frequency. Once chosen, the other elements in the series tank circuit are determined accordingly. The tuning ratio is closely related to the relative ratio of C_{v} to C_{e} . It is concluded that the SRF of capacitor C_{e} plays an important role in the frequency control and should be carefully chosen for higher frequency applications, such as in the 5.8 GHz ISM band. *
References
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9. Philips Products Data Book, September 1993.
10. P.R. Gray and R.G. Mayer, Analysis and Design of Analog Integrated Circuit, John Wiley & Sons Inc., New York, 3rd edition, 1997, Chapter 3, pp. 215217.
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