Chapter 2 Modeling
2.2 Frequency Domain Model
A model of the PLL dynamics that links deviations in the divide value to phase deviations at the PLL output is now derived. Our analysis is based on the princi-ple of superposition; in particular, we will subtract out the nominal condition that Foutc = NnomFref when defining all phase variables of interest. Given these variable definitions, we will derive equations that describe the behavior of the PFD and Di-vider components. Block diagrams of all the PLL components are then presented, and combined to form the overall PLL model.
2.2.1 Definition of Phase/Frequency Signals
We begin our modeling effort by defining instantaneous frequency signals associated with the divider, PLL, and reference outputs. The instantaneous frequency of the divider output during time interval k is defined to be
Fdiv[k] = 1 tk− tk−1
,
where tk is the time at which the kth rising edge of the divider output, Div (t), occurs;
Figure 2.2 illustrates the relevant notation. Substitution of tk = kT + ∆tk into the above expression yields
Fdiv[k] = 1
T + ∆tk− ∆tk−1
. (2.1)
2.2. FREQUENCY DOMAIN MODEL 45
Alternatively, the divider output can be expressed in terms of the instantaneous frequency of the PLL output, which is assumed to be relatively constant during the time intervals marked by tk. Specifically, we define the PLL output frequency during time interval k as Fout[k], and write
Fdiv[k] = Fout[k]
N [k] . (2.2)
Finally, the reference frequency is defined as Fref = 1/T.
Ref(t)
T
Div(t)
kT (k+1)T tk tk+1
Dtk Dtk+1
Figure 2.2: Definition of tk.
Given the above frequency variables, we now derive expressions for phase devia-tions at the reference, divider, and PLL outputs in recursive form. These signals are considered to be continuous-time in nature, but will be defined only at sample times tk. By definition, the phase deviation of the reference source is zero, so that
Φref[k] = 0. (2.3)
Samples of the phase deviations at the divider output are defined by Φdiv[k]− Φdiv[k− 1] = 2πFdiv[k]− 1
T
(T + ∆tk− ∆tk−1); (2.4) Similarly, samples of the PLL output phase deviation, Φout[k], are defined by
Φout[k]− Φout[k− 1] = 2πFout[k]− Nnom
1 T
(T + ∆tk− ∆tk−1) (2.5)
2.2.2 Derivation of PFD Model
The relationship between Φref[k], Φdiv[k], and E(t) will be briefly examined here;
further modeling details linking these variables are given in Chapter 9. We begin by defining a phase error signal as
Φe[k] = Φref[k]− Φdiv[k].
Substitution of Equations 2.3 and 2.4 into the above expression yields Φe[k] = 2π∆tk− ∆tk−1
T + Φe[k− 1] (2.6)
with the aid of Equation 2.1. To obtain a nonrecursive expression for Φe[k], we first relate its value to an initial time sample i:
Φe[k] = 2π
k m=i+1
∆tm− ∆tm−1
T + Φe[i]. (2.7)
Assuming initial conditions are zero, we obtain Φe[k] = ∆tk
T 2π. (2.8)
We relate Φe[k] to E(t) by constructing a new signal, ˆΦe(t), as Φˆe(t) =
∞ k=−∞
Φe[k]δ(t− kT ); (2.9)
this signal is an impulse train that is modulated by the instantaneous phase error samples, Φe[k]. (The ‘hat’ symbol serves as a reminder that ˆΦe(t) is a modulated impulse train rather than a continuous signal.) As explained in Chapter 9, the error signal generated by the PFD output can be approximated as
E(t)≈ T π
Φˆe(t) + Espur(t), (2.10) where Espur(t) is a square wave of period T with fifty percent duty cycle. By defining signals ˆΦdiv(t) and ˆΦref(t) as
Φˆdiv(t) =
∞ k=−∞
Φdiv[k]δ(t− kT ), ˆΦref(t) =
∞ k=−∞
Φref[k]δ(t− kT ) = 0, (2.11) E(t) is described in terms of the divider and reference phase signals as
E(t)≈ T
π( ˆΦref(t)− ˆΦdiv(t)) + Espur(t). (2.12) Note that the impulses forming ˆΦdiv(t) are referenced to time kT , while the samples in Φdiv[k] are referenced to time tk. Our PFD model therefore ignores time jitter caused by ∆tk; the impact of this jitter will be discussed in Chapter 9.
2.2. FREQUENCY DOMAIN MODEL 47
2.2.3 Derivation of Divider Model
We now relate the PLL and divider output phase deviations by linearizing the action of the divider. To do so, we first decompose the divide value at time increment k as
N [k] = Nnom + n[k],
where n[k] represents the instantaneous divide value deviation from its nominal set-ting of Nnom. (Note that both n[k] and Nnom may be non-integer in value.) All approximations to follow are based on the assumption that|n[k]| Nnom for all k.
Proceeding with the derivation, we first combine Equations 2.2 and 2.4 to obtain Φdiv[k]− Φdiv[k− 1] = 2π
Fout[k]
N [k] − 1 T
(T + ∆tk− ∆tk−1); (2.13) This expression can be approximated as
Φdiv[k]−Φdiv[k−1] ≈ 2π
1 Nnom
1− n[k]
Nnom
Fout[k]− 1 T
(T +∆tk−∆tk−1). (2.14) Since (Fout[k]/Nnom)(T + ∆tk− ∆tk−1)≈ 1, Equation 2.14 can be reduced to
Φdiv[k]− Φdiv[k− 1] ≈ 2π Nnom
Fout[k]− Nnom1 T
(T + ∆tk− ∆tk−1)− n[k]. (2.15) Finally, we use Equation 2.5 to express the above relationship as
Φdiv[k]− Φdiv[k− 1] ≈ 2π
Nnom (Φout[k]− Φout[k− 1] − n[k]) . (2.16) To obtain a non-recursive version of Equation 2.16, we first relate values at time k to those at an initial time sample i
Φdiv[k]− Φdiv[i]≈ 2π Nnom
Φout[k]− Φout[i]− k
m=i+1
n[m]
.
By setting i = 0 and assuming initial conditions are zero, we obtain Φdiv[k] ≈ 2π
Nnom
Φout[k]− k
m=1
n[m]
. (2.17)
To allow closed loop analysis of the overall PLL, it is advantageous to recast Equation 2.17 in terms of impulse trains as
Φˆdiv(t)≈ 2π Nnom
Φˆout(t)− t
−∞ˆn(τ )dτ
, (2.18)
where
Φˆout(t) =
∞ k=−∞
Φout[k]δ(t− kT ), ˆn(t) = ∞
k=−∞
n[k]δ(t− kT ), and ˆΦdiv(t) is defined in Equation 2.11.
2.2.4 Modeling of Divider Sampling Operation
So far, we have examined phase deviation signals within the PLL only at sample points set by tk. However, the output phase deviation of the PLL, Φout(t), that is generated by the VCO is assumed to be continuous-time in nature. We now discuss a frequency domain model that links this continuous-time signal to the impulse train representation, ˆΦout(t), used in the divider model.
Φout(t) and ˆΦout(t) are stochastic in nature, and therefore have undefined Fourier transforms. We seek the frequency-domain relationship between these two signals; its derivation will be obtained by temporarily assuming that these signals are determin-istic in nature, and that their Fourier transforms are well defined as
Φout(f ) =
∞
−∞Φout(t)e−j2πftdt, Φˆout(f ) =
∞
−∞
Φˆout(t)e−j2πftdt.
Since the impulses in ˆΦout(t) effectively sample e−j2πft when forming ˆΦout(f ), we obtain
Φˆout(f ) =
∞ k=−∞
Φˆout[k]e−j2πfT k.
It is a property of the Fourier transform [61] that Φout(f ) and ˆΦout(f ) are related as Φˆout(f ) = 1
T
∞ n=−∞
Φout(f − n T).
Thus, ˆΦout(f ) is composed of copies of Φout(f ) that are scaled in magnitude by 1/T and shifted in frequency from one another with spacing 1/T . We will assume that the bandwidth of Φout(f ) is much smaller than 1/T , so that negligible aliasing occurs between the copies of Φout(f ) within ˆΦout(f ). In addition, it is assumed that the continuous-time, lowpass filtering performed by the VCO and loop filter reduces the influence of the high frequency components in ˆΦout(f ) to negligible levels. (The influence of ˆΦout(f ) is manifested within the PLL through the signal ˆΦe(t) by the PFD output, which passes through the loop filter and VCO dynamics before influencing Φout(t).)
These concepts are illustrated for a general signal, x(t), in Figure 2.3. The draw-ing suggests that the overall dynamics of the PLL are influenced primarily by the
‘baseband’ copy of Φout(f ) within ˆΦout(f ). As shown at the bottom of the figure, we model the conversion between these signals in the frequency-domain as a simple scaling operation of the continuous-time signal by 1/T . Although this assumption will break down when trying to perform noise analysis at frequencies close to 1/T , it will prove useful and reasonably accurate when performing closed loop analysis for most frequencies of interest in our application. Note that the double outline of the box in the figure is meant to serve as a reminder that the model is an approximation since copies of the baseband signal are produced.
2.2. FREQUENCY DOMAIN MODEL 49
Impulse Train Modulator
DynamicsCT
T1 X(f) X(f) T
x(t)
x[k]
'Baseband' copy of X(f) has dominant effect
on PLL dynamics
1T 1T
2T 2
0 T f
f
f
f
1T
T
x(t) x(t)
T1 x(t) x(t)
Approx.
x(t)
Impulse Train Modulator
x[k]
Figure 2.3: Frequency domain modeling of discrete-time signals in PLL.
2.2.5 Overall Model
Figure 2.4 displays linearized, frequency-domain models for each of the PLL compo-nents. We briefly review the significant characteristics of each block.
The divider effectively samples the continuous-time output phase deviation, Φout(t), and then divides its value by Nnom. The output phase of the divider, ˆΦdiv(t), is di-rectly influenced by ˆΦn(t), which is formed as the integration of deviations in the divider value, ˆn(t). The integration of ˆn(t) is a consequence of the fact that the divider output is a phase signal, whereas ˆn(t) causes an incremental change in the divider output frequency. Since ˆn(t) and ˆΦn(t) consist of modulated impulse trains, the integration operation between them is modeled as the transfer function 1/(1−D), where D = e−j2πfT.
The PFD can be considered as a translator between the discrete-time phase error, Φe[k], and the continuous-time signal, E(t), that is sent into the loop filter. In other words, the PFD can be viewed as a DT to CT converter. Assuming that phase detection is implemented digitally as an XOR gate, the DT to CT conversion amounts to creating a square-wave output whose instantaneous duty cycle is a function of Φe[k].
As Chapter 9 explains, the resulting waveform, E(t), can be expressed as the sum of an impulse train modulated by Φe[k], and a square wave with constant duty cycle, Espur(t). In the PFD model shown in Figure 2.4, we have defined a perturbation
KV jf
vIN(t) Fout(t)
Fvn(t) H(f) vIN(t)
N1NOM Fout(t)
2p 1 - D1 n(t)
Fdiv(t)
T
Fout(t) 1
( D e -j2pfT )
Fn(t) N[k]
Multi-Modulus Divider
Out(t) Div(t)
Ref(t) PFD
Div(t)
E(t)
T 1
-1
Tp
Fdiv(t) Fref(t)
DN(t)
LoopFilter
E(t) VIN(t)
VIN(t) Out(t)
PLL Block Small-Signal Model
E(t)
E(t) Fe(t) e(t)
Figure 2.4: Linearized models of PLL components.
source, DN (t), that consists of the sum of Espur(t) and jitter noise from the reference frequency, divider logic, and PFD [46]. Espur(t) will cause spurious noise at frequencies that are multiples of 1/T , while the jitter noise is expected to be broadband in nature.
As a side remark, it should be noted that the PFD model assumes phase deviations are small so that the frequency detection capability of the PFD need not be considered.
As for the remaining components, the loop filter is modeled as a frequency domain transfer function, H(f ), and the VCO is represented by an integrator with gain Kv (Hz/V) that produces an output phase signal. The VCO includes an accompanying noise source, Φvn(t), to model phase noise that occurs in a practical implementation of a VCO. In the context of analysis in this thesis, we will assume that the spectral density of Φvn(t) decreases with increasing offset frequency from the carrier at -20 dB/dec [62, 63].
Each of the component models are combined to form the overall PLL linearized model shown in Figure 2.5.