Chapter 10

Case study: phase-locked loops

Open any book on frequency standards and you meet the same figure: boxes labeled oscillator, multiplier, detector, arrows chasing each other in a ring. This chapter is about reading those diagrams — what each box is for, and how the closed ring reshapes every noise you met in chapters 1–9. The diagrams here are alive: click the blocks.

Why lock anything?

Because no single oscillator has every virtue. A quartz crystal is magnificently quiet close to the carrier but comes at one fixed, low frequency — and multiplying it up to microwave frequencies multiplies its phase noise too, ruining its floor far from the carrier. A microwave VCO is tunable and clean far out, but its phase wanders badly close in. A phase-locked loop welds the two into a single output that follows the reference below a chosen loop bandwidth and the VCO above it: each oscillator is trusted exactly where it is good. That one sentence is the whole chapter; the demo in the noise-shaping section below lets you find the best weld point yourself.

Anatomy of a phase-locked loop

Four blocks and a ring of arrows. The VCO's output is compared against the reference in a phase detector; the phase error is filtered and fed back to the VCO's tuning voltage so that the error shrinks. Optionally a $\div N$ divider sits in the feedback path so a low reference frequency can discipline a high output frequency. Click each block (or tab to it) to see its job and its signal:

Noise shaping: the loop bandwidth decides whom to trust

Now the mathematics, compactly. Around the ring, phase error is converted to volts ($K_d$), filtered ($F$), and turned back into phase by the VCO — which, being an integrator from volts to phase, contributes a factor $1/(if)$. The open-loop gain is therefore

$$ G(f) \;=\; \frac{K}{i f}\, F(f), \qquad K = K_d K_0 \ (\times\, 1/N \text{ with a divider}), $$

and standard feedback algebra (the same $1/(1+G)$ bookkeeping as chapter 9's servo section) gives the closed-loop output phase as a weighted blend of the two oscillators' phases:

$$ \varphi_{\mathrm{out}} \;=\; H\,\varphi_{\mathrm{ref}} \;+\; (1 - H)\,\varphi_{\mathrm{vco}}, \qquad H = \frac{G}{1+G}. $$

To dock the notation explicitly onto chapter 9: its disturbance suppression $1/(1+G)$ is this chapter's $1-H$, and its sensor-noise injection $G/(1+G)$ is this chapter's $H$ — same algebra, with the "sensor" here being the reference-plus-phase-detector chain. Because of the VCO's built-in $1/f$, $|G|$ is large at low $f$ and small at high $f$: $H$ is a low-pass (the loop follows the reference, but only slowly) and $1-H$ is a high-pass (the loop cannot fix fast VCO wiggles). Note the flip side: inside the bandwidth the loop writes the reference's phase noise — and the divider's, and the phase detector's — straight onto the VCO. For the workhorse integrator-plus-lead ("type 2") loop filter the closed-loop response takes the textbook second-order form (with $s = i\,2\pi f$ and $x = f/f_n$):

$$ H(s) = \frac{2\zeta\omega_n s + \omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}, $$

$$ |H|^2 = \frac{1 + 4\zeta^2 x^2}{(1-x^2)^2 + 4\zeta^2 x^2}, \qquad |1-H|^2 = \frac{x^4}{(1-x^2)^2 + 4\zeta^2 x^2}, $$

with natural frequency $f_n \approx$ the loop bandwidth $f_L$ and damping $\zeta$. The composite spectrum is then computed exactly like a chapter-9 noise budget — in linear power units, never in dB:

// closed-loop composite (linear units, then 10·log10 for dBc/Hz)
const x = f / fn, z2 = 4 * zeta * zeta * x * x;
const den = (1 - x * x) ** 2 + z2;
const H2 = (1 + z2) / den;                       // |H|²  : low-pass
const E2 = x ** 4 / den;                         // |1−H|²: high-pass
const Ltot = N * N * Lref(f) * H2 + Lvco(f) * E2;

Three discoveries the sliders hand you. Too narrow: the VCO's close-in wander leaks through $|1-H|^2$ and the phase error explodes — at $f_L = 100\,$Hz it exceeds a radian, and a real loop would be slipping cycles. Too wide: you copy the multiplied reference's noise floor, $20\log_{10}N$ above the bare crystal's, out to offsets where the VCO was already quieter. Low $\zeta$: the loop resonates at $f_n$ and $|H|$ peaks (about $+5.8\,$dB at $\zeta = 0.3$), stacking a bump onto the output — when you see a bump on a beat note, you are usually looking at somebody's loop bandwidth.

The 20 log N tax

Why does multiplying a beautiful crystal wreck its floor? Phase is counted in cycles, and a frequency multiplier by definition produces $N$ output cycles per input cycle — so every phase excursion, deterministic or random, is multiplied by $N$ too: $\varphi \to N\varphi$, hence $S_\varphi \to N^2 S_\varphi$, hence

$$ \mathcal{L}(f) \;\longrightarrow\; \mathcal{L}(f) + 20\log_{10} N \quad \text{(everywhere, at every offset } f\text{)}. $$

This is the rule chapter 6 leaned on when it declared frequency multiplication "noise-neutral" in Leeson's model ($\mathcal{L} \propto \nu_0^2$ exactly cancels the $+20\log_{10}N$) — here is where the tax itself comes from. Worked number: a 5 MHz OCXO with $\mathcal{L} = -120$ dBc/Hz at 1 kHz, multiplied to 10 GHz ($N = 2000$, $20\log_{10}2000 \approx 66$ dB), arrives at $-54$ dBc/Hz. This single line is why microwave synthesis is hard, and it is exactly what the $N$ selector in the demo above does: the whole reference curve rides up by $20\log_{10}N$, dragging the optimal loop bandwidth down with it. The converse is a gift: frequency dividers reduce phase noise by $20\log_{10}N$ — down to the divider's own additive noise floor, which is why a good divider is one of the quietest boxes in a synthesizer.

An atomic clock is a lock loop

Now the classic diagram this chapter promised to teach you to read: the atomic frequency standard. A quartz oscillator is multiplied up to the caesium hyperfine transition near 9.19 GHz, the atoms are interrogated, and a control voltage steers the quartz so the interrogation stays centered on the atomic line. The output of the clock is the quartz — the atoms never emit your signal; they only referee it. Click the blocks:

The S-curve: how a symmetric line steers a loop

The atomic resonance is a symmetric curve: its height tells you how far $|\,\nu - \nu_0|$ you are from center, but not on which side — useless for feedback, which needs a signed error. The fix is chapter 8's lock-in trick: dither the interrogation frequency across the line at $f_m$ and demodulate the atomic response at $f_m$. Sitting left of center, a positive dither swing increases the signal; sitting right of center, it decreases it — the component at $f_m$ flips sign as you cross the line. The demodulated output versus detuning is an antisymmetric S-curve, linear through zero: a proper error signal, conjured out of a symmetric bump.

Play with the dither depth. The center slope — the discriminator sensitivity, in chapter 8's language literally volts per hertz — is greatest when the dither swings by about half a linewidth (the numerical optimum here is an amplitude $a \approx 0.35\,\Delta\nu$, i.e. swinging roughly between the half-maximum points; anything from $0.2$ to $0.6\,\Delta\nu$ is within about 20% of it). Too small a dither and the modulated signal vanishes into detector noise; too large and the slope collapses — you are averaging over the whole line, effectively broadening it.

And here the whole course lands in one formula. Within the loop bandwidth, the clock output's fractional frequency stability is set by the atoms and the detection SNR, not by the quartz:

$$ \sigma_y(\tau) \;\approx\; \frac{\Delta\nu}{\nu_0}\, \frac{1}{\mathrm{SNR}}\, \tau^{-1/2}. $$

The discriminator splits the line to $\delta\nu \approx \Delta\nu/\mathrm{SNR}$ in one second (chapter 8: noise referred to the input via the S-curve slope), and successive interrogations average down like white frequency noise — the $\tau^{-1/2}$ regime you met on every Allan-deviation plot in chapter 7 is this loop at work. High $Q = \omega_0/\Delta\omega$ and high SNR: that is the entire recipe for an atomic clock, and the reason the definition of the second lives in a box like this.

The same diagram everywhere

Once you can read one of these rings, you can read them all. Five roles, four labs:

The moral: the loop algebra — $H$, $1-H$, the servo bump, the $20\log_{10}N$ bookkeeping — is identical in every column. All the physics hides in the discriminator: its sensitivity (slope, V/Hz) and its noise floor set what the lock can possibly deliver, exactly as the atomic line's $Q$ and SNR set a clock's $\sigma_y(\tau)$.

Exercises

Where you started, where you are

Ten chapters ago this course opened with one equation and three knobs: $V(t) = A\cos(2\pi\nu_0 t + \varphi)$. Everything since has been that equation under stress. The knobs wiggled deliberately gave sidebands (chapter 2); wiggled randomly they needed ensembles (chapter 3) and spectral densities (chapter 4); the spectra turned out to be power laws with personalities (chapter 5), which on the phase knob became $\mathcal{L}(f)$, linewidths and jitter (chapter 6), compressed into $\sigma_y(\tau)$ for clocks (chapter 7), priced in per-root-hertz for sensors (chapter 8), stacked into budgets and filter functions for LIGO (chapter 9), and finally reshaped at will by a feedback ring (this chapter). If one sentence survives: noise is a spectral density; every measurement is a filter function; every quoted number is their integral. The next time an Allan plot, a dBc/Hz spec or a lock-loop block diagram crosses your desk, it should read like prose. Go twiddle the real knobs.