Model And Phase-Noise Perorance Model, Part 2 By understanding the basic sources o phase noise, it is possible to accurately odel a PLL with the help o coercial CAE progras. Eric Drucker PLL Consultants, 7701 56th Ave. NE, Seattle, WA 98115-6301; (206) 525-0674, FAX: (425) 290-1600, e-ail: linerc@sprintail.co. Signal Ideal receiver or spectru analyzer Ideal phase odulator Low-requency (baseband) receiver or spectru analyzer PHASE-LOCKED loops (PLLs) and their iportance to odern counications were detailed in the irst part o this article series (see Microwaves & RF, Noveber 1999, p. 69). In that article, a basic PLL odel was presented. The loop dynaics were odeled to deterine the open- and closed-loop transer unctions and the various loop paraeters (such as open-loop gain-crossover, phase argin, closed-loop bandwidth, etc.). An exaple was presented to illustrate these concepts. In Part 2, the basics o phase noise will be reviewed. Using the atheaticalanalysis sotware MathCAD, along with the previous exaple, it will be possible to show how the various noise sources in a PLL can be odeled. A novel ethod using PSPICE to odel the noise sources will also be shown. An aplitude and phase-odulated sinusoidal signal can be written as ollows: V(t) = V [ 1 + v (t)] 0 a { sin[ 2πct + θ(t) ]} (6) where: V 0 = the aplitude, c = the carrier requency, L( ) single-sideband phase noise Noise dbc 1-Hz bandwidth c c + S ( ) double-sideband phase noise 1 Hz Log scale 1 MHz 12. A spectru analyzer can be used to evaluate singlesideband (SSB) or double-sideband phase noise. 73 v a (t) = the aplitude-odulation (AM) coponent, and (t) = the phase-odulation (PM) coponent. For the purposes o this discussion, the AM coponent will be disregarded. V(t) = V { sin[2π t + θ(t)] } (7) Modeling a sinusoidal-angle odulation with a rate o, yields: θ(t) = sin(2πt) ; β = 0 c V() t = Vo [ sin(2πct) + βsin (2π t) ] (9) For 1 (sall-angle odulation) and applying trigonoetric identities: V() t = Vo [ sin(2πct) + β {sin[2 π(c + )] 2 sin[2 π( )]}] (10) c where: = the odulation requency, = the peak requency-odulation (8)
Rando walk FM ( -4 ). Environental actors such as echanical shock, vibration, and teperature. Flicker FM ( -3 ). Resonator noise and/or active coponent noise in oscillators. White FM ( -2 ). Broadband noise shaped by resonator Q in oscillators. Flicker phase ( -1 ). Active coponent noise. L( ) White phase ( 0 ). Broadband noise ro apliier stages and coponents. Note that licker phase and FM noise are dierent by 20 db/decade. The sae is or white phase and requency noise. -4 rando walk FM -3 licker FM -2 white FM -1 licker phase Toward DC or 0 oset w.r.t. carrier requency 0 white phase Log scale 13. Phase noise consists o several coponents, including rando-walk FM, licker-noise FM, white-noise FM, licker phase noise, and white phase noise. (FM) deviation, and = the odulation index. This shows that a sall-angle deviation gives rise to sidebands on each side o the carrier at an aplitude o /2. Extending this idea, noise can be treated as an ininite nuber o single FM sidebands. The distribution o these noise sidebands as a unction o oset requency can be expressed in dierent ways (Fig. 12). One way is to use an ideal receiver or spectru analyzer at RF with a 1-Hz resolution-bandwidth ilter. The total power o the signal would irst be easured and, since the noise is sall, this is essentially equal to the carrier power. Then the receiver would be tuned to a particular oset ( ) ro the carrier, and the phasenoise power is easured. The ratio o these two easureents, expressed in decibels, is the noralized power-spectral density (PSD) in one sideband reerred to the carrier at a requency oset. This is known as the singlesideband (SSB) phase noise, L( ), with units o Hertz 1 or expressed in decibels relative to the carrier level as dbc/hz. A plot o the phase noise as unction o the oset is coonly shown in data sheets in order to characterize oscillators and requency synthesizers. S() (a) Feedback phase-noise oscillator odel -1 Phase noise Resonator Noiseless apliier (d) Phase-noise-to-requency-noise transoration S() 1/ Phase noise Frequency noise -3-2 S () -1 0-2 1/ /2Q 1/ /2Q Another ethod is to deodulate the signal with an ideal phase deodulator. The output o the phase deodulator is the baseband phase noise and can be analyzed with a low-requency spectru analyzer, again with a 1-Hz resolution-bandwidth ilter. The resulting plot as a unction o baseband or oset requency is the double-sided phase-noise spectru, S ( ), expressed in (radians) 2 /Hz. The doublesideband phase noise is twice that o (or 3 db ore than) the SSB phase noise. One can integrate the area under the double-sideband phase-noise curve, over a speciic bandwidth ( 1 to 2 ) to obtain the root-ean-square (RMS) phase noise and, by extension, the RMS requency noise. Fro the RMS phase or requency noise, the L() (c) Leeson's Equation L() = FkTB 1 2P 3 (b) Output phase noise Low Q -3 L() -2 FkT/2P 1/ /2Q High Q -3-1 FkT/2P /2Q 1/ c 2 1/ 1 4QL 2 + 2 ( c ) 2 QL + 1/ + 1 Q = the resonator loaded Q, P = the resonator power, 1/ = the licker-noise corner, = the oset ro carrier, c = the RF requency, F = the oscillation noise igure, k = Boltzan constant, and T = the teperature. 14. An oscillator can be odeled as a eedback syste (a) that produces noise as a unction o oscillator Q (b). The phase noise can be described in ters o Leeson s equation (c) and transored to requency noise (d). ( 74
Input spectru Desired signal (a) Translation o LO phase noise LO phasenoise Mixer spectru Desired signal Phase noise 110 dbc/hz d a1 c (b) LO phase noise asks saller signal (c) Adjacent-channel easureent Signal generator 1 at adjacent channel Mixer Aplitude Signals at RF input to ixer Aplitude d LO a1 Aplitude Interering signal energy Mixer output spectru Aplitude 1 2 1 LO 2 LO 1 Phase noise asks the lower aplitude signal Local oscillator with phase noise Aplitude Signal generator 2 at desired channel 2 Receiver under test tuned to 2 15. An LO s phase noise can aect a receiver s adjacent-channel rejection (a) by asking low-level signals (b). Adjacent-channel perorance can be evaluated with a pair o signal generators (c). Reerence oscillator Sxo() R Srd() Phase detector Sd() Spd() SF(s)() N Loop ilter F(s) VCO Svco() Output o PLL Sxo() = the reerence oscillator phase noise Srd() = the reerence divider phase noise Spd() = the phase-detector phase noise Sd() = the ain divider phase noise SF(s)() = the loop-ilter phase noise SVCO() = the VCO phase-noise density 16. A variety o noise sources aect the perorance o a PLL. 10-MHz reerence oscillator S xo () R = 1/10 Phase detector K p = 0.5/rad Loop ilter F(s) R1 = 5.62k Reerence r = 1 MHz Sd() C3 = 0.068 C2 = 0.47 R2 = 1.13k N = 1/1000 VCO + K v = 10 MHz/V S vco () Output 17. In this exaple, the PLL is assued to have only three noise sources VCO noise, reerence oscillator noise, and ain divider noise. 1/s RMS tie jitter can be coputed. When looking at phase-noise plots, it should be noted that a zero requency oset or DC is the carrier. The integrated phase noise in ters o RMS radians can be expressed as eq. 11: θ while the integrated requency noise in ters o RMS Hz can be expressed as eq. 12: 2 RMS = S ( ) d (12) φ 2 1 When the phase noise is plotted in decibels versus log requency, various regions o the phase-noise curve can be identiied. These regions have slopes o 0, 1/ (10 db/decade), 1/ 2 (20 db/decade), 1/ 3 (30 db/decade), etc. (Fig. 13). The lat or zero-slope region corresponds to white phase noise o theral origin. This theral, resistive, or Johnson noise has a Gaussian aplitude distribution, constant with requency. Apliier noise igure is a aniestation o this theral noise. Close to DC or a zero-requency oset, there is a region o 1/, or licker noise. This is believed to coe ro irregularities in the seiconductor structure. Typically, the 1/ corner is between 1 and 10 khz. Frequency dividers and apliiers exhibit only 0 and 1/ slope regions. Oscillators can have all regions. The general phase-noise equation shown below (with the subscript dropped) is typically used or double-sideband noise [S ( )] expressed in power: k o = 2 S φ( )d (11) RMS 1 S φ () = k1 k2 k3 k + + + + 2 3 0.5 4 4 0.5 (13) An oscillator can be odeled as a eedback syste consisting o a resonator and a noiseless apliier. Phase noise is injected at the input to the apliier (Fig. 14a). This injected phase noise consists o a lat region and a 1/ region. This noise is shaped by the resonator, which has a 20-dB/decade 76
60 70 80 90 100 110 120 130 140 150 160 1 10 100 1 10 3 1 10 4 1 10 5 1 10 6 1 10 7 start i stop Frequency slope on either side o the center requency. This lat phase noise becoes 1/ 2 in nature and the 1/ phase noise becoes 1/ 3 in nature, within the resonator bandwidth, at the output o the oscillator (Fig. 14b). One could express this noise in ters o requency or FM noise (Fig. 14d). [It should be noted that the requency-to-phase transoration is an integration (20 db/ VCO noise 20-dB slope 30-dB slope Reerence noise Main divider noise 18. This graph shows the phase-noise plots or the three noise sources o the exaple in Fig. 17. decade) and, conversely, phase to requency transoration is dierentiation (+20 db/decade)]. The 1/ 2 phase noise, (20 db/decade), appears as lat FM noise and the 1/ 3 phase noise (30 db/decade) appears as licker FM noise (10 db/decade). Leeson s odel (Fig. 14c) describes the phase noise o an oscillator as a unction o the resonator-loaded quality actor (Q) or bandwidth, oscillation requency, noise igure, power, and oset ro the carrier. The phase noise iproves with higher Q (narrower resonator bandwidth) and power. A lower noise igure and 1/ corner also iproves the noise. Depending on the relative position o the 1/ corner and the resonator bandwidth, two cases arise: 1. In the low Q case, typical o ost RF/icrowave oscillators using inductive-capacitive (LC) tanks, transission lines, ceraic and dielectric resonators and yttriu-iron garnets (YIGs), the 1/ corner is inside (closer to the carrier or DC ) the resonator bandwidth. This gives rise to phase noise plot consisting o three dierent regions a lat noise region, a 1/ 2 region, and then a 1/ 3 region as shown in Fig. 14b. 2. Crystal oscillators and suraceacoustic-wave (SAW) resonator oscillators have typically very high Qs and, consequently, the phase-noise plot looks slightly dierent. As beore, there is a lat region, but then the 1/ corner is reached irst and then the phase noise increases at a rate o 10 78
db/decade. Since the resonator has very high Q, this iplies a very narrow bandwidth less than the licker corner. Thereore, the region closest to the carrier is the 1/ 3 region. Even the explanation is soewhat sipliied since 1/ 4 sloped regions have been observed very close to the carrier. I a PLL is used as a local oscillator (LO) in a receiver, the phase noise o the LO can degrade the adjacent-channel rejection o the receiver by a process known as reciprocal ixing. In Fig. 15a, an input spectru consisting o the desired signal and a adjacent-channel signal is ixed with an LO. I the LO is only a pure sinusoid, IDC In 1 1 Dn1 DNOISE1 + Vn1 0 VDC 6.283e7 R_k2_vco1 1.208e7 0 VDC 20-dB/dec noise 30-dB/dec noise R_k2_vco2 1.208e7 6.283e5 R_k0_vco1 38.19k Flat noise R_k0_vco2 38.19k the interediate-requency (IF) output o the ixer would just be a shited replica o the input spectru. O course, the IF would have a ilter to reject the adjacent channels. The LO phase noise will ix with the unwanted signals in adjacent channels, producing energy that appears in the IF passband, coincident with the desired signal. An exaple, using the Global Syste For Mobile Counications (GSM), assues that the irst interering signal is spaced 600 khz away and that the detection bandwidth is 200 khz. I the average LO phase noise is 110 dbc/(hz) 0.5 600 khz away, the total noise power in the 200-kHz channel with respect to the carrier is: 110 dbc / Hz P = = 200 khz 110 + 10log(200 khz) = 57 dbc vn_vco 19. The phase noise o a VCO can be odeled using a siple scheatic diagra in PSPICE sotware. This approxiates the noise as being lat across the 200-kHz-wide channel. I the desired signal is 100 db and the undesired signal is 60 db, the signal-to-noise ratio (SNR) is [100 db 0 dbc (LO carrier)] [60 db 57 dbc (LO noise)] = 17 db. This procedure can be repeated with additional interering signals to deterine the worst-case phase-noise requireent. In a spectru analyzer (Fig. 15b), when an attept is ade to resolve two closely spaced signals with widely diering aplitudes, the phase noise o the LO asks the weaker o the two signals. To easure the adjacent-channel selectivity o a receiver, two signal generators are used (Fig. 15c). The aplitude o the in-channel generator is set at the desired sensitivity level and the aplitude o the adjacent or o-channel generator is increased until the sensitivity decreases by a known aount. The phase noise ro the adja- 80
cent-channel generator that spills into the desired channel could cause the receive selectivity to be worse than expected. The phase noise o the ochannel generator will be easured rather than the in-channel signal. Phase noise can also have an adverse eect in clock-recovery systes, radar, and digital counications systes. All o the eleents in the PLL contribute to the overall phase noise. The noise echaniss or crystal and RF/icrowave oscillators were previously discussed. Another signiicant source o noise in PLLs is the dividers and phase detectors. There are a nuber o dierent types o phase detectors, including ixers or ultipliers, saple-and-hold devices, digital exclusive OR gates, and the ost coon, digital lip-lop phase detectors. The lip-lip phase detector, in addition to providing phase inoration, also has an intrinsic echanis to provide proper steering so the loop can achieve lock. The ain disadvantage o the liplop phase detector is that it suers a nonlinear region or dead zone close to a zero phase oset. However, various design techniques can itigate this proble. Other types o phase detectors only provide phase inoration, and additional circuitry is necessary or steering and acquisition. LOGIC DEVICES USED FOR PHASE DETECTORS AND DIVIDERS INTRODUCE PHASE NOISE, AND DIFFERENT LOGIC FAMILIES HAVE DIFFERENT PHASE- NOISE CHARACTERISTICS. 82 The diode ixer phase detector has the best phase noise and is used in critical applications. Logic devices used or phase detectors and dividers introduce phase noise and dierent logic ailies have dierent phase-noise characteristics as a unction o the operating requencies. ECL has a noise loor o approxiately 145 to 150 dbc/(hz) 0.5, while advanced copleentary-etal-oxide-seiconductor (CMOS) logic has a noise loor between 155 and 165 dbc/(hz) 0.5 depending on the input (I) and output (O) operating requency. The 1/ or licker corner is at an oset requency o between several hundred Hertz and 10 khz. In general, aster logic and larger voltage swings give rise to better phase noise. Assuing that the noise is ainly generated in the transition region between logic 0 and 1, with the aster rise tie, then less tie is spent in the transition region, resulting in lower noise levels. Since advanced CMOS logic has a +5- VDC swing versus approxiately 800 V or ECL, this would explain the noise iproveent when using advanced CMOS logic. It is diicult to easure the phase noise o dividers and (concluded on p. 117)
(continued ro p. 82) phase detectors due to the low noise levels involved. Soe anuacturers o single-chip PLLs provide noise-loor results. Other sources o noise in the PLL include operational apliiers (opaps) used or loop ilters and noise ro power supplies. These noise sources are shown in Fig. 16. The loop operates on these various noise sources. It is possible to write the Laplace transer unction, substituting s = j2 ro each o these noise sources to the output. The agnitude o the transer unction squared ultiplied by the phase-noise equation o the source, expressed in power, provides the output phasenoise power o that source at the output o the loop. By superposition, it is possible to power su the eects o all the individual noise sources in order to produce a coposite output phasenoise curve. MODELING NOISE Using the previous exaple (Fig. 17), assue that the PLL has only three noise sources voltage-controlled-oscillator (VCO) noise, reerence-oscillator noise, and ain-divider noise. It is desirable to express these noise sources in ters o double-sideband phase-noise power [(radians 2 /Hz]. The VCO has a noise loor o 155 db/hz, a 1/ corner requency o 5 khz, and a speciied noise o approxiately 110 db/hz at an oset requency o 10 khz. The coeicients o the phasenoise equation (equation 13) were anually adjusted in the MathCAD progra to yield the speciied phase noise at the particular osets ro the carrier. These are shown in the accopanying sidebar (see Noise Equations or Noise Sources ) on page 74 with soe representative values or the noise at particular osets. The ain divider has a loor o 155 dbc/_hz and a 1/ or licker corner o 1 khz. The 10-MHz reerence noise was NOISE CAN BE MODELED USING PSPICE ELEMENTS, BUT IT IS NECESSARY TO TRANSFORM THE NOISE EQUATION FOR THE VARIOUS NOISE SOURCES IN THE PLL. obtained ro the data sheet or a coercial 10-MHz teperature-copensated crystal oscillator (TCXO). The our coeicients in the reerence-noise equation were experientally deterined using asyptotic lines with 0, 10-, 20-, and 30-dB/decade slopes in MathCAD to best approxiate the noise plot ro the anuacturer s data sheet. The phase-noise plots, in decibels, o these three sources are displayed in Fig. 18. In Part 1, PSPICE was used to odel the loop dynaics o a PLL. Noise can also be odeled by using PSPICE eleents. First, it is necessary to transor the noise equation or the various noise sources in the PLL (VCO noise, reerence noise, etc.), into PSPICE eleents. Resistors produce lat noise and the diode odel in PSPICE has a ter or 1/ or licker noise (10 db/decade). By integrating the lat resistor noise, it is possible to produce 1/ 2 (20 db/decade) noise and by integrating the diode noise (1/), it is possible to produce 1/ 3 (30- db/decade) noise. Each ter in the noise equation can be represented by one o the PSPICE eleents entioned, or a cobination o eleents. This concept will be illustrated by odeling the VCO noise in the exaple (Fig. 19). Next onth, the third and inal installent o this article series on odeling PLL dynaics and noise will show how to cobine the dierent noise sources to review what has been learned and to show how to integrate the knowledge o phase-noise contributors to produce a inal phasenoise curve or the exaple loop. 117