Fundamentals of PLLs (III)

Transcription

1 Phase-Locked Loops Fundamentals of PLLs (III) Ching-Yuan Yang National Chung-Hsing University Department of Electrical Engineering

2 Phase transfer function in linear model i (s) Kd e (s) Open-loop transfer function System transfer function Error transfer function F (s) Ko s o (s) o (s) K d K o F (s) (loop gain) e (s) s (s) K d Ko F (s) G(s) H (s) o i ( s) 1 G ( s) s K d K o F ( s) (s) s E (s) e 1 H ( s) i ( s) s K d Ko F (s) G(s) 3-1 Ching-Yuan Yang / EE, NCHU

3 Steady-state phase errors Final-value theorem: lim() y t lim() t s0 sy s 1 lim() y nlim 1() z Y z z n z1 for Laplace transformer for -transformer Steady-state phase error: lim() x e tlim s0 s i() s s K K F() s d o 3- Ching-Yuan Yang / EE, NCHU

4 Phase offset Consider a step change of input phase: s i() s lim() lim t 0 e s x s0 s Kd KoF () s F(0) > 0 The loop eventually will track out any change of input phase. There is no steady-state phase error resulting from a step change of input phase in any PLL. 3-3 Ching-Yuan Yang / EE, NCHU

5 Frequency offset Consider a step change of input frequency: i()() t t i s s v lim() e tlim x s0 s K K F()(0) s K K F d o d o K o K d F(0) = DC gain = K DC [rad/s] called velocity constant Those familiar with servo terminology will recognize it as the velocity-error coefficient. may be due to an actual difference between the transmitter and receiver frequencies or it may be due to a Doppler shift. Static phase error: [rad] also called velocity error, or loop stress v K DC Another view: vd vc (VCO) vc vd F(0) (loop filter) e (PD) vd : DC o/p of PD K K o To produce the necessary vc requires the phase error e [rad] K K F(0) d o d 3-4 Ching-Yuan Yang / EE, NCHU

6 Steady-state velocity error between type-1 and type- PLLs In a type-1 PLL, the DC gain is finite, so static phase error is unavoidable. Static phase error impairs the performance of the PLL. In a type- PLL, the DC gain is infinite because of the integrator in the loop filter [whereby F(0) = ], so static phase error is zero. Note: No physical analog integrator has infinite DC gain, but the DC gain in most practical PLLs can be made large enough to reduce the static phase error to insignificance. 3-5 Ching-Yuan Yang / EE, NCHU

7 Frequency ramp Consider i(t) = t i(t) = t / nd-order type- PLL: Phase error i(s) = /s 3 s i ( s ) e (s) s n s n Acceleration error a lim e (t ) lim t s 0 s n s n n Physical insight: apply a DC voltage vd to the integrator of loop filter. vt Kv vc (t ) vc (0) d the rate of VCO changed freq. o d 1 1 vd K d e and n Ko Kd 1 e n What forms of F(s) is needed to reduce a to zero? 0 s 0 s s K K F ( s ) o d a lim F ( s) Y (s) and Y (0) 0 s type-3 PLL Tpye-3 PLL is useful in tracking signals from satellites or missiles. 3-6 Ching-Yuan Yang / EE, NCHU

8 Steady-state acceleration error between type- and type-3 PLLs Type- PLL requires large n. A large bandwidth to handle a rapidly changing input frequency In type-3 PLL, the input frequency rate can be accommodated in a loop with small bandwidth. 3-7 Ching-Yuan Yang / EE, NCHU

9 Transient response to a frequency step of nd -order type- PLL Two-path proportional-plus-integral configuration: K F() s K where K K 1 1 s Small damping (large integral-path gain) leads to quick reversal of the transient, whereas large damping (small integral-path gain) leads to very slow reversal. For larger, the peak error approaches /K, which is the steady-state error of a type 1 PLL. 3-8 Ching-Yuan Yang / EE, NCHU

10 Theoretical spectrum of oscillator output 3-9 Ching-Yuan Yang / EE, NCHU

11 Noise spectrum Power spectral density (PSD): The spectrum shows how much power the signal carries at each frequency. More specifically, the PSD, S X ( f ), of a noise waveform x(t ) is defined as the average power carried by x(t) in a one-hertz bandwidth around f. S X (f) is expresses in V /Hz. The spectrum shows the power carried in a small bandwidth at each frequency, revealing how fast the waveform is expected to vary in the time domain Ching-Yuan Yang / EE, NCHU

12 Spectrum power Two-sided and one-sided noise spectra Since S X ( f ) is an even function of f for real x(t), the total power carried by x(t) in the frequency range [ f 1 f ] is equal to f f f P 1 f1, f S X f df S f X f df S X f df f1 f1 Folded white spectrum 3-11 Ching-Yuan Yang / EE, NCHU

13 VCO Noise in Wireless Systems VCO noise has a negative impact on system performance. Receiver lower sensitivity, poorer blocking performance. Transmitter increased spectral emissions (output spectrum must meet a mask requirement). Noise is characterized in frequency domain. 3-1 Ching-Yuan Yang / EE, NCHU

14 VCO Noise in High Speed Data Links VCO noise also has a negative impact on data links. Receiver increases bit error rate (BER) Transmitter increases jitter on data stream (transmitter must have jitter below a specified level) Noise is characterized in the time domain Ching-Yuan Yang / EE, NCHU

15 Noise Sources Impacting VCO Extrinsic noise: Noise from other circuits (including PLL) Intrinsic noise: Noise due to the VCO circuitry 3-14 Ching-Yuan Yang / EE, NCHU

16 VCO Model for Noise Analysis We will focus on phase noise (and its associated jitter) Model as phase signal in output sine waveform out (t ) cos f ot out (t ) Using a familiar trigonometric identity out (t ) cos( f ot )cos out (t ) sin( f ot )sin out (t ) Given that the phase noise is small cos out (t ) 1, sin out (t ) out (t ) out (t ) cos( f ot ) sin( f ot ) out (t ) 3-15 Ching-Yuan Yang / EE, NCHU

17 - Calculation of Output Spectral Density Calculate autocorrelation Take Fourier transform to get spectrum Note that * symbol corresponds to convolution In general, phase spectral density can be placed into one of two categories Phase noise out (t) is non-periodic Spurious noise out (t) is periodic 3-16 Ching-Yuan Yang / EE, NCHU

18 Output Spectrum with Phase Noise Suppose input noise to VCO (vn(t)) is bandlimited, non-periodic noise with spectrum Svn(f). In practice, derive phase spectrum as K S out ( f ) v Svn ( f ) f Resulting output spectrum Sout ( f ) S sin ( f ) S sin ( f ) S out 3-17 Ching-Yuan Yang / EE, NCHU

19 Measurement of Phase Noise in dbc/hz Sout(f) 1 S out(f) dbc/hz fo fo f Definition of L(f) Spectral density of noise L ( f ) 10 log Power of carrier Units are dbc/hz. For this case out ( f ) L ( f ) 10 log 10 log out ( f ) Valid when out(t) is small in deviation (i.e., when carrier is not modulated, as currently assumed) 3-18 Ching-Yuan Yang / EE, NCHU

20 Single-Sided Version Definition of L(f) remains the same Spectral density of noise Power of carrier Units are dbc/hz. For this case So, we can work with either one-sided or two-sided spectral densities since L(f) is set by ratio of noise density to carrier power Ching-Yuan Yang / EE, NCHU

21 Output Spectrum with Spurious Noise Suppose input noise to VCO is d vn (t ) spur cos( f spur t ) Kv out (t ) K v vn (t )dt Resulting output spectrum dspur f spur sin( f spur t ) Sout ( f ) S sin ( f ) S sin ( f ) S out 1 d spur f spur 1 d spur f spur 3-0 Ching-Yuan Yang / EE, NCHU

22 Measurement of Spurious Noise in dbc 1 dspur f spur Definition of dbc Power of tone 10 log Power of carrier We are assuming double sided spectra, so integrate over positive and negative frequencies to get power. (Either single or double-sided spectra can be used in practice). For this case dspur dspur f spur 10 log dbc 0 log f spur 3-1 Ching-Yuan Yang / EE, NCHU

23 Phase noise in oscillators FkT 10 log P sig Measured in db below carrier per unit bandwidth. FkT Phase noise: L ( f ) 10 log Psig f1/ f 3 f f Q f Note: Leeson assumed that F( f) was constant over frequency. Hajimiri, IEEE JSSC, Mar Ching-Yuan Yang / EE, NCHU

24 Phase Noise of A Practical Oscillator FkT 10 log Psig Phase noise drops at -0 db/decade over a wide frequency range, but deviates from this at: Low frequencies slope increases (often -30 db/decade) High frequencies slope flattens out (oscillator tank does not filter all noise sources) Frequency breakpoints and magnitude scaling are not readily predicted by the analysis approach taken so far. 3-3 Ching-Yuan Yang / EE, NCHU

25 Phase-noise propagation in a PLL o, W o ( f ) i, W i ( f ) i e Ko s v o, W o ( f ) Phase-noise spectral density caused by the internal phase noise of VCO: Output phase-noise spectral density: W o o ( f ) W o ( f ) E ( f ) The phase error e due to o is o, so W e o ( f ) W o o ( f ) W o ( f ) E ( f ) (A) Phase-noise spectral density caused by input phase noise: The tracked phase-noise spectrum W o i ( f ) W i ( f ) H ( f ) The untracked phase-noise spectrum (phase-error spectrum) W e i ( f ) W i ( f ) E ( f ) Untracked jitter: (A) + (B) Wu ( f ) W ( f ) E ( f ) (B) where W ( f ) W i ( f ) W o ( f ) and the subscript u indicates untracked phase noise. 3-4 Ching-Yuan Yang / EE, NCHU

26 Bandwidth trade-off E( f ) has a highpass frequency response. An increase in PLL bandwidth K shifts the highpass corner to a higher frequency and reduces the integrated untracked phase jitter. This is opposite to the effect of bandwidth on phase jitter caused by additive noise. H( f ) has a lowpass frequency response. Total phase jitter cause by additive noise and phase noise: 0 '()()()() o n 0 W f H f df W f E f df 3-5 Ching-Yuan Yang / EE, NCHU

27 Delay-locked loops Vin Vout Vcont Voltage-Controlled Delay Line Vout Vin Phase Detector Charge Pump Low-Pass Filter Vc Delay line much less noisy than VCOs. First-order system (for first-order LPF), thus very stable. But input and output frequencies must be equal. 3-6 Ching-Yuan Yang / EE, NCHU

28 Phase transfer function in linear model (DLL) i (s) Kd e (s) F (s) K VCDL o (s) o (s) K d KVCDL F ( s ) (loop gain) e (s) (s) K d KVCDL F ( s ) G (s) H (s) o i ( s ) 1 G ( s ) 1 K d KVCDL F ( s ) (s) 1 E (s) e 1 H ( s) i ( s) 1 K d KVCDL F ( s ) Open-loop transfer function G ( s ) System transfer function Error transfer function 3-7 Ching-Yuan Yang / EE, NCHU

29 3-8 Ching-Yuan Yang / EE, NCHU Transfer function of DLL Transfer function of PD/CP/LPF Close-loop transfer function Discussion The closed-loop transfer can be used to determine how out settles if in experiences a change. n, In practice R P may not be need because the loop contains only one pole at the origin.

30 Skew reduction Skew between data and buffered clock Use of a PLL to eliminate skew Use of a DLL to eliminate skew 3-9 Ching-Yuan Yang / EE, NCHU