IEEE Solid-State Circuits Magazine - Spring 2017 - 9

the net negative resistance in Figure 4(b),
following an envelope given by
exp (t/x), where x = R n / (2L 1) and R n
is the absolute value of the net negative
resistance. For example, a 10-MHz crystal oscillator with a Q of 5,000 can take
roughly 0.5 ms to settle. This issue poses several difficulties. In low-power applications that operate with a low duty
cycle-as in sensors-the start-up time
translates to a higher power consumption. Also, communication systems
that come out of the sleep mode cannot begin operation until the settling
is completed. Finally, the simulation of
the oscillator becomes a very lengthy
task, especially if the circuit must reach
steady state for its phase noise to be
computed accurately.

Drive-Level Dependency
Crystals behave peculiarly if they
remain inactive: their equivalent series
resistance rises considerably. The
series resistance falls back to it original value after the crystal vibrates for
some time. This effect is called drivelevel dependency. A crystal oscillator that is turned on after a period of
inactivity may fail unless the negative
resistance is strong enough. As a rule
of thumb, we select this resistance
about four times R S in Figure 4(a).

Oscillation at Overtones
Actual crystals also exhibit resonances
at higher frequencies (overtones)
that are approximately harmonically
related to the first. Thus, the topology
of Figure 4(c) can oscillate at an overtone, a property exploited in highfrequency designs. On the other hand,
low-frequency oscillators must avoid a
solution at overtones. This is possible
by inserting a resistor in series with
the output of the inverter in Figure 4(c)
so as to reduce the loop gain at higher
frequencies. This resistor can also limit
the crystal's power dissipation, which,
if excessive, could cause damage.

Questions for the Reader
1) Estimate the oscillation frequency
of Figure 2(a) if R 1 and R 2 are large.

C1

S1

Vin

A

C2

S4

Cp1

X

B

-
+

S3

A0

Vout

Cp2

Figure 6: An integrator circuit including parasitics.

C1

S1

S2

C2

X

Vin

-
A0

S4

Vout

+

S3

Figure 7: A noninverting integrator.

2) How does the finite output impedance of M 1 and M 2 in Figure 4(c)
affect the oscillator's performance?

Answers to Last Issue's Questions
1) In the circuit of Figure 6, C p2 appears in series with C 2 when S 3
turns off. Does the charge injected
by S 1 corrupt the sampled value
in this case?
No, it does not. The charge
injected by S 1 is later removed
by S 4 .
2) Given that the op amp in Figure 7 is
placed in an inverting configuration, how do we intuitively explain the noninverting operation
of the integrator?
The front-end passive sampling circuit in fact inverts the
signal. This can be seen by noting
that, if S 2 is absent, then the voltage generated on the right plate
of C 2 in the hold mode is equal
to -Vin .

References

[1] R. Bechmann, "Piezoelectricity - frequency control," in Proc. Annu. Symp. Frequency Control, May 1964, pp. 43-92.
[2] W. Cady, "The piezoelectric resonator,"
Proc. IRE, vol. 10, pp. 83-114, Apr. 1922.
[3] J. Luscher, "Oscillator circuit including a
quartz crystal operating in parallel resonance," U.S. Patent 3 585 527, June 15,
1971.
[4] R. Walton, "Electronically controlled timepiece using low power MOS transistor
circuitry," U.S. Patent 3 664 118, Sept. 9,
1970.
[5] C. Fonjallaz and E . Vittoz, "Circuits
electroniques pour montres-bracelet a
quartz," in Proc. Int. Congress Chronometry, 1969, pp. B244-1.
[6] S. Eaton, "Micropower crystal-controlled
oscillator design using RCA COS/MOS inverter," RCA application note ICAN-6539,
1971.
[7] M. P. Forrer, "Survey of circuitry for wristwatches," Proc. IEEE, vol. 60, pp. 1047-
1054, Sept. 1972.
[8] E . Vittoz, M. Deg rauwe, a nd S. Bitz,
"High-performance crystal oscillator
circuits: theory and application," IEEE J.
Solid-State Circuits, vol. 23, pp. 774-783,
June 1988.

IEEE SOLID-STATE CIRCUITS MAGAZINE

S p r i n g 2 0 17

9



Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Spring 2017

IEEE Solid-State Circuits Magazine - Spring 2017 - Cover1
IEEE Solid-State Circuits Magazine - Spring 2017 - Cover2
IEEE Solid-State Circuits Magazine - Spring 2017 - 1
IEEE Solid-State Circuits Magazine - Spring 2017 - 2
IEEE Solid-State Circuits Magazine - Spring 2017 - 3
IEEE Solid-State Circuits Magazine - Spring 2017 - 4
IEEE Solid-State Circuits Magazine - Spring 2017 - 5
IEEE Solid-State Circuits Magazine - Spring 2017 - 6
IEEE Solid-State Circuits Magazine - Spring 2017 - 7
IEEE Solid-State Circuits Magazine - Spring 2017 - 8
IEEE Solid-State Circuits Magazine - Spring 2017 - 9
IEEE Solid-State Circuits Magazine - Spring 2017 - 10
IEEE Solid-State Circuits Magazine - Spring 2017 - 11
IEEE Solid-State Circuits Magazine - Spring 2017 - 12
IEEE Solid-State Circuits Magazine - Spring 2017 - 13
IEEE Solid-State Circuits Magazine - Spring 2017 - 14
IEEE Solid-State Circuits Magazine - Spring 2017 - 15
IEEE Solid-State Circuits Magazine - Spring 2017 - 16
IEEE Solid-State Circuits Magazine - Spring 2017 - 17
IEEE Solid-State Circuits Magazine - Spring 2017 - 18
IEEE Solid-State Circuits Magazine - Spring 2017 - 19
IEEE Solid-State Circuits Magazine - Spring 2017 - 20
IEEE Solid-State Circuits Magazine - Spring 2017 - 21
IEEE Solid-State Circuits Magazine - Spring 2017 - 22
IEEE Solid-State Circuits Magazine - Spring 2017 - 23
IEEE Solid-State Circuits Magazine - Spring 2017 - 24
IEEE Solid-State Circuits Magazine - Spring 2017 - 25
IEEE Solid-State Circuits Magazine - Spring 2017 - 26
IEEE Solid-State Circuits Magazine - Spring 2017 - 27
IEEE Solid-State Circuits Magazine - Spring 2017 - 28
IEEE Solid-State Circuits Magazine - Spring 2017 - 29
IEEE Solid-State Circuits Magazine - Spring 2017 - 30
IEEE Solid-State Circuits Magazine - Spring 2017 - 31
IEEE Solid-State Circuits Magazine - Spring 2017 - 32
IEEE Solid-State Circuits Magazine - Spring 2017 - 33
IEEE Solid-State Circuits Magazine - Spring 2017 - 34
IEEE Solid-State Circuits Magazine - Spring 2017 - 35
IEEE Solid-State Circuits Magazine - Spring 2017 - 36
IEEE Solid-State Circuits Magazine - Spring 2017 - 37
IEEE Solid-State Circuits Magazine - Spring 2017 - 38
IEEE Solid-State Circuits Magazine - Spring 2017 - 39
IEEE Solid-State Circuits Magazine - Spring 2017 - 40
IEEE Solid-State Circuits Magazine - Spring 2017 - 41
IEEE Solid-State Circuits Magazine - Spring 2017 - 42
IEEE Solid-State Circuits Magazine - Spring 2017 - 43
IEEE Solid-State Circuits Magazine - Spring 2017 - 44
IEEE Solid-State Circuits Magazine - Spring 2017 - 45
IEEE Solid-State Circuits Magazine - Spring 2017 - 46
IEEE Solid-State Circuits Magazine - Spring 2017 - 47
IEEE Solid-State Circuits Magazine - Spring 2017 - 48
IEEE Solid-State Circuits Magazine - Spring 2017 - 49
IEEE Solid-State Circuits Magazine - Spring 2017 - 50
IEEE Solid-State Circuits Magazine - Spring 2017 - 51
IEEE Solid-State Circuits Magazine - Spring 2017 - 52
IEEE Solid-State Circuits Magazine - Spring 2017 - 53
IEEE Solid-State Circuits Magazine - Spring 2017 - 54
IEEE Solid-State Circuits Magazine - Spring 2017 - 55
IEEE Solid-State Circuits Magazine - Spring 2017 - 56
IEEE Solid-State Circuits Magazine - Spring 2017 - 57
IEEE Solid-State Circuits Magazine - Spring 2017 - 58
IEEE Solid-State Circuits Magazine - Spring 2017 - 59
IEEE Solid-State Circuits Magazine - Spring 2017 - 60
IEEE Solid-State Circuits Magazine - Spring 2017 - 61
IEEE Solid-State Circuits Magazine - Spring 2017 - 62
IEEE Solid-State Circuits Magazine - Spring 2017 - 63
IEEE Solid-State Circuits Magazine - Spring 2017 - 64
IEEE Solid-State Circuits Magazine - Spring 2017 - 65
IEEE Solid-State Circuits Magazine - Spring 2017 - 66
IEEE Solid-State Circuits Magazine - Spring 2017 - 67
IEEE Solid-State Circuits Magazine - Spring 2017 - 68
IEEE Solid-State Circuits Magazine - Spring 2017 - 69
IEEE Solid-State Circuits Magazine - Spring 2017 - 70
IEEE Solid-State Circuits Magazine - Spring 2017 - 71
IEEE Solid-State Circuits Magazine - Spring 2017 - 72
IEEE Solid-State Circuits Magazine - Spring 2017 - 73
IEEE Solid-State Circuits Magazine - Spring 2017 - 74
IEEE Solid-State Circuits Magazine - Spring 2017 - 75
IEEE Solid-State Circuits Magazine - Spring 2017 - 76
IEEE Solid-State Circuits Magazine - Spring 2017 - 77
IEEE Solid-State Circuits Magazine - Spring 2017 - 78
IEEE Solid-State Circuits Magazine - Spring 2017 - 79
IEEE Solid-State Circuits Magazine - Spring 2017 - 80
IEEE Solid-State Circuits Magazine - Spring 2017 - 81
IEEE Solid-State Circuits Magazine - Spring 2017 - 82
IEEE Solid-State Circuits Magazine - Spring 2017 - 83
IEEE Solid-State Circuits Magazine - Spring 2017 - 84
IEEE Solid-State Circuits Magazine - Spring 2017 - 85
IEEE Solid-State Circuits Magazine - Spring 2017 - 86
IEEE Solid-State Circuits Magazine - Spring 2017 - 87
IEEE Solid-State Circuits Magazine - Spring 2017 - 88
IEEE Solid-State Circuits Magazine - Spring 2017 - 89
IEEE Solid-State Circuits Magazine - Spring 2017 - 90
IEEE Solid-State Circuits Magazine - Spring 2017 - 91
IEEE Solid-State Circuits Magazine - Spring 2017 - 92
IEEE Solid-State Circuits Magazine - Spring 2017 - 93
IEEE Solid-State Circuits Magazine - Spring 2017 - 94
IEEE Solid-State Circuits Magazine - Spring 2017 - 95
IEEE Solid-State Circuits Magazine - Spring 2017 - 96
IEEE Solid-State Circuits Magazine - Spring 2017 - Cover3
IEEE Solid-State Circuits Magazine - Spring 2017 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2023
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2022
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2021
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_spring2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_winter2020
https://www.nxtbook.com/nxtbooks/ieee/mssc_fall2019
https://www.nxtbook.com/nxtbooks/ieee/mssc_summer2019
https://www.nxtbook.com/nxtbooks/ieee/mssc_2019summer
https://www.nxtbook.com/nxtbooks/ieee/mssc_2019winter
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018fall
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018summer
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018spring
https://www.nxtbook.com/nxtbooks/ieee/mssc_2018winter
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2017
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2016
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2015
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_winter2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_fall2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_summer2014
https://www.nxtbook.com/nxtbooks/ieee/solidstatecircuits_spring2014
https://www.nxtbookmedia.com