Resonator Design for Low Phase Noise Oscillators 1. Tapped Inductor Oscillator.ColpittsTapped LLoLoQuQuRpVc2. Clapp OscillatorResonator Design for Low Phase Noise Oscillators Phase noise can be estimated by a simplified version of Leeson’s equation: 202102sig ukTFlogPQωω⎡⎤⎛⎞⎢⎥⎜⎟Δ⎢⎥⎝⎠⎣⎦2sig res BIAS p (1) This is a noise power (single sideband) to carrier power ratio normalized to a 1 Hz bandwidth. Units are dBc/Hz. We saw that Vres = 2 IBIAS Rp is an approximation of the voltage across the resonator for the Colpitts oscillator. Qu = Rp/Xp = Bp/Gp for a simple parallel LRC resonant circuit. So, how can we modify the circuit to improve the noise to carrier ratio? From the equation (1), increasing Psig and Qu would help. The former increases carrier power, whereas the latter reduces noise generated by losses in the resonator and also does a better job of bandpass filtering the noise. We increase signal power by increasing voltage or current. 22PVorI orRα Inductor: Let’s assume that the inductor limits the Qu. This is typically true at frequencies below 10 GHz or so. How can we increase Qu or Rp at a given frequency? Qu = ωL/Rs for a series LR. S. Long January 19, 2005 1For a coil inductor (whether on a magnetic core or in air) with N turns, we can assume that the series resistance increases in proportion with length of the wire, therefore with N. The inductance will increase in proportion to N2. Thus, Qu increases in proportion to N. What about Rp? 22pssXLR2RRω== and thus should increase as N3. This is an ideal version and in reality may not scale as favorably. Skin effect will cause Rs to increase with frequency for example. Also, if we consider a spiral inductor on an RFIC, the inner turns have less area than the outer ones. They contribute less inductance per turn than a solenoidal coil, so the scaling is not obvious and requires E/M simulation to determine Qu and Rp. Another consideration is what Q definition should be used? There are several: 00sQL/REnergy StoredQPower DissipateddQdωωωωθωω==×==− (2abc) Analysis of oscillators has shown that the last definition, the phase slope with frequency, correlates best with phase noise performance. While these definitions may predict the same Q for simple resonators (series or parallel RLC), they do not agree very well with more complicated topologies or with transmission line resonators. How can we improve Psig? Any increase in IBIAS or Vres will help, but our device may begin to clip or breakdown. There are limits to how much voltage and current a device can handle safely. Simply increasing Rp will increase Vres, but may reach these limits in the device. So, a better approach is to modify the topology so that more power can be put into the resonator without exceeding the limits of the transistor. There are many such attempts in the literature of oscillators to do this, but two approaches will be briefly reviewed here. 1. Tapped inductor. 2. Clapp oscillator. S. Long January 19, 2005 21. Tapped Inductor Oscillator. Vres N1IBIASN1IBIASVC C1 C2 Here we see a modified version of the Colpitts. The tapped capacitor provides the feedback as before, but now there is a tapped inductor, an autotransformer with turns ratio N. • This has the effect of reducing the collector voltage, VC = Vres/(N+1). • It also reduces the load resistance at the collector. RL = Rp/(N+1)2. If the tapping ratio was 1:1, then VC = Vres/2 and RL = Rp/4. Thus, we can now increase IBIAS by a factor of 4, increasing the Psig by 16 times, without increasing the voltage swing at the collector. Of course, the noise contribution of the transistor generally also increases with IBIAS, so the net result is a 4 times improvement in phase noise to carrier ratio (6 dB). Scaling to higher N values may be beneficial, but it depends on how Rp scales with N. This can only be determined accurately by measurement or E/M simulation of the inductor. S. Long January 19, 2005 3Colpitts Tapped L Lo Lo Qu Qu Rp j2(1)ppRRN=+ Vc j(1resCVVN=+ ) S. Long January 19, 2005 42. Clapp Oscillator CCC2C1LRSZ(jω)REIBIASVCCCCC2C1LRSZ(jω)REIBIASVCC In this case, the resonator is modified by adding a capacitor CC in series with the inductor L. This allows the inductance to be increased (Qu, Rp higher) from its Colpitts value, L0. To keep the resonant frequency the same, the imaginary part of the impedance of the series LRC resonator must be the same as that of L0. If we increase the inductance by a factor P1, ()001/CjLP C jLωωω−= . (3) From this, we can determine CC: 201(1CCLPω=)− . (4) To determine what effect the series RLC resonator has on the load presented to the collector, RL, we must first find Z(jω), then Y(jω), then determine RL from 1/Re{Y(jω)}. Define (5) 0usQ L/R PL /Rωω==s uQ is the unloaded Q andsRcorresponds to the series resistance of the Clapp inductor, 1 W. Hayward, Introduction to Radio Frequency Design, Ch. 7, American Radio Relay League, 1994. S. Long January 19, 2005 5L = PL0. Eliminating CC, we get for Z: ()()000() 1/uCuuLP jQPLZj j PL CQQωωωωω+=+ − = (6) Y = 1/Z, so ()01/Re ( )uLuuQPRLYj Q Pωωω⎛⎞==+=⎜⎟⎝⎠0LQP (7) because P/Qu << 1. To determine what effect this has on the oscillator, we can use the assumption above about the how series resistance and inductance scale with number of turns2. If we want an inductor P times larger than the Colpitts inductor, we need to increase the number of turns by a factor of P. Thus, 0ssLPL R PR==. (8) Then, 0uusPLQ Q (Colpitts ) PPRω==. (9) First observation: The effective unloaded Q of the Clapp resonator is higher than the Colpitts when the scaling of resistance is included in the calculation. It can be shown that the phase slope also increases by the same amount. What about RL? Equation (7) described RL in terms of the unloaded Q of the Clapp inductor, L. But, if we are comparing with the Colpitts, we should refer RL to Qu of L0, not L. For the Colpitts, the parallel equivalent resistance is Rp. 200psuLupRX/R LQRLQ /P R / Pωω==== Observation #2: The loading that the collector sees with the Clapp resonator is reduced by a factor of Pbelow the Colpitts. Therefore, IBIAS can be increased by a factor of P while maintaining the same collector