TFE4188 - Lecture 9

Oscillators

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Goal

Why

Introduction to Crystal Oscillators

Introduction to VCOs

Introduction to Relaxation-oscillators

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Why

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I just want the most precise clock that can be made !!!

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Atomic clocks

Cesium standard

The second is defined by taking the fixed numerical value of the cesium frequency Cs, the unperturbed ground-state hyper-fine transition frequency of the cesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1

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Microchip 5071B Cesium Primary Time and Frequency Standard

< 5E-13 accuracy high-performance models
Accuracy levels achieved within 30 minutes of startup
< 8.5E-13 at 100s high-performance models
< 1E-14 flicker floor high-performance models

"Ask for a quote" => The price is really high, and we don't want to tell you yet

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Rubidium standard

Rubidium standard, use the rubidium hyper-fine transition of 6.8 GHz (6834682610.904 Hz)

The MAC is a passive atomic clock, incorporating the interrogation technique of Coherent Population Trapping (CPT) and operating upon the D1 optical resonance of atomic Rubidium Isotope 87.

A rubidium clock is basically a crystal oscillator locked to an atomic reference.

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Crystal oscillators

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Assuming zero series resistance

Z_{in} = \frac{s^2 C_F L + 1}{s^3 C_P L C_F + s C_P + s C_F}

Divide top and bottom by \(s\) and the shape is easier to see:

Z_{in} = \frac{1}{s}\cdot\frac{L C_F s^2 + 1}{L C_F C_P s^2 + C_F + C_P}

See Crystal oscillator impedance for a detailed explanation, or the interactive version where the motional and static elements are sliders and the pulling is worked out for you.

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Negative transconductance compensate crystal series resistance

Long startup time caused by high Q

Can fine tune frequency with parasitic capacitance

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Controlled Oscillators

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Ring oscillator

t_{pd} \approx R C

R \approx \frac{1}{gm} \approx \frac{1}{\mu_n C_{ox} \frac{W}{L} (VDD - V_{th})}

C \approx \frac{2}{3} C_{ox} W L

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t_{pd} \approx \frac{2/3 C_{ox} W L}{\frac{W}{L} \mu_n C_{ox}(VDD - V_{th})}

\(f = \frac{1}{2 N t_{pd}} = \frac{\mu_n (VDD-V_{th})}{\frac{4}{3} N L^2}\)

K_{vco} = 2 \pi \frac{\partial f}{\partial VDD} = \frac{2 \pi \mu_n}{\frac{4}{3} N L^2}

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Capacitive load

f = \frac{\mu_n C_{ox} \frac{W}{L} (VDD - V_{th})}{2N\left(\frac{2}{3}C_{ox}WL + C\right)}

K_{vco} = \frac{2 \pi \mu_n C_{ox} \frac{W}{L}}{2N\left(\frac{2}{3}C_{ox}WL + C\right)}

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Realistic

I = C \frac{dV}{dt}

f \approx \frac{ I_{control} + \frac{1}{2}\mu_p C_{ox} \frac{W}{L} (VDD - V_{control} - V_{th})^2}{C \frac{VDD}{2} N}

K_{vco} = 2 \pi \frac{\partial f}{\partial V_{control}}

K_{vco} = - 2 \pi \frac{\mu_p C_{ox} \frac{W}{L} \left(VDD - V_{control} - V_{th}\right) }{C\frac{VDD}{2}N}

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Digitally controlled oscillator

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Differential

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LC oscillator

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f \propto \frac{1}{\sqrt{LC}}

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Relaxation oscillators

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\[V_1 = I R\]

I = C \frac{dV}{dt}

dt = \frac{C V_2}{I} = \frac{C I R}{I}

f = \frac{1}{dt} = \frac{1}{RC}

f_o = \frac{1}{2}f = \frac{1}{2RC}

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Summary

The precision ladder: atomic clocks, then crystals (ppm), then LC (phase-noise kings on chip), then rings, then RC relaxation - each rung cheaper and noisier
A crystal is a mechanical resonator with Q in the tens of thousands; the Pierce circuit keeps it ringing with one inverter
Ring oscillators are small, tune over decades, and follow every millivolt of supply - which is why the PLL supply-controls one on purpose
Current starving and capacitive load make the ring controllable; the varactor does the same for the LC tank
The relaxation oscillator charges C to IR and resets: the cheap always-on clock for waking things up
An oscillator's frequency stability over temperature and supply, not its schematic, decides where it may be used

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Would you like to know more?

Crystal oscillators

The Crystal Oscillator - A Circuit for All Seasons [@razavi17]

High-performance crystal oscillator circuits: theory and application [@vittoz88]

Ultra-low Power 32kHz Crystal Oscillators: Fundamentals and Design Techniques [@xu21]

A Sub-nW Single-Supply 32-kHz Sub-Harmonic Pulse Injection Crystal Oscillator [@kim21]

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CMOS oscillators

The Ring Oscillator - A Circuit for All Seasons [@razavi19]

A Study of Phase Noise in CMOS Oscillators [@razavi96]

An Ultra-Low-Noise Swing-Boosted Differential Relaxation Oscillator in 0.18-um CMOS [@lee20]

Ultra Low Power Frequency Synthesizer

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Thanks!

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