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Corel Ventura R/Microwave Circuit Design for Wireless Applications. Ulrich L. Rohde, David P. Newkirk Copyright © 2000 John Wiley & Sons, Inc. ISBNs: 0-471-29818-2 (Hardback); 0-471-22413-8 (Electronic) 5 RF/WIRELESS OSCILLATORS 5-1 INTRODUCTION TO FREQUENCY CONTROL Practically all modern telecommunication and test equipment uses frequency-control tech- niques based on frequency synthesis, the production of the many frequencies involved in a radiocommunication systems modulation, transmission, reception, and demodulation func- tions through the combination and mathematical manipulation of a very few input frequen- cies. Although the frequency synthesis techniques now ascendant in wireless systemsphase-locked loops and, less commonly, direct digital synthesizersare funda- mentally different, all are ultimately based on R" oscillators. This chapter covers oscillator theory, evaluation, and design. Two types of oscillators are needed in a phase-locked-loop system ("igure 5-1). One, typically a crystal oscillator, generates the synthesizers reference signal. As "igure 5-1 reflects, the reference oscillator may be built into the synthesizer IC in highly integrated systems. The other oscillator, a voltage-controlled oscillator (VCO), is varied in frequency by the system to produce the synthesizers output signal. Although designing good oscillators remains somewhat like black magic or a special art, we will show that mathematics and CAD tools, applied in conjunction with practical experience, can keep the oscillator design process well under control. 5-2 BACKGROUND The oscillator was probably first discovered by people who wanted to build an amplifier back in the vacuum-tube days. Its modern equivalent is shown in "igure 5-2. Here we see an active device, a bipolar transistor or field-effect transistor, with a tuned input and output circuit. Under ideal circumstances, the voltage at the input is 180° out of phase and the amplifier is stable. Because of the unavoidable feedback capacitance, a certain amount of energy is transferred from the output to the input at about 90° out of phase. If one of the tuned circuits is detuned to the point where the phase shift is 180°, oscillation will occur. This means that the small energy from the output will be amplified, brought back to the input in phase and 716 Figure 5-1 Block diagram of a modern, integrated frequency synthesizer. In this case, the designer has control over the VCO and the loop filter; the reference oscillator is part of the chip. In most cases (up to 2.5 GHz), the dual-modulus prescaler is inside the chip. 717 718 RF/WIRELESS OSCILLATORS Figure 5-2 Schematic diagrams of RF connections for common oscillator circuits (dc and biasing circuits not shown) [1]. further amplified, causing the stage to take off and oscillate. Unless there is some mechanism to limit the amplitude of the oscillation at the input or output, its amplitude can theoretically increase to a level sufficient to destroy the device through breakdown effects or thermal runaway. The criterion for oscillation was first described by Barkhausen [see Eq. (5-6) in the next section]. The conditions for oscillation exist when the feedback gain is high enough to cancel all losses and the difference between forward gain and reverse gain is less than zero. This implies also that oscillator circuits ("igure 5-2) provide a negative resistance, which is responsible for oscillation. The following mathematical treatment explains this. 5-3 OSCILLATOR DESIGN 719 5-3 OSCILLATOR DESIGN A phase-locked loop generally has two oscillators: the oscillator at the output frequency and the reference oscillator. The reference oscillator at times can be another loop that is being mixed in, and the voltage-controlled oscillator (VCO) is controlled by either the reference or the oscillator loop. The VCO is one of the most important parts in a phase-locked-loop system, the performance of which is determined only by the loop filter at offsets inside the loop bandwidth, and only by the quality of the VCO design at offsets outside the loop bandwidth. To some designers, VCO design appears to be magic. Shortly, we will go through the mathematics of the oscillator and some of its design criteria, but the results have only limited meaning. This is due to component tolerances, stray effects, and, most of all, nonlinear performance of the oscillator device, which can be modeled with only a certain degree of accuracy. However, after building oscillators for awhile, a certain feeling will be acquired for how to do this, and certain performance behavior will be predicted on a rule-of-thumb basis rather than on precise mathematical effort. "or reasons of understanding, we will deal with the necessary mathematical equations, but we consider it essential to explain that these are only approximations. 5-3-1 Basics of Oscillators An electronic oscillator is a device that converts dc power to a periodic output signal (ac power). If the output waveform is approximately sinusoidal, the oscillator is referred to as sinusoidal. There are many other oscillator types normally referred to as relaxation oscilla- tors. "or application in frequency synthesizers, we will explore only sinusoidal oscillators for reasons of purity and noise-sideband performance. All oscillators are inherently nonlinear. Although the nonlinearity results in some distor- tion of the signal, linear analysis techniques can normally be used for the analysis and design of oscillators. "igure 5-3 shows, in block diagram form, the necessary components of an oscillator. It contains an amplifier with frequency-dependent forward loop gain G(jω) and a frequency-dependent feedback network H(jω). The output voltage is given by V G( jω) = in (5-1) Vo 1 + G( jω)H( jω) Figure 5-3 Block diagram of an oscillator showing forward and feedback loop components. 720 RF/WIRELESS OSCILLATORS "or an oscillator, the output Vo is nonzero even if the input signal Vi = 0. This can only be possible if the forward loop gain is infinite (which is not practical), or if the denominator 1 + G(jω)H( jω) = 0 (5-2) at some frequency ωo. This leads to the well-known condition for oscillation (the Nyquist criterion), where at some frequency ωo ( ω ) ( ω ) = − (5-3) G j o H j o 1 That is, the magnitude of the open-loop transfer function is equal to 1: ( ω ) ( ω ) = (5-4) |G j o H j o | 1 and the phase shift is 180°: ( ω ) ( ω ) = (5-5) arg[G j o H j o ] 180° This can be more simply expressed as follows: If in a negative-feedback system, the open-loop gain has a total phase shift of 180° at some frequency ωo, the system will oscillate at that frequency provided that the open-loop gain is unity. If the gain is less than unity at the frequency where the phase shift is 180°, the system will be stable, whereas if the gain is greater than unity, the system will be unstable. This statement is not correct for some complicated systems, but it is correct for those transfer functions normally encountered in oscillator design. The conditions for stability are also known as the Barkhausen criteria, which state that if the closed-loop transfer function is V µ o (5-6) = Vi 1 − µβ the system will oscillate provided that µβ = 1. This is equivalent to the Nyquist criterion, the difference being that the transfer function is written for a loop with positive feedback. Both versions state that the total phase shift around the loop must be 360° at the frequency of oscillation and the magnitude of the open-loop gain must be unity at that frequency. The following analysis of the relatively simple oscillator shown in "igure 5-4 illustrates the design method. The linearized (and simplified) equivalent circuit of "igure 5-4 is given in "igure 5-5. hrb has been neglected, and 1/hob has been assumed to be much greater than the load resistance RL and is also ignored. Note that the transistor is connected in the common-base configuration, which has no voltage phase inversion (the feedback is positive), so the conditions for oscillation are ( ω ) ( ω ) = (5-7) |G j o H j o 1| and ( ω ) ( ω ) = (5-8) arg[G j o H j o ] 0° 5-3 OSCILLATOR DESIGN 721 Figure 5-4 Oscillator with capacitive voltage divider. The circuit analysis can greatly be simplified by assuming that 1 hibRE (5-9) << ω( + ) + C2 C1 hib RE and also that the Q of the load impedance is high. In this case the circuit reduces to that of "igure 5-6, where VoC1 (5-10) V = + C1 C2 and 2 h R C + C R = ib E 1 2 (5-11) eq + C hib RE 1 "igure 5-7 shows how the input impedance of an oscillator circuitin this case, a BJT Colpitts oscillator minus its resonatorsatisfies this condition for oscillation. Then the forward gain Figure 5-5 Linearized and simplified equivalent circuit of Figure 5-4. 722 RF/WIRELESS OSCILLATORS Figure 5-6 Further simplification of Figure 5-4, assuming high-impedance loads. h α ( ω) = fb = (5-12) G j ZL ZL hib hib and C1 H( jω) = (5-13) + C1 C2 where = 1 = 1 + 1 + 1 + 1 YL (5-14) ZL jωL Req RL jωC Figure 5-7 Calculated input impedance of a 200-MHz BJT Colpitts oscillator with its resonator removed. (Figure 5-46 shows an equivalent graph with the resonator present.) 5-3 OSCILLATOR DESIGN 723 A necessary condition for oscillation is that arg[G( jω)H( jω)] = 0° (5-15) Since H does not depend on frequency in this example, if arg(GH) is zero, the phase shift of the load impedance ZL must also be zero.
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