Analog
Design
Kevin
Aylward B.Sc.
Power
- Accuracy - Frequency Limit
Of
CMOS Amplifiers
Back to Contents
Abstract
This paper addresses the theoretical
limit of any CMOS amplifier with regards to the constraints of Power (current),
Accuracy, and Frequency. Suitable modifications to the theory are easily made
for bipolar amplifies, and are addressed in another paper.
Overview
Analogue design is always one of performance tradeoffs. One can rest
assured that any tweak of a design that improves one aspect of its performance
will always result in another aspect of its performance being inferior. It
would therefor seem, that it should be possible to identify some key
performance features, and construct some design equations that expressed such
design limitations. This paper shows that for a given CMOS process, there is an
inherent limit that links together, power (current), accuracy and speed (gain
bandwidth), thereby making it much clearer as to how to more optimally effect such
design compromises in practice.
The initial motivation for this paper was based largely on the
realization that principles behind the Heisenburg uncertainty relation
regarding time and energy might be applied to electronic design. Clearly, it
was not anticipated that the quantum mechanical limit itself would impact a
typical analogue design, and that is indeed the case. However, its analysis
certainly highlights some of the fundamental issues involved.
As a starter then, the Heisenburg time-energy relation is given by:
ΔE.Δt≥
h
2π
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamyraiaac6cacqqHuoarcaWG0bGaeyyzIm7aaSaaaeaacaWGObaabaGaaGOmaiabec8aWbaacaqGGaaaaa@4104@
where ΔE is the rms uncertainty in energy and Δt is the time that the
energy is measured for.
With a little hand waving (appendix C), the following relation can be
derived:
P
0
≥
h
2π
.
f
2
σ
where
P
0
is power, σ is the error and f is frequency
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FC6@
The details of this result are not overly important, the key point
being that this indicates that there is a minimum power for a signal or device,
subject to a constraint of accuracy and speed of operation. Plugging in some
numbers gives us nW's of power for GHz, 20bit signals, so the quantum limit is
obviously no practical limit in terms of real designs. However, it does
illustrate the basic connection between power, speed and accuracy. That is,
there is always a relation of the form:
G(P,σ,f)=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaacIcacaWGqbGaaiilaiabeo8aZjaacYcacaWGMbGaaiykaiabg2da9iaaicdaaaa@3EAC@
PAFL Power - Accuracy -
Frequency Limit
It is first noted, that this derivation relies on suitable
approximations that are usually reasonable valid in most situations. The
purpose of the derivation is to formulate a reasonable "best case"
such that the real circuit will generally always be worst. This allows an
immediate determination of whether a given specification is achievable from the
outset.
Accuracy:
In appendix A, it is shown that, given a Width and Length of a mosfet,
a relation can be formed relating the area of a mosfet to how accurately it
matches another mosfet. This relation is:
σ=
α
WL
- 1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyypa0ZaaSaaaeaacqaHXoqyaeaadaGcaaqaaiaadEfacaWGmbaaleqaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabccacaqGXaaaaa@4536@
where σ is the relative error and α is a constant.
This should be quite an intuitive equation. It simply says that the
matching error is inversely proportional to the square root of the total area
of the mosfet. Typically, manufactures characterize their process and
empirically determine the value for alpha. It can therefor be taken as a known
constant, just as any other process characteristic.
Frequency:
The starting point for this is the well-known capacitor current
equation:
i
c
=C
d
V
o
dt
=Cω
V
o
-2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaadoeadaWcaaqaaiaadsgacaWGwbWaaSbaaSqaaiaad+gaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0Jaam4qaiabeM8a3jaadAfadaWgaaWcbaGaam4BaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabkdaaaa@482F@
This equation represents an output voltage swinging across a capacitor
load. If one considers a mosfet single transistor amplifier driven by a pure
voltage source, the main capacitance across the transistors output is the gate
drain capacitance and the drain bulk/substrate capacitance. The gate source
capacitance will usually have negligible effect, assuming low gate resistance,
with this type of voltage drive. The drain source capacitance can usually be
ignored as well. It might be argued that a cascode connection might be faster,
however, it should be noted that cascodes generally only offer an advantage
when there is significant resistance in the gate drive that results in
significant Miller roll off.
There are a two principal drain gate capacitances. One is proportional
to Width X Length, but in the normal, active constant current saturation
region, this one is negligible. The
second capacitance is the overlap capacitance, and is proportional to the of
the W of the device, so that C in (2) above can be expressed as:
C=(
C
gdf
+
C
gbf
)W - 3,
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadEgacaWGKbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadEgacaWGIbGaamOzaaqabaGccaGGPaGaam4vaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabccacaqGGaGaae4maiaabYcacaqGGaaaaa@4970@
where Cgdf, Cgbf are the respective specific
capacitance's of the gate and drain bulk/substrate per unit length
respectively.
Substituting (3) into (2)
i
c
=(
C
gdf
+
C
gbf
)Wω
V
o
-4
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadEgacaWGKbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadEgacaWGIbGaamOzaaqabaGccaGGPaGaam4vaiabeM8a3jaadAfadaWgaaWcbaGaam4BaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabsdaaaa@4B4C@
Squaring (1) and Substituting into (4)
i
c
=(
C
gdf
+
C
gbf
)ω
V
o
α
2
L
σ
2
-5
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadEgacaWGKbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadEgacaWGIbGaamOzaaqabaGccaGGPaGaeqyYdCNaamOvamaaBaaaleaacaWGVbaabeaakiaabccadaWcaaqaaiabeg7aHnaaCaaaleqabaGaaGOmaaaaaOqaaiaadYeacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabwdaaaa@5283@
Power (current):
In appendix B it is shown that the small signal output current from a
mosfet amplifier, when biased with a current I, is given by:
i
d
=
V
i
2KI
W
L
- 6
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGKbaabeaakiabg2da9iaadAfadaWgaaWcbaGaamyAaaqabaGcdaGcaaqaaiaaikdacaWGlbGaamysamaalaaabaGaam4vaaqaaiaadYeaaaaaleqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeiiaiaabAdacaqGGaGaaeiiaaaa@45B3@
where K is a process constant and Vi is the small signal
input voltage.
Since this current is the current that charges the load current,
equations (5) and (6) can be equated to each other:
V
i
2KI
W
L
=(
C
gdf
+
C
gbf
)ω
V
o
.
α
2
L
σ
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakmaakaaabaGaaGOmaiaadUeacaWGjbWaaSaaaeaacaWGxbaabaGaamitaaaaaSqabaGccqGH9aqpcaGGOaGaam4qamaaBaaaleaacaWGNbGaamizaiaadAgaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaWGNbGaamOyaiaadAgaaeqaaOGaaiykaiabeM8a3jaadAfadaWgaaWcbaGaam4BaaqabaGccaqGGaGaaeOlamaalaaabaGaeqySde2aaWbaaSqabeaacaaIYaaaaaGcbaGaamitaiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaGccaqGGaaaaa@52CC@
I=
(
C
gdf
+
C
gbf
)
2
2K
.
(2πGBW)
2
WL
.
α
4
σ
4
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maalaaabaGaaiikaiaadoeadaWgaaWcbaGaam4zaiaadsgacaWGMbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaam4zaiaadkgacaWGMbaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaam4saaaacaGGUaWaaSaaaeaacaGGOaGaaGOmaiabec8aWjaadEeacaWGcbGaam4vaiaacMcadaahaaWcbeqaaiaaikdaaaaakeaacaWGxbGaamitaaaacaGGUaGaaeiiamaalaaabaGaeqySde2aaWbaaSqabeaacaqG0aaaaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaI0aaaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaaaaa@5778@
I=
(
C
gdf
+
C
gbf
)
2
2K
.
(2πGBW)
2
.
α
2
σ
2
- 7
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maalaaabaGaaiikaiaadoeadaWgaaWcbaGaam4zaiaadsgacaWGMbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaam4zaiaadkgacaWGMbaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaam4saaaacaGGUaGaaiikaiaaikdacqaHapaCcaWGhbGaamOqaiaadEfacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaaiOlaiaabccadaWcaaqaaiabeg7aHnaaCaaaleqabaGaaeOmaaaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGGaGaae4naaaa@57C4@
Letting M=
2π(
C
gdf
+
C
gbf
)
2K
in (7) gives
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaabwgacaqG0bGaaeiDaiaabMgacaqGUbGaae4zaiaabccacaWGnbGaeyypa0ZaaSaaaeaacaaIYaGaeqiWdaNaaiikaiaadoeadaWgaaWcbaGaam4zaiaadsgacaWGMbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaam4zaiaadkgacaWGMbaabeaakiaacMcaaeaadaGcaaqaaiaaikdacaWGlbaaleqaaaaakiaabccacaqGPbGaaeOBaiaabccacaqGOaGaae4naiaabMcacaqGGaGaae4zaiaabMgacaqG2bGaaeyzaiaabohaaaa@573E@
I=
(GBW.M
α
σ
)
2
-8
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaacIcacaWGhbGaamOqaiaadEfacaGGUaGaamytamaalaaabaGaeqySdegabaGaeq4WdmhaaiaabccacaqGPaWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeioaaaa@469E@
Thus, it can be seen that the minimum operating current of a mosfet
amplifier depends on the square of the product of the amplifier's
gain-bandwidth and accuracy, and with the process other fixed characteristics.
If the transistor operates in the linear region such that Cgd becomes a
strong function of W X L, such that Cgd=WL.Cp, than the equation becomes:
I=
(L.GBW.M
α
σ
)
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaacIcacaWGmbGaaiOlaiaadEeacaWGcbGaam4vaiaac6cacaWGnbWaaSaaaeaacqaHXoqyaeaacqaHdpWCaaGaaeiiaiaabMcadaahaaWcbeqaaiaabkdaaaGccaqGGaGaaeiiaiaabccaaaa@4570@
Appendix A
Derivation of :
i
d
=
V
i
2KI
W
L
- 6
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGKbaabeaakiabg2da9iaadAfadaWgaaWcbaGaamyAaaqabaGcdaGcaaqaaiaaikdacaWGlbGaamysamaalaaabaGaam4vaaqaaiaadYeaaaaaleqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeiiaiaabAdacaqGGaGaaeiiaaaa@45B3@
It is well known that a reasonable accurate large signal design
equation for mosfets in the saturation region is given by:
I
d
=
W
L
K
2
(
V
gs
−
V
t
)
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGKbaabeaakiabg2da9maalaaabaGaam4vaaqaaiaadYeaaaWaaSaaaeaacaWGlbaabaGaaGOmaaaacaGGOaGaamOvamaaBaaaleaacaWGNbGaam4CaaqabaGccqGHsislcaWGwbWaaSbaaSqaaiaadshaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@445E@
hence:
d
I
d
d
V
gs
=
W
L
K(
V
gs
−
V
t
)=
W
L
K.
2
I
d
L
WK
=
2
I
d
K
W
L
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5777@
Appendix B
This appendix does the hand waving argument on the quantum limit of
power accuracy and frequency from:
ΔE.Δt≥
h
2π
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamyraiaac6cacqqHuoarcaWG0bGaeyyzIm7aaSaaaeaacaWGObaabaGaaGOmaiabec8aWbaacaqGGaaaaa@4104@
The uncertainty in energy can be expressed by:
ΔE=σE
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamyraiabg2da9iabeo8aZjaadweaaaa@3BA7@
Where E is the
energy being measured, or equivalently the energy being produced. Note that
Δt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamiDaaaa@3843@
is not an uncertainty in the measurement of
time, but the time period that the (system) energy remains in a given state.
However, by assumption this energy (state) is changing over time, such that the
time that the system can be measured for, can be no greater time than the time
that the energy changes by another
ΔE
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamyraaaa@3814@
.
This energy can be related to the power via:
E=PΔt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2da9iaadcfacqqHuoarcaWG0baaaa@3AE8@
hence:
P.
(Δt)
2
σ≥
h
2π
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaac6cacaGGOaGaeuiLdqKaamiDaiaacMcadaahaaWcbeqaaiaaikdaaaGccqaHdpWCcqGHLjYSdaWcaaqaaiaadIgaaeaacaaIYaGaeqiWdahaaiaabccaaaa@43B8@
or
P≥
h
2π
f
2
σ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabgwMiZoaalaaabaGaamiAaaqaaiaaikdacqaHapaCaaWaaSaaaeaacaWGMbWaaWbaaSqabeaacaaIYaaaaaGcbaGaeq4Wdmhaaaaa@3FA6@
© Kevin Aylward 2013
All rights reserved
The information on the page may be
reproduced
providing that this source is acknowledged.
Website last modified 30th August
2013
www.kevinaylward.co.uk