β: This English translation is in beta — the Traditional-Chinese original is the authoritative version.

Quadrature generation and coupled-oscillator phase noise

Where this page sits (up front): this is an advanced design page. It takes the need every SerDes/transceiver designer faces daily — generating a clock pair with a $90^\circ$ phase offset (quadrature, i.e. I/Q) — and explains it inside this site's existing ISF + generalized Adler machinery. Prerequisites: first understand the ISF of [P1] (isf_definition, effective_isf), the generalized-Adler injection locking of [P3] (paper_003), and the ILFD / frequency division of [P4] (paper_004). Without those three pages, the equations here will look like they appear out of nowhere.

Quadrature (a pair of signals I and Q with a $90^\circ$ phase offset) is a basic building block of modern transceivers: image-reject mixers, single-sideband modulation, 4-phase sampling in half-rate SerDes, phase detection in CDRs — all of them need a clean I/Q pair with small phase error. The problem: generating quadrature itself carries a phase-noise cost, and the cost structure differs completely between generation methods. This page answers:

What this page answers: (1) What does the phase-noise cost of each of the three mainstream quadrature generation methods look like? (2) In a coupled QVCO (coupled quadrature VCO), why is coupling strength ↔ I/Q phase error ↔ phase noise a triangular trade-off? (3) For the coupled oscillator pair, is their noise common-mode or differential-mode correlated, and how does that determine the famous $\sim 3$ dB? (4) How do you rigorously wire the "coupling injection" back to the ISF / generalized Adler of [P3]?

Physical intuition (conclusion first): a coupled QVCO is just "two oscillators injection-locking each other" — A's output injects into B, B's output injects into A. Once you see it that way, the entire [P3] machinery applies immediately: the coupling current $i_{c}$ sees an effective ISF $\tilde\Gamma$, "pulls" the other oscillator's phase through the harmonics $c_n$ of $\tilde\Gamma$, and the locking dynamics obey the generalized Adler equation ([P3] Eq.(30)). The stronger the coupling, the more tightly the two phases are bound (smaller phase error), but the coupling device also injects extra noise and pulls the frequency away from the tank resonance — that is where the triangular trade-off comes from.

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1. The three quadrature generation methods and their phase-noise costs

Put the three routes side by side first. Each route's "cost" is different; understanding the differences matters more than memorizing conclusions.

(a) Coupled QVCO (coupled quadrature VCO)

Two identical LC VCOs, with a pair of coupling transistors injecting A's output into B and B's output (inverted) into A, forcing the two to lock at a fixed $90^\circ$ phase offset.

• How it generates quadrature: under the constraint "total loop phase $=0$", the coupling arrangement forces the two oscillators to $90^\circ$ (derivation in Section 4; this is the steady-state solution of the [P3] generalized Adler equation in the "mutual injection" case).
• Phase-noise cost: in theory it can be better than a single oscillator — differential/anti-phase coupling of two identical VCOs averages out the uncorrelated noise portion (a potential $\sim 3$ dB benefit, Section 3); but the coupling device's own noise gets in, and strong coupling pulls the frequency off the tank peak, lowering the effective $Q$ and raising phase noise instead. Whether the net is a gain or a loss depends on the coupling strength (the triangular trade-off of Section 2).

(b) Divide-by-2 (÷2) from a $2f_0$ source (ILFD / static divider)

Run a $2f_0$ oscillator, then divide by 2. A master-slave flip-flop divider natively outputs two clocks $90^\circ$ apart (because $\div 2$ maps one input period onto an output half-period $=90^\circ$); or use an ILFD (injection-locked frequency divider) — inject $2f_0$ into an $f_0$ oscillator and lock via the 2nd harmonic of the ISF down to $f_0$ (this is exactly the ILFD of paper_004, subharmonic locking). The full lock-range derivation ($\omega_L=I_{inj}\vert\tilde\Gamma_N\vert/2$) and the "half-wave symmetry can't divide by 2" payoff have their own page: injection_locked_division.

• How it generates quadrature: the two complementary outputs of the $\div 2$ (or the four nodes of a differential ÷2) are naturally $90^\circ$ apart; quadrature accuracy is set by circuit symmetry, independent of tank detuning — its biggest advantage over the QVCO.
• Phase-noise cost: an ideal $\div N$ improves phase noise by $20\log_{10}N$ dB. Reason: division divides the phase by $N$, so the phase error is divided by $N$ too, and the phase power ($\propto\phi^2$) by $N^2$:

$$\mathcal{L}_{out}(\Delta f)=\mathcal{L}_{2f_0}(\Delta f)-20\log_{10}N$$

For $\div 2$ that is $-20\log_{10}2=-6.02$ dB. But this only accounts for the "source" noise; you still have to build a clean $2f_0$ oscillator first (high-frequency VCOs usually have lower $Q$, smaller $q_{max}$, and are inherently noisier), and the divider (especially a CML latch) adds its own noise floor. The net benefit is the "$-6$ dB improvement" minus "the noisier $2f_0$ source + divider noise". This formula is a standard frequency-synthesis result (external literature, not among the five source PDFs), but its physical basis — the phase being divided by $N$ — is consistent with this site's [P1] phase definition; the ILFD locking mechanism is the subharmonic injection locking of [P4].

Dimension / order-of-magnitude check: if a $2f_0=10$ GHz source has $\mathcal{L}(1\text{MHz})=-110$ dBc/Hz, an ideal $\div 2$ gives $5$ GHz, $\mathcal{L}=-116$ dBc/Hz. But if that 10 GHz VCO's $\Gamma_{rms}/q_{max}$ is more than 6 dB worse than a VCO built directly at 5 GHz, the $\div 2$ gains nothing — a common practical trap.

(c) RC-CR polyphase filter

Purely passive: an RC low-pass on one path (phase $-45^\circ$), a CR high-pass on the other (phase $+45^\circ$); at $\omega=1/RC$ the two paths differ by $90^\circ$. Cascading multiple stages (polyphase) widens the bandwidth.

• How it generates quadrature: passive phase shift, $90^\circ$ set by $R$, $C$ matching; no second oscillator needed.
• Phase-noise cost: a passive network generates no new phase noise (in theory), but it has two costs: (1) insertion loss — each RC-CR stage attenuates $\sim 3$ dB, requiring a buffer stage whose thermal noise becomes additive noise (degrading SNR, raising the far-out noise floor); (2) quadrature accuracy is sensitive to the absolute values of $R$, $C$ and to frequency — off $1/RC$ you get I/Q phase and amplitude error, requiring multiple stages and calibration. It does not change the close-in $1/f^2$, $1/f^3$ phase noise (that is set by the PLL/VCO), only raising the far-out floor slightly via buffer additive noise.

The three methods compared

Dimension	coupled QVCO	$\div 2$ (ILFD/static)	RC-CR polyphase
Source of quadrature	coupling forces $90^\circ$	$\div 2$ natively $90^\circ$	passive phase shift $\pm45^\circ$
What sets quadrature accuracy	tank detuning + coupling symmetry	circuit symmetry (independent of detuning)	$R,C$ matching + frequency
Close-in PN	can beat or lose to a single VCO (depends on coupling)	$-20\log_{10}N$ (minus the noisier source)	unchanged (VCO-determined)
Extra noise sources	coupling device	high-frequency source + divider	buffer additive noise
Main trap	strong coupling pulls frequency, drops $Q$	$2f_0$ source inherently noisier	insertion loss + narrow bandwidth
Link back to this site's machinery	[P3] generalized Adler (Section 4)	[P4] ILFD (subharmonic)	purely linear network (no ISF involved)

The qualitative comparison and design trade-offs of the three methods are oscillator-design common knowledge (external literature, not among the five source PDFs; standard references such as Razavi's transceiver texts, Behbahani polyphase 1999). The ILFD locking dynamics and the $\div 2$ subharmonic injection mechanism belong strictly to [P4] (see paper_004); the $20\log_{10}N$ improvement of $\div N$ is a standard frequency-synthesis result.

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2. Coupled QVCO: coupling strength ↔ I/Q phase error ↔ phase noise

This is the core triangular trade-off of QVCO design. Define the coupling strength first, then show how the three quantities pull against each other.

Definition of the coupling strength $m$

Let each VCO's core (−$G_m$) transconductance sustain its own oscillation, providing current $I_{core}$; the coupling transistors inject the other oscillator's signal into this one with injection-current amplitude $I_{c}$. Define the coupling factor

$$m\equiv\frac{I_{c}}{I_{core}}$$

$m$ is the dimensionless "how strong the coupling is relative to the core". As $m\to 0$ the two decouple and run independently; large $m$ binds them tightly.

How the three pull against each other (physics)

(i) coupling ↑ → I/Q phase error ↓: the coupling provides the restoring force that pulls the pair back to $90^\circ$. Any mismatch that makes the two free-running frequencies differ (process variation in $L$, $C$, $g_m$) tries to push the phase offset away from $90^\circ$; the stronger the coupling, the larger the restoring force and the smaller the residual phase error. Intuition and order of magnitude (external literature, standard QVCO result):

$$\Delta\phi_{IQ}\ \approx\ \frac{Q}{m}\,\frac{\Delta\omega_0}{\omega_0}\quad\text{(order of magnitude; stronger coupling }m\text{ shrinks the error; larger }Q\text{ enlarges it)}$$

where $\Delta\omega_0/\omega_0$ is the relative detuning of the two tanks (caused by mismatch). An equivalent form is $\Delta\phi_{IQ}\approx\Delta\omega_0/\omega_L$, where $\omega_L=\dfrac{m\,\omega_0}{2Q}$ is the Adler lock range of the mutual injection (the [P3] Eq.(35) form; external QVCO literature). Intuition: stronger coupling (large $m$) widens the lock range and strengthens the restoring force, so the residual I/Q error shrinks; but a sharper tank (large $Q$) makes the lock range narrower, so the same mismatch is harder to pull back to $90^\circ$ and the I/Q phase error is larger.

Numerical example (order of magnitude): take $m=0.3$, $Q=10$, tank mismatch $\Delta\omega_0/\omega_0=0.1\%$; then

$$\Delta\phi_{IQ}\approx\frac{Q}{m}\,\frac{\Delta\omega_0}{\omega_0}=\frac{10}{0.3}\times0.001\approx0.033\ \text{rad}\approx1.9^\circ.$$

That is, for a sharp high-$Q$ ($Q=10$) tank, even with mismatch squeezed to $0.1\%$, coupling of $m\sim0.3$ only brings the I/Q error down to about $1.9^\circ$; to go smaller you must strengthen the coupling (larger $m$) or reduce the mismatch (smaller $\Delta\omega_0/\omega_0$). Unit check: $\dfrac{(\text{dimensionless})}{(\text{dimensionless})}\times(\text{dimensionless})=$ dimensionless $=$ rad ✓. Conversely, to get within $0.5^\circ\approx8.7\times10^{-3}$ rad (same $\Delta\omega_0/\omega_0=0.1\%$), you need $Q/m\lesssim 8.7$, i.e. $m\gtrsim Q/8.7\approx1.15$ — for a $Q=10$ tank that means quite strong coupling, which is exactly the quantitative bound behind "high-$Q$ QVCOs are hungrier for coupling strength".

(ii) coupling ↑ → phase noise ↑ (two mechanisms):

• The coupling device injects its own noise: like the core devices, the coupling transistors have thermal/flicker noise; it hits the tank node directly, sees an effective ISF, and contributes extra phase noise (quantified via the ISF in Section 4). The stronger the coupling (larger coupling device, larger current), the larger this noise.
• The coupling pulls the oscillation frequency off the tank peak, lowering the effective $Q$: the coupling injection is a current out of phase with the tank voltage, so the oscillator must shift away from the tank resonance to keep the total loop phase $=0$ (this is the steady state of injection pulling). Off resonance → the tank's phase slope at the operating frequency (i.e. the effective $Q$) shrinks → by Leeson/ISF, phase noise $\propto 1/Q^2$ rises. This is the main phase-noise penalty of a strongly coupled QVCO.

(iii) net trade-off: so QVCO design is finding the sweet spot between "coupling strong enough to suppress I/Q phase error" and "coupling so strong it raises phase noise". Practical experience (external literature): there is an optimum $m$ (typically of order $m\sim 0.2\!-\!0.5$, topology-dependent); too small and the I/Q error and unlock risk grow, too large and the effective-$Q$ degradation dominates the phase noise.

Parallel vs series coupling (the two hookups)

	parallel coupling	series coupling
How the coupling device connects	in parallel with the core −$G_m$ device at the tank node	in series between the core device and the tail
Current allocation	coupling and core compete for the same tank-node current	coupling current flows through the core, sharing the bias current
Phase-noise intuition	coupling device adds noise directly; larger frequency pulling	coupling burns no extra current, smaller frequency pulling → generally better PN
Cost	noisier but simpler to design	tighter headroom, harder to design

Series coupling usually has better phase noise: it opens no extra noisy current branch into the tank, has smaller frequency pulling, and preserves more of the effective $Q$. This is one of the core conclusions of the QVCO literature (external literature, not among the five source PDFs; standard references such as the Andreani QVCO series, Romanò parallel vs series QVCO). This site does not redraw their schematics.

Design knobs (coupled QVCO):

1. Coupling strength $m$: trade I/Q error (wants large $m$) against phase noise (wants small $m$) → pick the optimum in between.
2. Coupling hookup: series generally beats parallel (smaller frequency pulling, no extra current).
3. Reduce coupling-device noise: use large-size, low-$g_m$ / low-flicker coupling transistors to cut the $\Gamma_{rms}$ they inject.
4. Reduce mismatch: good layout, common-centroid → smaller I/Q error at the same $m$, allowing weaker coupling.

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3. Noise correlation: common-mode vs differential-mode and the ~3 dB

A QVCO is "two identical oscillators", so you must distinguish which noise is correlated between the two and which is not — that determines whether the famous $\sim 3$ dB is a gain or a loss.

Bookkeeping of the two noise types

Think of the second oscillator (Q) as a "replica" of the first (I). For the phase-noise power of each of the I/Q paths:

• Differential-mode / uncorrelated noise (each VCO core device's own thermal noise, independent between the two): the phase noise of the two is uncorrelated. When you bind the two strongly into one oscillating system, the output phase is set jointly by both — two uncorrelated power contributions average, and the output phase noise is $\sim 3$ dB ($10\log_{10}2$) lower than a single oscillator. This is the QVCO's theoretical "two oscillators buy 3 dB" benefit.

$$\mathcal{L}_{QVCO}\ \approx\ \mathcal{L}_{single}-10\log_{10}2\ =\ \mathcal{L}_{single}-3.01\ \text{dB}\quad\text{(uncorrelated core noise only)}$$

• Common-mode / correlated noise (noise hitting both oscillators simultaneously, or shared by the coupling path — e.g. shared tail, shared bias, coupling devices): the perturbations on the two are fully correlated, and averaging does not reduce it — correlated power gets no $\sqrt{N}$ averaging. This noise cancels part of the 3 dB benefit.

How to do the accounting (key caveat)

The honest version of the $\sim 3$ dB: "two oscillators → $-3$ dB" holds only for uncorrelated noise, and only if the coupling devices introduce no significant correlated noise and do not drag down the effective $Q$. In practice:

1. Uncorrelated thermal noise of the core devices: enjoys $-3$ dB ✓.
2. Noise injected by the coupling devices: it is added, and often correlated (shared coupling path) → cancels part of the benefit.
3. Effective $Q$ reduced by coupling-induced frequency pulling: phase noise $\propto 1/Q^2$ rises → eats some more.

So a real QVCO is not necessarily 3 dB better than a single oscillator — often only 1–2 dB, and under strong coupling even worse. The $\sim 3$ dB is the upper bound of "averaging two uncorrelated sources", not a guaranteed value (external literature, not among the five source PDFs).

Accounting mnemonic: correlated ↔ amplitudes add (total power $\propto N^2$, no reduction relative to a single source) → no averaging benefit; uncorrelated ↔ powers add (total power $\propto N$) → enjoys $-10\log_{10}N$. This is the same correlation-accounting logic as the Section 1 $\div N$ improvement of $20\log_{10}N$ (phase divided by $N$, a deterministic correlated scaling) — two faces of one bookkeeping rule.

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4. Wiring the coupling injection back to the ISF / generalized Adler ([P3])

This is the mathematical core of the page: the locking dynamics of a coupled QVCO are just the [P3] generalized Adler equation applied to "mutual injection". We invent no new theory; we use the equations of this site's paper_003 directly.

Step 1: the effective ISF seen by the coupling current

The coupling current $i_{c}(t)$ injected from A into B, like any injection current, drives B's phase through the unit-bearing ISF $\tilde\Gamma(x)=\Gamma(x)/q_{max}$ (units rad/C, [P3] Eq.(26), p.2113). In other words, the coupling injection is nothing special — it sees B's effective $\tilde\Gamma$ at the injection node, and "receives" the other oscillator's signal through the Fourier harmonics $c_n$ of $\tilde\Gamma$ ([P1] Eq.(12)): the better the harmonic alignment, the more effective the coupling (locking).

The coupling device's own noise $i_{c,n}(t)$ travels the same $\tilde\Gamma$, so its phase-noise contribution follows the [P1] recipe: $\mathcal{L}\propto\Gamma_{rms}^2/q_{max}^2$ ([P1] Eq.(21), p.185, with $\Gamma$ replaced by the effective ISF seen by the coupling injection). This is the quantitative outlet for Section 2's "the coupling device injects extra noise".

Step 2: generalized Adler for mutual injection

Now write B's injection into A as well. For oscillator A, the relative phase $\theta_A=\phi_A-\phi_{ref}$ obeys the time-averaged generalized Adler equation of [P3] ([P3] Eq.(30), p.2113, with a plus sign in front of the averaged term):

$$\frac{d\theta_A}{dt}=(\omega_{0,A}-\omega_{ref})+\frac{1}{T}\int_{T}\tilde\Gamma\big(\omega_{ref}t+\theta_A\big)\,i_{c,B\to A}(t)\,dt$$

where $i_{c,B\to A}$ is the coupling current B injects into A (proportional to B's output $\propto\cos(\omega_{ref}t+\theta_B)$). Write the symmetric equation for B. Key point: a coupled QVCO is nothing but two such Adler equations coupled to each other — $\theta_B$ appears in A's equation and $\theta_A$ in B's. Rearranged into the lock characteristic form of [P3] ([P3] Eq.(33), p.2114):

$$\frac{d\theta_A}{dt}=(\omega_{0,A}-\omega_{ref})+\Omega(\theta_A-\theta_B)$$

$\Omega(\cdot)$ is the coupling-induced average frequency shift as a function of the relative phase difference (the stronger the coupling, the wider the range of $\Omega$, the firmer the lock).

Step 3: steady-state solution → why $90^\circ$

The QVCO's coupling arrangement (A injects into B in phase, B injects into A inverted, differing by a minus sign) means that in steady state ($d\theta_A/dt=d\theta_B/dt=0$, both at the same frequency), the two Adler equations are self-consistent only at the phase difference

$$\theta_A-\theta_B=\pm 90^\circ$$

— the plus or minus sign (I leading or lagging Q) is one of two symmetric stable solutions, decided at start-up. The $90^\circ$ is not patched in; it is the steady-state constraint of the mutually injecting Adler equations. Any tank detuning ($\omega_{0,A}\ne\omega_{0,B}$) nudges this $90^\circ$ off a little, and how much is resisted by the slope of $\Omega$ (i.e. the coupling strength) — this is exactly the differential-equation version of Section 2's $\Delta\phi_{IQ}\propto Q/m$.

Order-of-magnitude / dimension check: $\Omega(\theta)$ has the same units as $(\omega_0-\omega_{ref})$, rad/s ✓. Strong coupling → wide range of $\Omega$ → the same detuning $(\omega_{0,A}-\omega_{0,B})$ needs only a tiny phase shift to be compensated by $\Omega(\theta_A-\theta_B)$ → small I/Q error. Weak coupling → narrow $\Omega$ → once the detuning grows there is no solution (unlock: the two no longer share a frequency, the phase slips periodically — exactly the injection pulling of [P3]).

Step 4: stitching the three pieces into one picture

Design phenomenon (Sections 2, 3)	Corresponding [P3] / ISF quantity
Coupling strength $m$	width of the range of $\Omega(\theta)$ (= lock range; [P3] Eq.(33))
I/Q phase error vs mismatch	detuning divided by the slope of $\Omega$ (strong coupling → steep slope → small error)
Extra phase noise from the coupling device	effective $\Gamma_{rms}$ seen by the coupling injection, via [P1] Eq.(21)
Strong coupling pulls frequency, drops $Q$	steady state $\theta^\*\ne 0$ → operating frequency off the tank peak
$90^\circ$ appears naturally	steady-state constraint of the anti-phase mutually injecting Adler equations

One sentence: a coupled QVCO = two oscillators locking each other via the generalized Adler of [P3]; the coupling current travels the ISF of [P1], so "locking (quadrature)" and "extra phase noise" are two faces of the same $\tilde\Gamma$ — fully consistent with [P3]'s point that "the same ISF accounts for both phase noise and injection locking".

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Validity and failure conditions

Condition	When it holds	What happens when it fails
Weak-to-moderate coupling, phase linearity	generalized Adler ([P3] Eq.(30)) holds, $90^\circ$ steady-state solution exists	strong coupling → amplitude modulation (needs the APF of [P4]), the ISF itself is altered
Two nearly identical VCOs, small mismatch	I/Q error $\propto Q/m$, the $\sim 3$ dB bound is approachable	large mismatch → large I/Q error, even unlock (pulling)
Coupling-device noise / correlation negligible	$-3$ dB (averaging of uncorrelated core noise) holds	correlated coupling/shared noise → cancels the 3 dB, possibly worse
Operating frequency still near the tank peak	effective $Q$ preserved, PN not penalized by $1/Q^2$	strong coupling pulls frequency → effective $Q$ drops, close-in PN rises
Clean $\div 2$ source	the $-20\log_{10}N$ improvement is a net gain	$2f_0$ source too noisy / divider noise → the improvement is eaten up

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Key takeaways

• The three quadrature generation methods have different cost structures: the coupled QVCO pays with coupling-device noise + pulling-induced $Q$ drop (net value depends on coupling strength); $\div 2$ (ILFD) ideally improves by $-20\log_{10}N$ but needs a clean $2f_0$ source first; RC-CR polyphase generates no new close-in PN, paying instead with insertion loss + buffer additive noise + narrow bandwidth.
• The coupled-QVCO triangular trade-off: coupling ↑ → I/Q phase error ↓ (stronger restoring force), but phase noise ↑ (coupling device injects noise + pulling drops the effective $Q$) → an optimum coupling strength $m$ exists. Series coupling generally beats parallel (smaller frequency pulling, no extra current).
• The $\sim 3$ dB is an upper bound, not a guarantee: only the uncorrelated core-device noise enjoys $-10\log_{10}2$; coupling/shared noise is correlated, canceling part of the benefit; the pulling-induced $Q$ drop eats some more → real QVCOs are often only 1–2 dB better, and under strong coupling even worse.
• Wiring back to [P3]: coupled QVCO = two mutually injecting generalized Adler equations ([P3] Eq.(30), p.2113); the coupling current travels the unit-bearing ISF $\tilde\Gamma=\Gamma/q_{max}$ ([P3] Eq.(26)); the $90^\circ$ is the steady-state constraint of anti-phase mutual injection; the lock range = the range of $\Omega(\theta)$ ([P3] Eq.(33)). Locking and extra phase noise are two faces of the same $\tilde\Gamma$.
• Honesty note: the QVCO topology comparison, the $\Delta\phi_{IQ}$ order-of-magnitude formula, the $\sim 3$ dB, and the parallel/series conclusion are all external literature (not among the five source PDFs); the locking dynamics are strictly tied to [P3], and the $\div 2$/ILFD subharmonic mechanism strictly to [P4].

Further reading

• Generalized Adler / injection locking (source of this page's locking dynamics): paper_003 ([P3] Eq.(30))
• ILFD / frequency division / subharmonic locking (the $\div 2$ route): paper_004
• How quadrature is used in SerDes (half-rate sampling, CDR phase detection): serdes_clocking_connection
• Effective ISF $\Gamma_{eff}=\Gamma\cdot\alpha$ (the ISF seen by the coupling injection): effective_isf
• ISF of real topologies (tank/tail noise of the cross-coupled LC VCO): real_oscillator_topologies
• Geometry of phase vs amplitude noise (why strong coupling requires looking at amplitude): phase_vs_amplitude_noise

External literature (not in the five downloaded PDFs)

• [E-Andreani-QVCO] P. Andreani, A. Bonfanti, L. Romanò, C. Samori, "Analysis and Design of a 1.8-GHz CMOS LC Quadrature VCO," IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1737–1747, Dec. 2002. (Authoritative analysis of the QVCO coupling-strength vs phase-noise vs I/Q-error triangular trade-off; basis of Sections 2 and 3 of this page. Volume/issue/pages verified.)
• [E-Romano-QVCO] L. Romanò, S. Levantino, C. Samori, A. L. Lacaita, "Multiphase LC Oscillators," IEEE Trans. Circuits Syst. I, vol. 53, no. 7, pp. 1579–1588, Jul. 2006 (and the related parallel-vs-series QVCO literature). (Basis of the parallel vs series coupling phase-noise comparison. Volume/issue/pages verified.)
• [E-Behbahani-PPF] F. Behbahani, Y. Kishigami, J. Leete, A. A. Abidi, "CMOS Mixers and Polyphase Filters for Large Image Rejection," IEEE JSSC, vol. 36, no. 6, pp. 873–887, Jun. 2001. (Basis of the RC-CR polyphase filter design and insertion-loss/bandwidth trade-off. Volume/issue/pages verified.)
• $\div N$ improvement of $20\log_{10}N$ dB: standard frequency-synthesis result (see any PLL / frequency-synthesis text, e.g. Razavi RF Microelectronics). Not in the five PDFs; its physical basis (phase divided by $N$) is consistent with this site's [P1] phase definition.