Oscillators (I)
Professor Jri Lee
台大電子所 李致毅教授
Electrical Engineering Department
National Taiwan University
Outline
Barkhausen Criteria
Ring Oscillators
LC Oscillators
Colpitts Oscillator
Tuning Techniques
BarkhausenCriteria
H (jω0 ) ≥ 1, ∠ H (jω0 ) = 180°
Poles locate at the y-axis.
Necessary but not sufficient.
Loop gains are typically chosen to be at least 2
or 3 to ensure oscillation. Oscillators adjust
themselves to satisfy the criteria.
Evolution of Ring Oscillators
Barkhausen criteria is achievable only when the
number of stage is greater than 2.
Three-Stage Ring Oscillator
A03
H (s ) =
(1 + s
)3
ω0
A0 = gmRD , ω0 = 1
RDCL
⇒ tan-1(ωosc
ωosc
⇒
ω0 ) = 60°
= 3ω 0
A0
1+
ω
2
osc
=1
ω20
A0 ≥ 2
Tuning can be included by adjusting RD or CL.
Three-Stage Ring Oscillator
Vout
=
Vin
Vout
A03
H (s ) =
(s ) =
Vin
(1 + s
)3
ω0
Vout ∝ exp(
A03
(1 + s
)3
ω0
A03
1+
(1 + s
)3
ω0
s1 = (− A0 − 1)ω0
s2,3
A0 − 2
A 3
ω0t ) cos( 0
2
2
⎡ A0 (1 ± j 3 ) ⎤
=⎢
− 1⎥ ω0
2
⎥⎦
⎢⎣
ω 0t )
Large Signal Analysis
A0 3ω0
2
≠
6TD
In reality, the signal “saturates” to make the
average loop gain equal to unity.
The actual oscillation frequency is determined by
large signal behavior.
Four-Stage Ring Oscillators
A04
H (s ) =
(1 + s ) 4
ω0
ωosc = ω0
A0 ≥ 2
The number of stages in a ring oscillator is determined
by speed, power and noise performance.
CMOS Realizations without Resistors
Overcome voltage
headroom issue.
If poly resistors are available then use them.
Quality Factor of Inductors
Definitions
(i) Q =
ω0
∆ω
(ii) Q = 2π
(iii) Q =
Energy Stored
Energy Dissipated Per Cycle
ω0 dφ
⋅
2 dω
Three definitions are equivalent.
Equivalent Circuit of an LC Tank
LS = LP
ωLS
RS =
Q
RP = Q ⋅ ωLP
Q is a function of inductance and frequency.
Q is around 3-5 for large inductors (>5nH). Smaller
inductors typically achieve a higher Q.
LC Oscillators
≡
Barkhausen criteria can be easily satisfied.
Minimum phase noise point.
Oscillation frequency = ω1.
Another Approach to Oscillation
Cross-coupled differential pair can be recognized as
negative resistors.
One Port Oscillators
Three possible realizations
As long as a negative equivalent resistance can be
found, an oscillator can be made.
Colpitts Oscillator
RpLp s (gm + C2 s )
Vout
=
Iin
RpC1C2Lp s 3 + (C1 + C2 )Lp s 2 + [ gmLp + Rp (C1 + C2 )]s + gmRp
C1
C2 2
gmRp =
(1 + ) ≥ 4
C2
C1
ωR =
(@ C1 = C2 )
1
CC
Lp (Cp + 1 2 )
C1 + C2
(if Cp included)
Oscillates under proper conditions.
Voltage-Controlled Oscillator
Tradeoffs:
Center Frequency
Tuning Linearity
Tuning Range
Output Amplitude
Power Dissipation
Supply Rejection
Phase Noise
Ring VCO – Output Swing Control
Replica biasing “servos” the on-resistance to vary
the frequency while maintaining the constant output
swing.
Ring VCO – Delay Variation by Positive Feedback
Current steering topology keeps the swing constant.
It may suffer from voltage headroom issue.
Ring VCO – Current-Folding Topology
Folding the current saves one stacking stages.
Ring VCO – Delay Variation by Interpolation
Stability issue?
Large tuning range
Oscillation Frequency of Simple LC Oscillator
Rp = Q ⋅ ωosc L = 1
gm
ωosc =
1
Cp CGS
2L( +
)
2
2
Oscillation Frequency of LC Oscillator
If Cp is insignificant compared with CGS, then
ωosc ≈
=
1
LCGS
1
1
CGS
gmQωosc
= QωT ωosc
⇒ ωosc = Q ⋅ ωT
An LC oscillator could theoretically operate at
arbitrarily high frequency as long as the inductor
provides a sufficiently high Q.
In reality, neither Cp is negligible nor can an on-chip
inductor achieve very high Q, nor negligible Cp.