Injection Locked Oscillators

Posted on Edited on In

image-20260718192646572

image-20260718184425262

Rotating Phasor

image-20260718192825536

image-20260718212046942

  • \(E_1\): externally applied forcing signal.
  • \(E\): oscillator-generated component after being influenced by \(E_1\).
  • \(E_g\): total signal resulting from the oscillator-generated and injected contributions.
  • \(\alpha(t)\) is the phase difference between the internal oscillator phasor \(E\), after being influenced by \(E_1\), and the injected phasor \(E_1\)
  • \(\phi(t)\) is the angle between that internal oscillator phasor \(E\) and the resultant phasor \(E_g=E+E_1\)

Although \(E\) is called the "internally generated signal," it is not the free-running, unaffected signal. The coupling is implicit: \[ E_1 \rightarrow E_g=E+E_1 \rightarrow H(j\omega) \rightarrow E \] Therefore, \[ \boxed{E\text{ depends dynamically on }E_1.} \]

Adler's Equation

image-20260718212423404

image-20260718212543675

\[ \boxed{\frac{\mathrm{d}\alpha}{\mathrm{d}t} = \Delta\omega_0 - \frac{E_1}{E}\frac{\omega_0}{2Q}\sin\alpha} \] Define the injection-locking strength as \[ K = \frac{E_1}{E}\frac{\omega_0}{2Q}. \] Then Eq. (3) becomes the standard Adler form \[ \boxed{ \frac{\mathrm{d}\alpha}{\mathrm{d}t} = \Delta\omega_0-K\sin\alpha } \]

In steady state \[ \frac{\mathrm{d}\alpha}{\mathrm{d}t} = 0 \]

When free-running frequency is equal to injected frequency, in a steady state, the large oscillation aligns in phase with the small injected current

image-20260718222735209

image-20260718213042113

KCL plus the inductor law \[ C\dot v + \frac{v}{R} + i_L = \frac{\pi}{4}I_1\,\mathrm{sgn}(v) + \epsilon I_1\sin(\omega_{inj}t),\qquad L\,\dot i_L = v, \] with \(I_1\equiv\tfrac{2}{\pi}I_0\)

"inject at \(\omega_0\)" must mean the measured free-running frequency — we measure it from an \(\epsilon=0\) run first

image-20260718231802240

phase estimates the instantaneous phase of waveform v relative to the reference \(\sin(\omega_{\rm ref}t)\)

It acts like a simple lock-in detector:

1
N = int(round(2*np.pi / wref / dt))

Computes the number of samples in one reference period.

1
2
Ic = moving_average(2*v*sin(wref*t))
Qc = moving_average(2*v*cos(wref*t))

The one-period moving average extracts the components of v aligned with the reference sine and cosine while suppressing harmonics.

For

\[ v(t)=A\sin(\omega_{\rm ref}t+\phi) \]

the averages are approximately

\[ I_c=A\cos\phi,\qquad Q_c=A\sin\phi \]

so:

1
np.arctan2(Qc, Ic)

returns \(\phi\), in radians between \(-\pi\) and \(\pi\)

[credits to Claude Fable 5]

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
"""
Neat two-level ODE check of the slide (Dw0 = 0), one eps, one figure (3 panels).

Level 1 (phase ODE, eq.3):   da/dt = -B sin(a),   B = (E1/E) * w0/(2Q)
Level 2 (raw circuit ODE):   C dv/dt + v/R + iL = Ip*sgn(v) + eps*I1*sin(w_osc*t)
                             L diL/dt = v
Both start at a0 = 90 deg (v0 ~ cos, injection ~ sin) and are compared with
the slide's closed form  tan(a/2) = exp(-Bt)*tan(a0/2)  and with a0*exp(-Bt).
Two waveform panels show v0(t) right after injection turns on (90 deg ahead
of i_n) and in steady state ((1+eps)*I1*R*sin, in phase with i_n).
"""
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

# ---- parameters ------------------------------------------------------------
w0  = 2*np.pi                 # 1/sqrt(LC)  (period = 1 s)
Q, R = 10.0, 1.0
C, L = Q/(w0*R), R/(w0*Q)
I1  = 1.0                     # limiter fundamental (= (2/pi)*I0)
Ip  = np.pi/4 * I1            # square-wave amplitude -> fundamental I1
eps = 0.10                    # E1/E
B   = eps*w0/(2*Q)            # slide's B
a0  = np.pi/2
dt, T = 2e-3, 200.0

# ---- Level 2: full circuit ODE, no averaging -------------------------------
def circuit(t, y, eps, winj):
    v, iL = y
    dv = (Ip*np.tanh(v/0.01) + eps*I1*np.sin(winj*t) - v/R - iL)/C
    return dv, v/L

def run(eps, winj, T):
    t = np.arange(0, T, dt)
    s = solve_ivp(circuit, (0, T), [I1*R, 0.0], args=(eps, winj),
                  t_eval=t, method="LSODA", rtol=1e-7, atol=1e-9, max_step=5e-3)
    return t, s.y[0]

def phase(t, v, wref):        # phase of v relative to sin(wref*t)
    N = int(round(2*np.pi/wref/dt)); box = np.ones(N)/N
    Ic = np.convolve(2*v*np.sin(wref*t), box, "same")
    Qc = np.convolve(2*v*np.cos(wref*t), box, "same")
    return np.arctan2(Qc, Ic), N

# limiter harmonics shift the autonomous frequency slightly below 1/sqrt(LC)
# (Groszkowski), so measure w_osc once and inject there ("w0" of the slide):
t, v = run(0.0, w0, 150.0)
th, N = phase(t, v, w0)
tt, thh = t[N:-N], np.unwrap(th[N:-N])
w_osc = w0 + np.polyfit(tt[tt > 50], thh[tt > 50], 1)[0]

t, v = run(eps, w_osc, T)                     # injection ON at t = 0
a_ckt, N = phase(t, v, w_osc)

# ---- Level 1: integrate the phase ODE itself -------------------------------
a_ode = solve_ivp(lambda t, a: -B*np.sin(a), (0, T), [a0],
                  t_eval=t, rtol=1e-10, atol=1e-12).y[0]

# ---- slide's closed form and small-angle limit -----------------------------
a_exact = 2*np.arctan(np.exp(-B*t)*np.tan(a0/2))
a_small = a0*np.exp(-B*t)

print(f"B = {B:.4f} rad/s (tau = 1/B = {1/B:.1f} s), "
      f"w_osc - w0 = {w_osc-w0:+.5f} rad/s")
print(f"alpha({T:.0f}s): circuit ODE {np.degrees(a_ckt[-2*N]):+.2f} deg | "
      f"phase ODE {np.degrees(a_ode[-1]):+.2f} deg | closed form "
      f"{np.degrees(a_exact[-1]):+.2f} deg")

Fundamental ILO (1st harmonic)

TODO 📅

Subharmonic ILO

TODO 📅

Multiplying ILO (MILO)

J. Im et al., "6.1 A 112Gb/s PAM-4 Long-Reach Wireline Transceiver Using a 36-Way Time-Interleaved SAR-ADC and Inverter-Based RX Analog Front-End in 7nm FinFET," 2020 IEEE International Solid-State Circuits Conference - (ISSCC), San Francisco, CA, USA, 2020, pp. 116-118 [https://sci-hub.ru/10.1109/ISSCC19947.2020.9063081]

—, "A 112-Gb/s PAM-4 Long-Reach Wireline Transceiver Using a 36-Way Time-Interleaved SAR ADC and Inverter-Based RX Analog Front-End in 7-nm FinFET," in IEEE Journal of Solid-State Circuits, vol. 56, no. 1, pp. 7-18, Jan. 2021 [https://sci-hub.ru/10.1109/JSSC.2020.3024261]

M. Erett et al., "16.7 A 126mW 56Gb/s NRZ wireline transceiver for synchronous short-reach applications in 16nm FinFET," 2018 IEEE International Solid-State Circuits Conference - (ISSCC), San Francisco, CA, USA, 2018, pp. 274-276 [https://sci-hub.ru/10.1109/ISSCC.2018.8310290]

TODO 📅

image-20261020002412076

image-20261020002156517

image-20261020002220507

Injection-Locked Frequency Divider (ILFD)

TODO 📅

reference

Bae, Woorham, and Deog-Kyoon Jeong. Analysis and Design of CMOS Clocking Circuits for Low Phase Noise. Institution of Engineering and Technology, 2020

Cowan, Glenn. (2024). Mixed-Signal CMOS for Wireline Communication: Transistor-Level and System-Level Design Considerations

Luong, H. C., & Yin, J. (2016). Transformer-based design techniques for oscillators and frequency dividers. Springer International Publishing

R. Adler, "A Study of Locking Phenomena in Oscillators," in Proceedings of the IRE, vol. 34, no. 6, pp. 351-357, June 1946 [https://sci-hub.jp/10.1109/JRPROC.1946.229930]

—, "A study of locking phenomena in oscillators," in Proceedings of the IEEE, vol. 61, no. 10, pp. 1380-1385, Oct. 1973 [https://sci-hub.jp/10.1109/PROC.1973.9292]

B. Razavi, "A study of injection locking and pulling in oscillators," in IEEE Journal of Solid-State Circuits, vol. 39, no. 9, pp. 1415-1424, Sept. 2004 [https://www.seas.ucla.edu/brweb/papers/Journals/RSep04.pdf]

A. A. Hafez and C. -K. K. Yang, "Analysis and Design of Superharmonic Injection-Locked Multipath Ring Oscillators," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 7, pp. 1712-1725, July 2013 [https://sci-hub.ru/10.1109/TCSI.2012.2230591]

Deog-Kyoon Jeong. "Topics in IC (Wireline Transceiver Design): Lec 4 - Injection Locked Oscillators" [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%204%20-%20Injection%20Locked%20Oscillators.pdf]

Min-Seong Choo. Review of Injection-Locked Oscillators [https://journal.theise.org/jse/wp-content/uploads/sites/2/2020/09/JSE-2020-0001.pdf]

Mozhgan Mansuri. ISSCC2021 SC3: Clocking, Clock Distribution, and Clock Management in Wireline/Wireless Subsystems

Aditya Varma Muppala. Oscillator Theory - Injection Locking [note, video1, video2]

Ali M. Niknejad. EECS 242 Lecture 26 Injection Locking [https://rfic.eecs.berkeley.edu/courses/ee242/pdf/eecs242_lect26_injectionlocking.pdf]

Tony Chan Carusone, 35 Injection Locked Oscillators [https://youtu.be/IgB2NRdUMVo]