A Time-to-Digital Converter Based AFC for Wideband Frequency

A Time-to-Digital Converter Based AFC for Wideband
Frequency Synthesizer
Deping Huang2,1, Wei Li1, Jin Zhou1, Ning Li1, Junyan Ren1, Jinghong Chen2
State Key Laboratory of ASIC & System, Fudan University Shanghai 201203, China
Department of Electrical Engineering, Southern Methodist University, Dallas, TX 75205 USA
Email: [email protected]
Abstract—The automatic frequency calibration (AFC) technique
is routinely used in the wideband frequency synthesizers which
contain multiple voltage-controlled oscillator (VCO) tuning
curves. In this paper, a counter-based AFC design method is
presented. The relationship between the AFC counting time and
the VCO tuning curve characteristic is quantitatively analyzed.
An AFC circuit which uses a time-to-digital converter (TDC) in
the counting process is developed. Simulation results show that
the proposed circuit significantly reduces the AFC calibration
time while preserving the calibration accuracy. The simulated
error-free AFC time of the proposed AFC is less than 3 μs.
To cover a wide frequency range and to achieve a
relatively low VCO gain (Kvco), a switched capacitor array is
often utilized in wideband LC VCOs. In such an approach, a
fixed varactor or a varactor array is continuously tuned by an
analog control voltage to achieve a fine frequency tuning,
while a binary-weighted array of switched capacitors is
digitally controlled to carry out coarse frequency tuning. In
doing so, the wide frequency range is accomplished by
multiple VCO tuning curves with each of the tuning curves
exhibiting a relatively smaller Kvco, thus ensuring the VCO
noise performance. In such a structure, an AFC circuit is
required to properly select one of the VCO tuning curves at
startup of the PLL. After that, the PLL takes over to adjust the
varactor control voltage performing the fine tuning. Several
AFC techniques have been proposed [1]-[3]. One method is to
monitor the VCO control voltage Vctrol [1]. When the correct
tuning curve is selected, the Vctrol should be within a predefined voltage range. However, in this approach the PLL
must first be settled before one can monitor the Vctrol voltage
for the AFC operation. This results in a long calibration time.
Another widely used method is to build a dedicated AFC loop
for coarse frequency calibration [2]-[3]. Fig. 1 shows a typical
implementation of a fractional-N PLL with a dedicated AFC
loop. During the AFC process, the VCO control node is
disconnected from the loop filter and is set at VDD/2. The
VCO frequency is detected and recorded as the number of the
VCO cycles in a counting interval. As shown in Fig. 1, the
Fig. 1. Block diagram of a fractional-N PLL with an AFC loop.
PLL reference clock is divided down by a factor of 2M and a
half period of the divided-down signal is used as the counting
interval. Thus, the length of the counting window is 2M-1×Tref,
where Tref is the period of the PLL reference clock. The result
is then compared with the expected number of VCO cycles,
2M-1×N.α, where 2M-1 indicates the number of PLL reference
clock cycles in the counting window and N.α is the division
ratio of the fractional-N PLL with N being the integer part and
α being the fractional part. Based on the comparison result, a
successive approximation algorithm is used to find out the
optimal VCO tuning curve. The calibration time in such a
counter-based frequency detection scheme, however, is
limited by the frequency detection accuracy, which is also
observed in the previous designs [2]-[3]. This paper
quantitatively analyzes the relationship between the AFC
calibration time and the VCO tuning curve characteristic. In
Section II, we first discuss the error mechanisms of the
counter-based AFC and then analyze the minimum required
AFC calibration time. Section III describes the proposed AFC
which adopts a TDC as a fractional counter to improve the
calibration accuracy. Section IV presents the simulation
results, and Section VI concludes the paper.
This work was sponsored by the National Natural Science Foundation of
China under Grant 61176029 and US National Science Foundation under
award 1040429.
978-1-4673-0219-7/12/$31.00 ©2012 IEEE
Fig. 2. (a) Counter-based frequency detection; (b) Frequency detection error.
Frequency Detection Accuracy
The frequency calibration accuracy of the AFC is mainly
determined by the frequency detection accuracy which is
based on cycle counting. As shown in Fig. 2(a), if there are N
cycles of signal under detection in a given counting window
TGATE, then the detected signal frequency fdetect is N/TGATE.
However, due to the initial phase uncertainty, the jitter in the
gating signal and the integer counter rounding effect, the
counting result for a signal with a frequency fdetect can be
round(fdetect·TGATE) ± 1 as indicated in Fig. 2(b), where the
round function returns the closest integer to the argument.
This causes a frequency detection error of ±1/TGATE, which is
also the frequency resolution that the counter-based frequency
detector can provide [3]. To reduce the error, long enough
counting time is required, which unfortunately increases the
synthesizer locking time.
B. Minimum Counting Time Requirement
Since the frequency detection accuracy of the AFC is
limited by the width of the counting window TGATE, it is
necessary to find out the minimum required counting time for
a given VCO tuning curve characteristic. Besides the
frequency detection error, the finite-precision numerical effect
of the division ratio 2M-1×N.α also causes an incorrect AFC
operation. Typically, only the integer part of 2M-1×N.α is
treated as the expected number of VCO cycles. Thus, the
worst-case numerical truncation error is 1/2M-1.
In Fig. 3, curves A and B are two VCO tuning curves with
fA and fB being their midpoint frequencies, respectively.
During the AFC operation, fA and fB are detected and then
compared with the target frequency fT. The curve with its
midpoint frequency closer to the target frequency will be
chosen. However, due to the frequency detection error and the
division ratio truncation error, the AFC may select a wrong
tuning curve. For example, if the target frequency fT is larger
than the rightmost frequency of tuning curve B (the gray
portion of curve A) and AFC chooses tuning curve B, then the
Fig. 3. Two worst-case scenarios causing incorrect AFC operation.
PLL cannot be locked. Two worst-case scenarios of incorrect
AFC operations are shown in Fig. 3. Fig. 3(a) depicts a
scenario where the frequency detection error causes the
detected frequencies to both exhibit a positive offset relative to
their true value, i.e., the detected frequencies are fA′ and fB′,
respectively; and the target frequency due to the division ratio
truncation error is fT′. In this case, the AFC will mistakenly
choose curve B if fT′ becomes closer to fB′ than fA′. Fig. 3(b)
depicts another worst-case scenario where the detected
frequency errors have opposite signs and there is no division
ratio truncation error. Then, if fA′ and fB′ cross over the
frequency (fA +fB)/2, the AFC will also mistakenly choose
tuning curve B. To avoid these two situations, |fA′ − fT′| < |fB′ −
fT′| should hold and thus the following two inequalities need to
be satisfied.
f step − 0.5 K vco ΔV + 2 / (2 M −1 Tref ) < 0.5 K vco ΔV − 2 / (2 M −1 Tref )
f step / 2 > 1/ (2M −1Tref )
In deriving the above inequalities, the frequency detection
error of 1/TGATE=1/(2M-1Tref) as discussed in Section II.A is
assumed. The worst-case frequency offset of fT due to the
division ratio truncation error as discussed in II.A is also
1/(2M-1Tref). Eq. (1) ensures that fT′ is closer to fA′ than fB′ in
Fig. 3(a) while (2) guarantees that fA′ and fB′ do not cross over
the frequency point (fA +fB)/2 in Fig. 3(b). From (1) and (2),
the minimum required counting window width can thus be
derived as
2 ⎫⎪
2 M −1 Tref = max ⎨
⎩⎪ K vco ΔV − f step f step ⎭⎪
Given a set of VCO tuning curves, (3) can be used to
derive the required AFC counting time. For instance,
assuming that fref = 40 MHz, KvcoΔV = 35 MHz, and the
overlapping ratio of the tuning curves defined as 1−fstep/
(KvcoΔV) is 50% and thus the fstep is 17.5 MHz, then from (3),
2M-1 needs to be larger than 9.1. Since M is an integer, M = 5 is
required, and thus the length of the counting window needs to
be 2M-1×Tref = 16×Tref.
Fig. 4 shows the proposed AFC circuit that uses a TDC to
detect the ratio between fvco and fref. There are two benefits of
using TDC for the frequency detection. First, the frequency
detection accuracy is increased because of the fractional
period estimation. Secondly, since the detected frequency
contains the fractional period information, the comparison in
the AFC is no longer limited to the integer part of 2M-1×N.α.
More fractional part of the division ratio can be included in the
comparison, thus the target frequency offset caused by
division ratio truncation is also reduced. These two
improvements help to significantly reduce the AFC time while
preserving the AFC accuracy.
The TDC in this work shown in Fig. 4 adopts a
configuration similar to that in [5], where an integer counter is
used in conjunction with a fractional counter. The fractional
counter implementation is shown in Fig. 5. The digital
fractional phase is obtained by passing the VCO clock through
a chain of inverters. In Fig. 5, the tr[n] is the time distance
between each counting edge and the last VCO edge. It is
normalized to the VCO period Tpvco. The Nfrac[n] is the
residual fractional VCO period in the counting window.
The smallest time interval Tres that can be resolved in the
fractional counter is the TDC inverter delay. For a typical 130nm CMOS technology, it is about 30 ps. The counting
resolution becomes Tres/Tvco, where Tvco is the VCO output
period. Therefore, the frequency detection resolution and
division ratio truncation error are both reduced to
According to the above analysis, it can be found that the
AFC frequency detection principle is quite similar to the
Time-to-Digital Converter in an All Digital PLL (ADPLL)
[4]. In the divider-less ADPLL, the TDC compares the DCO
output signal with the reference signal by computing the
number of DCO periods between two adjacent reference edges,
i.e. the ratio between the output frequency and the reference
frequency. This can be implemented by an integer counter, but
the resolution is only equal to the DCO period which is too
coarse for most applications. To improve the resolution, a
fractional counter computing the residual time distance
between the two reference edges after integer counting has
been used [4]. The fractional counter measures the residual
time distance between each reference edge and the last DCO
edge before that using a multi-phase DCO output signal. This
concept can be applied to the AFC design to reduce the
counting error due to the initial phase uncertainty and the jitter
of the VCO output.
Fig. 4. TDC-based AFC.
Fig. 5. Fractional counter implementation in the TDC-based AFC.
(Tres/Tvco)/(2M-1Tref). Eq. (3) for the TDC-based AFC can thus
be modified as
⎧⎪ 4Tres / Tvco
2T / T ⎫⎪
2 M −1 Tref = max ⎨
, res vco ⎬ (4)
f step ⎭⎪
⎪⎩ K vco ΔV − f step
Assume that Tres = 30 ps and the VCO output frequency is
5 GHz, for the same set VCO tuning curves in Section II B,
the minimum required 2M-1 is reduced to 1.4, which is much
smaller than that of the conventional AFC method.
It should be mentioned that the multi-phase counting
requires more hardware and thus more power consumption
than the conventional AFC approach. However, the AFC is
only activated at the startup of the PLL; therefore, there is no
power penalty during normal PLL operation.
The conventional integer counter-based AFC and the
TDC-based AFC have been designed in a 0.13 μm CMOS
technology to verify the above analysis. A simulation setup
shown in Fig. 6 is developed to find out the frequency
searching behavior of these two circuits. Figs. 7 and 8 show
the simulation results of the frequency calibration processes.
The VCO outputs 64 discrete frequency bands ranging from
3.5 GHz to 5 GHz. The fstep is about 23.5 MHz. The reference
clock frequency is 40 MHz. In the simulation, the calibration
accuracies of the two AFCs are compared for a given counting
time, which is set as 2×Tref. Fig. 7 shows the division ratio
truncation effect on the frequency calibration process. The
target frequency fT is 3778 MHz which is between two VCO
output frequencies of 3761.8 MHz and 3785.6 MHz. The
3785.6 MHz tuning curve should be selected since it is closer
x 10
Conventional AFC
TDC-based AFC
Target Frequency: 3778MHz
Fig. 6. Simulation setup.
to the target frequency. However, because of the division ratio
truncation error, the 3760 MHz is considered as the target
frequency in the integer counter-based AFC. As a result, the
AFC incorrectly chooses the 3761.8 MHz tuning curve. On
the other hand, the TDC-based AFC correctly chooses the
3785.6 MHz tuning curve as it has a smaller division ratio
truncation error.
A counter-based AFC design analysis is presented in this
paper. The minimum required counting time for a given set of
VCO tuning curves is quantitatively derived. To the authors’
knowledge, this is the first time that such a quantitative
analysis is developed. A TDC-based AFC method is proposed
to improve the frequency detection accuracy and reduce the
target frequency offset. The simulation results show that the
TDC-based AFC is more robust than the conventional integer
counter-based AFC method and greatly reduces the AFC time
while preserving the frequency calibration accuracy.
x 10
Fig. 7 Division ratio truncation effect on the AFC operation.
x 10
Conventional AFC
TDC-based AFC
Fig. 8 shows the initial phase uncertainty effect on the
calibration process. A delay element is deliberately introduced
to the VCO output in the simulation setup to adjust its phase.
The target frequency is set at 4015.6 MHz. Simulation finds
that when the delay ranges from 260 ps to 340 ps, the optimal
tuning curve cannot be selected in the case of the conventional
AFC scheme. This is because the 4023.6 MHz VCO output is
detected as 4040 MHz under this initial phase condition in the
conventional AFC method. This causes the AFC to mistakenly
determine that the 3999.8 MHz tuning curve is closer to the
target frequency. The TDC-based AFC, on the other hand, is
less sensitive to the counting signal initial phase uncertainty
due to its fractional phase estimation. Thus the frequency
point is accurately detected and 4023.6MHz frequency is
correctly identified to be closer to the target frequency. In
summary, the simulation results show that the TDC-based
AFC correctly chooses the optimal tuning curves with a
2×Tref counting window due to the improved frequency
detection accuracy and the smaller target frequency offset. The
conventional AFC, on the other hand, mistakenly selects the
suboptimal tuning curves in both simulations because of the
frequency detection error and the target frequency offset error.
Target Frequency: 4015.6MHz
x 10
Fig. 8 Effect of initial phase uncertainty on the AFC operation.
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