ORIGINAL ARTICLE

Mathematical models and simulations of phase noise

in phase-locked loops
Sethapong Limkumnerd1 and Duangrat Eungdamrong2

Abstract
Limkumnerd, S. and Eungdamrong, D.
Mathematical models and simulations of phase noise in phase-locked loops
Songklanakarin J. Sci. Technol., 2007, 29(4) : 1017-1028

Phase noises in Phase-Locked Loops (PLLs) are a key parameter for communication systems that
contribute the bit-rate-error of communication systems and cause synchronization problems. Accurate
predictions of phase noises through mathematical models are consequently desirable for practical designs
of PLLs. Despite many phase noise models derived from noise sources from electronic devices such as an
oscillator and a multiplier have been proposed, no phase noise models that include noises from loop filters
have specifically been investigated. This paper therefore investigates the roles of loop filters in phase noise
contribution. The major scopes of this paper is a detailed analysis and simulations of phase noise models
resulting from all components. i.e. a voltage-controlled oscillator, a multiplier and a filter. Two particular
second-order passive and active low-pass filters are compared. The results show that simulations of phase
noises without an inclusion of filter noises may not be accurate because the filter noises, particularly the
active filter, significantly contribute the total phase noise. Moreover, the passive filter does not significantly
dominate the phase noise at low offset frequency while the active filters entirely dominate. Therefore, the
passive filter is a more efficient filter for PLL circuit at low offset frequency. The phase noise models
presented in this paper are relatively simple and can be used for accurate phase noise prediction for PLL
designs.

Key words : phase-locked-loop, phase noise, voltage control oscillator, phase detector,
loop filter, main divider
1
M.Sc. student in Telecommunication, 2Ph.D.(Electrical and Computer Engineerging), Asst. Prof., School
of Communications Instrumentations and Control Sirindhorn International Institute of Technology,
Thammasat University, 131 Moo 5, Tiwanont Road, Bangkadi, Muang, Pathum Thani, 12000 Thailand.
Corresponding e-mail: thanyee@yahoo.com
Received, 28 November 2006 Accepted, 28 February 2007
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1018 Limkumnerd, S. and Eungdamrong, D.

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Phase Locked Loops (PLLs) are extensively important before practical implementations.
used for a variety of radio applications such as in Despite Leeson's Equation (1966) being first
frequency synthesizers or in carrier recovery and recognized as a classical phase noise prediction in
clock recovery circuits (Lathi, 1998). Such PLLs PLLs, such equation predicts only the single-
are closed-loop systems that utilize a negative sideband phase noise measured from the power
feedback to sustain a constant ratio of an output spectral density of the carrier and an amplitude
frequency to an input frequency (Rohde, 1997). variation may also be included. Moreover,
Generally, desirable performances of PLLs are not calculating phase noise using Leeson's equation
only high frequency but also low phase noise requires not only operation of all linear device but
(Rohde et al., 2005). The high frequency opera- also an oscillator that contains only a single
tions can be achieved through the use of high resonator (Razavi, 2001). Recently, Ducker (2000)
transition frequency (fT) technologies such as has proposed mathematical models of double-
Bipolar or BiCMOS devices. However, the phase sideband phase noise through two major noise
noises have significantly degraded the perform- sources, i.e. voltage-controlled oscillator and
ances of PLLs by reducing the signal-to-noise multiplier. Although Eric Ducker's equations are
ratio (SNR), increasing adjacent channel power relatively simple for phase noise prediction, noise
and reducing adjacent channel rejection (Kroupa, models of loop filter have not yet been studied.
2003). As a result, the degraded performances of As loop filters can be of various types and
the PLLs contribute the bit-rate-error of commu- orders, they may significantly generate noises
nication systems and cause synchronization leading to the total phase noise of PLLs and this
problems in clocked and sampled data digital paper therefore investigates the roles of loop filters
systems (Misra, 2001). in phase noise contribution through mathematical
Consequently, predicting phase noise models and simulations. The major scope of this
through mathematical models and simulations are paper is a detailed analysis and simulations of
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1019 Limkumnerd, S. and Eungdamrong, D.

phase noise models resulting from all components. are devices that convert a high output frequency
i.e. a voltage-controlled oscillator, a multiplier and a reference frequency, respectively, to a low
and using MATHEMATICA and MATLAB. Two frequency for a multiplication process at the PD.
particular second-order passive and active low With reference to Figure 1, the loop gain
pass filters are compared. The paper is organized L(s) (Thamsirianunt and Kwasniewski, 1997) in
as follows; Section 1 commences by literature s-domain can be expressed by a multiplication
reviews on basic PLLs. Sections 2 and 3 then between a forward loop gain G(s) = KpKvF(s)/s and
describe practical effects of phase noise in the a reverse loop gain H(s) = 1/N as
frequency domain and mathematical noise models
1  K × Kv × F(s) 
of all components, respectively. Finally, total phase L(s) = G(s) × H(s) =  p 
noises of PLLs are summarized and simulated in s N 
sections 4 and 5. (1)

where KP is a phase gain of the PD, Kv is a voltage

control sensitivity of the VCO and F(s) is the
transfer function of a loop filter. The consequent
closed loop transfer functions are generally
described separately for each component as will
be seen later in section 4.

Figure 1. Block diagrams of the basic phase lock 1.2 Operations

loop. The operations of The PLL can be
described in terms of a reference phase ΦREF and an
1. PLL Basics output phase ΦOUT as follows Zhang et al.(2003).
1.1 General Descriptions When ΦOUT ΦREF, the PD compares the phases
Figure 1 shows the block diagrams of a between the output phase ΦOUT/N and the reference
basic PLL. It can be seen from Figure 1 that the phase ΦREF/R and hence generates an error voltage
PLL is a closed-loop system and consists mainly e(s). In other words,
of five components, i.e. a phase detector (PD), a Φ Φ
loop filter, a voltage-controlled oscillator (VCO) e(s) = REF − OUT (2)
and dividers (Dai and Harjani, 2002; Shu et al., R N
2004 ). Firstly, the PD is typically a two-input and
The derivative of Equation (2) yields
one-output device that can be realized by a
specialized mixer. This PD comes in many con- d[e(s)] FREF FOUT
= − (3)
figurations including those with logic level inputs, dt R N
passive and active analog versions, and sampling
versions specifically used for high frequency Typically, the error voltage e(s) in
multiplications. Secondly, the loop filter is a low- Equation (2) subsequently enters the VCO as a
frequency circuit that filters the phase detector control voltage and must remain constant. Such
error voltage with which it controls the VCO error voltage e(s), however, consists of both DC
frequency. Although it can be active or passive, it and high-frequency components and the filtering
is usually analog and very simple. In extreme cases, process is significantly required. Therefore, the
it might be an entire microprocessor. Thirdly, the e(s) is then filtered through the lowpass filter with
VCO is the control element for a PLL in which a transfer function F(s) for suppressing the high-
the output frequency changes correspondingly with frequency components of the PD and presenting
the tuning voltage. Finally, the N and R dividers only the DC level for the oscillator. If the loop
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1020 Limkumnerd, S. and Eungdamrong, D.

gain is large enough, the error e(s) becomes a very the carrier (PSSB) to the carrier power (PC). In other
small value in steady state or is almost constant. words,
Equation (3) is consequently equal to Single − Sideband Phase Noise =
d[Constant] F F
= 0 = REF − OUT (4)  P 
dt R N 10 log(L( f m )) = 10 log C  (6)
 PSSB 
Thus,
On the other hand, the SΦ(fm) can be
FOUT =   FREF
N
 R (5) measured through the PSD of the modulation of
signal with the ideal phase modulator using the
spectrum analyzer at RF with a 1-Hz resolution-
When the PLL is locked, it produces an
bandwidth. Note that the SΦ(fm) is twice (3-dB more
output that has a small and constant phase error
than) the L(fm). Figure 3 shows plots of the SΦ(fm)
with respect to the input phase but the output
versus the log scale of the offset frequency fm
frequency is the same or linearly proportional to
consequently demonstrating the nonlinear decay
the reference frequency as shown in Equation (5).
and demon-strates various regions of the phase
noise depend-ing on the regions of slopes. Table 1
2. Phase Noise Analysis
summarizes types and slopes of noises in each
Phase noise is widely used to describe the
region. The SΦ(fm) can be described in dBc/Hz
characteristic randomness of frequency stability.
as a general phase noise equation that includes
Generally, two types are single- sideband L(fm) or
the noises in all regions of slopes as summarized in
double sideband SΦ(fm) phase noises. On the other
Table 1. In other words,
hand, the L(fm) can directly be measured through
the power spectral density (PSD) of the signal using
the spectral analyzer at RF with a 1-Hz resolution-
bandwidth filter. Figure 2 shows the measured
PSD of the carrier power and the single-sideband
phase noise power. As shown in Figure 2, the L(fm)
(Manassewitsch, 1987) can be described in decibels
relative to the carrier level as dBc/Hz as the ratio
of the noise power at an offset frequency fm from

Figure 3. Plots of double-sideband phase noises

in dBc/Hz versus offset frequency fm
in Hz.

Table 1. Summary of types and slopes of noises in

each region.

Regions Coefficients Types of Noise Slopes (dB)

1/f0 k0 White Noise 0
1/f1 k1 Flicker Noise -10
Figure 2. Power spectral density of the carrier 1/f2 k2 White FM -20
power and the single-sideband noise 1/f3 k3 Flicker FM -30
power. 1/f4 k4 Random Walk FM -40
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1021 Limkumnerd, S. and Eungdamrong, D.

Double - Sideband Phase Noise =

k k k k k 
SΦ ( f m ) = 10 log 00 + 11 + 22 + 33 + 44 
f f f f f 
(7)
where f is a frequency in Hz (Manassewitsch,
1987). Consequently, mathematical models of
noise sources in this paper are based on the double Figure 4. Block diagrams of the basic phase lock
sideband phase noise equation shown in Equation loop with an inclusion of noise sources.
(7).

3. Mathematical Noise Source Models

3.1 Typical Noise Models for VCO and
Multiplier
Figure 4 shows the block diagram of a
basic PLL with an inclusion of three noise models,
i.e. a noise model for the VCO NVCO(f), a noise
model for multiplier NMUL(f) and noise models Figure 5. Circuit configurations of the second-
NFIL1(f) and NFIL2(f) for passive filter and active order passive lowpass filter F1.
filters, respectively. With reference to Figure 4,
noise models for NVCO(f) and NMUL(f) are common fore, the noises models of NFIL1(f) and NFIL2(f) are
and described as follows. The noise model of particularly analyzed and compared as follows.
NVCO(f) is given by
3.2 Noise Model NFIL1(f) of Passive Filter
k2_VCO k3_VCO
NVCO ( f ) = k0_VCO + + (8) F1
f2 f3 Figure 5 shows the circuit configura-
tions of a second-order passive lowpass filter F1.
where k0_VCO, k2_VCO, k3_VCO are coefficients of white, As shown in Figure 5, the circuit is relatively
white FM and flicker FM noises in the VCO, simple and are formed a single resistor R2 and two
respectively. On the other hand, the noise model capacitors C2 and C3. The transfer function F1(s) in
NMUL(f) can be expressed in terms of the noise s-domain can be described in terms of an output
models of the dividers and the reference oscillator voltage Vout and an input current Iin as follows
for simplicity as (Ducker, 2000) Vout1 (s) 1 + sR1C1

F1 (s) = = (10)
 k1_ref k2_ref k3_ref  Iin (s) sR C C
 k1_md   k0_ref + f + f 2 + f 3  s(C1 + C2 )(1 + 1 1 2 )
2
NMUL ( f ) = N  k0_md +  +  C1 + C2
 f   R2 
  
  
In other words,
(9)
where k0_md and k1_md are coefficients of white and
F1 (s) =
(
kF1 1 + sτ F1_1 )
( )
flicker noises in the main divider, respectively. In (11)
addition, k0_ref, k2_ref, k3_ref and k4_ref are coefficients s 1 + sτ F1_2
of white, flicker, white FM and flicker FM noises
in the reference oscillator, respectively. However, where kF1 = 1/(C1+C2) is a constant. τF1_1 = R1C1
the noise model of the loop filter is not typically and τF1_2 = (R1C1C2)/(C1+C2) are time constants.
specified depending on the types of filters. There- Analytical treatments for F1(s) are shown in
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1022 Limkumnerd, S. and Eungdamrong, D.

Appendix A.1. In addition to the transfer function, Vout2 (s) 1 + sR1C1

the noises in the filter F1 may come from both F2 (s) = = (13)
Vin (s) sR1C1C2
capacitor and resistor. As the capacitor does not sR2 (C1 + C2 )(1 + )
significantly contribute noises, the noise of the loop C1 + C2
filters are therefore mainly from the resistor R1. In other words,
Typically, noise in resistor results from a random
motion of electrons in the resistor. The noise model
F2 (s) =
(1 + sτ )F2_1

sτ (1 + sτ )
for a resistor Pr(R) is assumed to be white noise (14)
and is equal to 4kTBR where k is the Boltzmann's F2_3 F2_2

constant, T is an absolute temperature in [K], B is

a bandwidth of the filter in [Hz] and R is an actual where τF2_1= R1C1, τF2_2 = (R1C1C2)/(C1+C2) and τF2_3
value of the resistor in [Ω]. Such Pr(r) is also a = R2(C1+C2) are time constants in addition the DC
power dissipated by the resistor and commonly gain of the active filter can be determined by the
known as a thermal noise. As a result, the noise integrator term, i.e kF2 = 1/R2(C1+C2). Analytical
model of the resistor dominates the passive filter treatments for F2(s) are shown in Appendix A.2. In
F1 and the can be described as NFIL1(f) = Pr(R) = terms of noise contributions, it can be seen from
4kTBR (Kroupa, 2003). In other words, Figure 5 that the noise sources result from not
only resistors as previously described in Equation
NFIL1 ( f ) = k0_ R (12) (12) but also the Op-Amp. Generally, the noise
from the Op-amp includes the flicker and the
where k0_R is a coefficient of the white noise thermal noises and is derived by experiments or
contributed by the resistor in the passive filter F1. given by the manufacturer. Consequently, the noise
model for the active filter F2 is given by
3.3 Noise Model NFIL2(f) of Active Filter k1_OA
N FIL1 (f ) = k 0_R + k 0_OA + (15)
F2 f
Figure 6 shows the circuit configura-
tions of a second-order active lowpass filter F2. As where k0_OA and k1_OA are coefficients of the white
shown in Figure 6, the circuit mainly consists of and flicker noises contributed by the operational
the operational amplifier (Op-Amp), two resistors amplifier in the active filter F2.
R1 and R2, and two capacitors C1 and C2. The
transfer function F2(s) in s-domain can be described 4. Mathematical Phase Noise Models
in terms of an output voltage Vout and an input 4.1 Typical Phase Noise Models for VCO
voltage Vin as follows and Multiplier
Typically, output phase noise (SΦ(f)) of
each noise source can be modeled by a multi-
plication between an input power spectral density
and a squared magnitude of a closed loop transfer
function (θ(s)) in which the input is varied
depending on the investigated noise sources. In
other words,
SΦ ( f ) = N( f ) × θ (s)
2
(16)

Referring to Figure 4, the closed loop transfer

Figure 6. Circuit configurations of the second- function of the VCO noise is given by
order active lowpass filter F2.
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1023 Limkumnerd, S. and Eungdamrong, D.

θVCO (s) =
1 1  Kv 
1 + L(s)
(17) θ R (s) =   × F2 (s) (23)
s  1 + L(s) 

Substituting Equations (16) with Equations (8) Meanwhile the closed loop transfer function for
and (17) yields the phase noise model of the VCO the Op-Amp is expressed as
SΦ_VCO(f) as
1  Kv 
SΦ_VCO ( f ) = NVCO ( f ) × θVCO (s)
2
(18) θOA (s) =   (24)
s  1 + L(s) 

In addition, the closed loop transfer function for By substituting Equations (16) with Equations (15)
the multiplier is given by and (23), the phase noise model of the resistor
L(s) (SΦ_R(f)) is given by
θ MUL (s) = (19)
1 + L(s) 2
SΦ_ R ( f ) = NR ( f ) × θ R (s) (25)

Similarly, substituting Equations (16) with Equa-

tions (9) and (19) yields the phase noise model of Similarly, substituting Equations (16) with Equa-
the multiplier SΦ_MUL(f) as tions (15) and (24) yields the phase noise model
of the Op-Amp (SΦ_OA(f)) as
2
SΦ_ MUL ( f ) = NMUL ( f ) × θ MUL (s) (20) 2
SΦ_OA ( f ) = NOA ( f ) × θOA (s) (26)

4.2 Phase Noise Model SΦ_FIL1(f) of Passive

Filter F1 As a result, the summation of Equations (25) and
As shown in Figure 5, the closed loop (26) yields phase noise model of the active low-
transfer function for the passive lowpass filter F1 pass filter F2 (SΦ_FIL2(f)) as follows
can be expressed as 2 2
SΦ_FIL2 ( f ) = NR ( f ) × θ R (s) + NOA ( f ) × θOA (s)
1  Kv 
θ FIL1 (s) =   (21) (27)
s  1 + L(s) 
4.4 Total Phase Noise Models
By substituting Equations (16) with Equations The total phase noise models can be
(12) and (21), the phase noise model of the passive generally determined by the summation of phase
lowpass filter F1(SΦ_FIL1(f)) is given by noise generated by all components in a PLL.
2
SΦ_FIL1 ( f ) = NFIL1 ( f ) × θ FIL1 (s) (22) Therefore, the total phase noise model using the
second-order passive lowpass filter can be modeled
through the summation of phase noise models
4.3 Phase Noise Model SΦ_FIL2(f) of Passive described in Equations (18), (20) and (22), i.e.
Filter F2 SΦ1(f) = SΦ_VCO(f) + SΦ_MUL(f) + SΦ_FIL1(f). In other
As the active low pass filter F2 is words, such SΦ1(f) is described in dBc/Hz as
formed by resistors, capacitors and Op-Amp, the
two major noise sources are therefore contributed SΦ1 ( f m ) = 10 log(SΦ_VCO ( f ) + SΦ_ MUL ( f ) + SΦ_FIL1 ( f ))
by the resistors and the Op-Amp. Therefore, the (28)
transfer function of the Op-Amp and the resistor
are different. As shown in Figure 4, the closed In the similar manner, the total phase noise model
loop transfer function for the resistor is given by using the second-order active lowpass filter can be
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1024 Limkumnerd, S. and Eungdamrong, D.

modeled through the summation of phase noise filter F2 shown in Figures 5 and 6, respectively. In
models described in Equations (18), (20) and (27), addition, Table 3 also summarizes values of noise
i.e. SΦ2(f) = SΦ_VCO(f) + SΦ_MUL(f) + SΦ_FIL2(f), and is coefficients used in simulations by Ducker (2000).
described in dBc/Hz as For purposes of comparison, both filters F1 and F2
have been designed to operate at the same corner
SΦ2 ( f m ) = 10 log(SΦ_VCO ( f ) + SΦ_ MUL ( f ) + SΦ_FIL2 ( f ))
frequencies. Figure 7 shows the simulated bode
(29) plots of such two filters. Comparisons of calculated
and simulated values DC gains and corner
5. Simulation Results frequencies are summarized in Table 4. It can be
Phase noises of the PLL have been simulated considered from Table 4 that the corner frequencies
using MATHEMATICA and MATLAB. Based on of both filters F1 and F2 are equal, i.e. f F1_1 = fF2_1 and
Drucker (2000), Table 2 summarizes values of f F1_2 = f F2_2. However, the DC gain of the filter F1 is
components of the passive filter F1 and the active has been constantly fixed while the DC gain of the
filter F2 can be tuned through the corner frequency
Table 2. Summary of values of components of the fF2_3 that behaves as an integrator and yields a -20
passive filter F1 and the active filter F2. dB/decade for the DC gain.
Figure 8 shows the simulated total phase
Components Values Units noise SΦ1(f) in [dBc/Hz] versus offset frequency
Resistors R1 5.62 kΩ in [Hz], running from 1 Hz to 1 GHz. Such total
R2 2.94 kΩ phase noise SΦ1(f) is the sum of SΦ_VCO(f), SΦ_MUL(f)
and SΦ_FIL1(f) as described in Equation (28). As
Capacitors C1 47 nF
C2 6.8 nF
shown in Figure 8, the total phase noises can be
considered in three regions of offset frequency.
First, at the offset frequency lower than approxi-
Table 3. Summary of noise coefficients used in mately 700 kHz, only the multiplier noise
simulations. dominates the total phase noise while both VCO
and filter noises are relatively low at low offset
Components Constants Values
frequency and do not significantly contribute to the
VCO k0_VCO 10-15.5 total noise. Secondly, between 700-kHz to 100-
k2_VCO 10-3 MHz offset frequency, the filter noise dominates
k3_VCO 100.7
Kv 107
Main Divider k0_md 10-15.5
k1_md 10-12.5
N 1000
Reference Divider R 10
Reference Oscillator k0_ref 10-15.8
k1_ref 10-12.7
k2_ref 10-9.86
k3_ref 10-7.82
Resistor k0_R1 10-12.64
k0_R2 10-12.92
Op-Amp k0_OA 10-17.045
k1_OA 10-16.02
Figure7. Simulated magnitude of filters F1 and F2
Phase Detector Kp 0.5
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1025 Limkumnerd, S. and Eungdamrong, D.

Table 4. Summary of calculated and simulated values DC gains and corner

frequencies of the filters F1 and F2.

Values
Filters Parameters
Calculated Simulated
F1 DC Gain kF1 dB 145.38 145.62
Corner Frequency f F1_1 = ω F1_1/2π = 1/τ F1_1 Hz 7.24 × 10 3
7.21 × 103
f F1_2 = ω F1_2/2π = 1/τ F1_2 Hz 5.73 × 104 5.74 × 104
F2 DC Gain kF2 dB 76.02 76.11
Corner Frequency f F2_1 = ω F2_1/2π = 1/τ F2_1 Hz 3.31 × 103 3.35 × 103
f F2_2 = ω F2_2/2π = 1/τ F2_2 Hz 7.24 × 103 7.21 × 103
f F2_3 = ω F2_3/2π = 1/τ F2_3 Hz 5.73 × 104 5.74 × 104

the total noise. Finally, at the offset frequency noise. Secondly, between 10-Hz to 800-kHz offset
higher than approximately 100 MHz, only the frequency, both the filter noises dominate the total
VCO noise dominates. noise. Finally, at the offset frequency higher than
Figure 9 shows the simulated total phase approximately 800 kHz, only the VCO noise
noise SΦ2(f) in [dBc/Hz] versus offset frequency dominates.
in [Hz], running from 1 Hz to 1 GHz. Such total Figure 10 shows the simulated total phase
phase noise SΦ2(f) is the sum of SΦ_VCO(f), SΦ_MUL(f) noises of the PLL in dBc/Hz versus offset fre-
and SΦ_FIL2(f) as described in Equation (29). As quency in [Hz]. As shown in Figure 10, the total
shown in Figure 9, the total phase noises can be phase noise SΦ1(f) decreases by -30 dBc/dec in the
considered in three regions of offset frequency. range of 1 to 35 Hz and gradually decreases until
First, at the offset frequency lower than approxi- 300 kHz offset frequency before decreasing with
mately 10 Hz, only the multiplier noise dominates the slope to -20 dBc/dec before it reaches the noise
the total phase noise while both VCO and filter floor of -155 dBc at approximately 400 MHz. On
noises are relatively low at low offset frequency the other hand, the total phase noise SΦ2(f) starts
and do not significantly contribute to the total at -20 dBc/dec at the low offset frequency range

Figure 8. The simulated total phase noise SΦ1(f) Figure 9. The simulated total phase noise SΦ2(f)
in [dBc/Hz] versus offset frequency in in [dBc/Hz] versus offset frequency in
[Hz]. [Hz].
Songklanakarin J. Sci. Technol. Simulations of phase noise in phase-locked loops
Vol. 29 No. 4 Jul. - Aug. 2007 1026 Limkumnerd, S. and Eungdamrong, D.

(2000) reach the noise floor at the same magnitude

of -155 dBc/Hz.

Conclusions

Detailed analysis and simulations of

mathematical phase noise models of phase-locked
loops have been presented. Unlike other existing
phase noise models in which the filter noises are
not included, this work has not only included the
filter noises for the phase noise model but also
compared the noise contribution between passive
Figure 10. Simulated total phase noises of the PLL and active loop filters. The results show that
in dBc/Hz versus offset frequency in simulations of phase noises without an inclusion
(Hz); (a) the total phase noise SΦ1(f) of filter noises may not be accurate because the
with noises from the passive filter, (b) filter noises, particularly the active filter, signific-
the total phase noise SΦ2(f) with noises antly contribute to the total phase noise rather than
from the active filter, (c) the total phase other components. Moreover, the passive filter
noise with no noises from the filter. does not significantly dominate the phase noise
at low offset frequency while the active filters
entirely dominate. Therefore, the passive filter is
about 1 to 10 Hz. Then the slope remains plateau a more efficient filter for PLL circuit at low offset
until it reaches to 1 kHz offset frequency and frequency. The phase noise models presented in
changes to -40 dBc/dec before it reaches the noise this paper are relatively simple and can be used for
floor of -155 dBc at approximately 10 MHz. In accurate phase noise prediction for PLL designs.
addition, plots of both SΦ1(f) and SΦ2(f) start at same
magnitude of -39 dBc/Hz at 1 Hz offset frequency. References
With reference to Figure 10, comparisons Dai, L. and Harjani, R. 2002. Design of Low-Phase-
between the simulated total phase noise with no Noise CMOS Ring Oscillators, IEEE Transac-
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Vol. 29 No. 4 Jul. - Aug. 2007 1028 Limkumnerd, S. and Eungdamrong, D.

Appendix

Appendix A.1: Analytical treatments for Equa- Appendix A.2: Analytical treatments for Equa-
tions (10) and (11) tions (13) and (14)
Vout1 (s) 1  1   1 1 
F1 (s) = =  R1 + sC1   R1 +
sC2 V (s) sC2 sC1 
Iin (s)   Fs (s) = out2 = 
Iin (s)  R2 
1 1 + sR1C1  
=  
sC2 sC1
 1 1 + sR1C1 
1 + sR1C1  sC sC1 
= 
2
s 2C1C2
=  R2 
1 1 + sR1C1  
+  
sC2 sC1

1 + sR1C1  1 + sR1C1 
=  s 2
C C 
sC1 + sC2 + s 2 R1C1C2 =
1  1 2 
R2  1 1 + sR1C1 
1 1 + sR1C1   sC + sC 
=    2 1 
s  C1 + C2 + sR1C1C2 
1  1 + sR1C1 
1 + sR1C1 =  
 sC1 + sC2 + s R1C1C2 
2
= R2
sR C C
s(C1 + C2 )(1 + 1 1 2 )
C1 + C2 1  1 + sR1C1 
=  
kF1 (1 + sτ F1_1 ) sR2  C1 + C2 + sR1C1C2 
=
s(1 + sτ F1_2 ) 1 + sR1C1
=
sR C C
sR2 (C1 + C2 )(1 + 1 1 2 )
where kF1 = 1/(C1+C2) is a constant. τF1_1 = R1C1 and C1 + C2
τF1_2 = (R1C1C2)/(C1+C2) are time constants
(1 + sτ F2_1 )
=
sτ F2_3 (1 + sτ F2_2 )

where τF2_1 = R1C1, τF2_2 = (R1C1C2)/(C1+C2) and

τF2_3 = R2(C1+C2) are time constants in addition the
DC gain of the active filter