Analog Design
Kevin Aylward B.Sc
Power - Accuracy - Frequency
Limit
Of Bipolar Amplifiers
Back to Contents
Abstract
This paper addresses the theoretical limit of any bipolar amplifier
with regards to the constraints of Power (current), Accuracy, and Frequency.
Suitable modifications to the theory are easily made for cmos 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 bipolar 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 an emitter area of a bipolar
transistor, a relation can be formed relating this area of a bipolar transistor
to how accurately it matches another bipolar transistor. This relation is:
σ=
α
A
- 1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyypa0ZaaSaaaeaacqaHXoqyaeaadaGcaaqaaiaadgeaaSqabaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeiiaiaabgdaaaa@444F@
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 bipolar transistor. 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 bipolar transistor single transistor amplifier driven
by a pure voltage source, the main capacitance across the transistors output is
the base collector capacitance and the collector substrate capacitance. The
base emitter capacitance, including the diffusion capacitance, will usually
have negligible effect, assuming low base resistance, with this type of voltage
drive. The collector emitter 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 base drive that results in a significant Miller
roll off.
It is now noted that bipolar transistor capacitance's are proportional
to the total area of the device, so that C in (2) above can be expressed as:
C=(
C
cbf
+
C
csf
)A - 3,
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadogacaWGIbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadogacaWGZbGaamOzaaqabaGccaGGPaGaamyqaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabccacaqGGaGaae4maiaabYcacaqGGaaaaa@4961@
where Ccbf, Ccsf are the respective specific
capacitance's of the base and collector substrate per unit area respectively.
Substituting (3) into (2)
i
c
=(
C
cbf
+
C
csf
)Aω
V
o
-4
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadogacaWGIbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadogacaWGZbGaamOzaaqabaGccaGGPaGaamyqaiabeM8a3jaadAfadaWgaaWcbaGaam4BaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabsdaaaa@4B3D@
Squaring (1) and Substituting into (4) for A
i
c
=(
C
bcf
+
C
csf
)ω
V
o
α
2
σ
2
-5
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaacIcacaWGdbWaaSbaaSqaaiaadkgacaWGJbGaamOzaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaadogacaWGZbGaamOzaaqabaGccaGGPaGaeqyYdCNaamOvamaaBaaaleaacaWGVbaabeaakiaabccadaWcaaqaaiabeg7aHnaaCaaaleqabaGaaGOmaaaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeynaaaa@51B9@
Power (current):
In appendix B it is shown that the small signal output current from a
bipolar transistor amplifier, when biased with a current I, is given by:
i
d
=
V
i
I
V
t
- 6
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGKbaabeaakiabg2da9iaadAfadaWgaaWcbaGaamyAaaqabaGcdaWcaaqaaiaadMeaaeaacaWGwbWaaSbaaSqaaiaadshaaeqaaaaakiaadccacaqGGaGaaeiiaiaabccacaqGTaGaaeiiaiaabAdacaqGGaGaaeiiaaaa@43BE@
where Vt = KT/q and Vi is the small signal input
voltage, and where K is boltzman's constant, q is the electronic charge and T
is the absolute temperature of the device.
Since this current is the current that charges the load current, equations
(5) and (6) can be equated to each other:
V
i
I
V
t
=(
C
bcf
+
C
csf
)ω
V
o
α
2
σ
2
-7
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakmaalaaabaGaamysaaqaaiaadAfadaWgaaWcbaGaamiDaaqabaaaaOGaeyypa0JaaiikaiaadoeadaWgaaWcbaGaamOyaiaadogacaWGMbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaam4yaiaadohacaWGMbaabeaakiaacMcacqaHjpWDcaWGwbWaaSbaaSqaaiaad+gaaeqaaOGaaeiiamaalaaabaGaeqySde2aaWbaaSqabeaacaaIYaaaaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqG3aaaaa@5496@
I=
V
t
(
C
bcf
+
C
csf
)
ω
V
o
V
i
α
2
σ
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaadAfadaWgaaWcbaGaamiDaaqabaGccaGGOaGaam4qamaaBaaaleaacaWGIbGaam4yaiaadAgaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaWGJbGaam4CaiaadAgaaeqaaOGaaiykamaalaaabaGaeqyYdCNaamOvamaaBaaaleaacaWGVbaabeaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaOGaaeiiamaalaaabaGaeqySde2aaWbaaSqabeaacaaIYaaaaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaaaaa@4F50@
I=
V
t
(
C
bcf
+
C
csf
)2π.GBW
α
2
σ
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaadAfadaWgaaWcbaGaamiDaaqabaGccaGGOaGaam4qamaaBaaaleaacaWGIbGaam4yaiaadAgaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaWGJbGaam4CaiaadAgaaeqaaOGaaiykaiaaikdacqaHapaCcaGGUaGaam4raiaadkeacaWGxbWaaSaaaeaacqaHXoqydaahaaWcbeqaaiaaikdaaaaakeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaaaa@4E66@
Letting M=2π
V
t
(
C
bcf
+
C
csf
)=
2πkT
q
(
C
bcf
+
C
csf
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaabwgacaqG0bGaaeiDaiaabMgacaqGUbGaae4zaiaabccacaWGnbGaeyypa0JaaGOmaiabec8aWjaadAfadaWgaaWcbaGaamiDaaqabaGccaGGOaGaam4qamaaBaaaleaacaWGIbGaam4yaiaadAgaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaWGJbGaam4CaiaadAgaaeqaaOGaaiykaiabg2da9maalaaabaGaaGOmaiabec8aWjaadUgacaWGubaabaGaamyCaaaacaGGOaGaam4qamaaBaaaleaacaWGIbGaam4yaiaadAgaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaWGJbGaam4CaiaadAgaaeqaaOGaaiykaaaa@5D14@
gives:
I=M.GBW.
α
2
σ
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaad2eacaGGUaGaam4raiaadkeacaWGxbGaaiOlamaalaaabaGaeqySde2aaWbaaSqabeaacaaIYaaaaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaaaaa@41AB@
PAFLimit
for bipolar transistor amplifies.
Thus, it can be seen that the minimum operating current of a bipolar
transistor amplifier depends on the product of the amplifier's gain-bandwidth
and square of the accuracy, along with the process capacitances.
Example
Some values are presented here in order to gain some insight into the
limit of a typical process.
C
cbf
=
C
csf
=1.0
ff
(um)
2
=1m
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBaaaleaacaWGJbGaamOyaiaadAgaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGJbGaam4CaiaadAgaaeqaaOGaeyypa0JaaGymaiaac6cacaaIWaWaaSaaaeaacaWGMbGaamOzaaqaaiaacIcacaWG1bGaamyBaiaacMcadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0JaaGymaiaab2gaaaa@4A67@
A
min
=4.0um×4um
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaaciGGTbGaaiyAaiaac6gaaeqaaOGaeyypa0JaaGinaiaac6cacaaIWaGaaeyDaiaab2gacqGHxdaTcaaI0aGaaeyDaiaab2gaaaa@4387@
Typical error σ = 10% = 0.1 at A=16u2, so that α =0.4u
Thus M is:
M=2π(1m+1m)25m=3.14×
10
-4
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaaikdacqaHapaCcaGGOaGaaGymaiaab2gacqGHRaWkcaaIXaGaaeyBaiaacMcacaaIYaGaaGynaiaab2gacqGH9aqpcaqGZaGaaeOlaiaabgdacaqG0aGaey41aqRaaeymaiaabcdadaahaaWcbeqaaiaab2cacaqG0aaaaaaa@4B1B@
so,
I=3.14×
10
−4
.GBW.
α
2
σ
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iaaiodacaGGUaGaaGymaiaaisdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaeyOeI0IaaGinaaaakiaac6cacaWGhbGaamOqaiaadEfacaGGUaWaaSaaaeaacqaHXoqydaahaaWcbeqaaiaaikdaaaaakeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaaaa@492F@
From which the following table can be constructed:
Desired Desired Required
GBW σ I
1GHz 0.1 5.02ua
100MHz 0.1 0.5ua
10Mhz 0.1 -
1GHz 0.01 502ua
100MHz 0.01 50.2ua
10Mhz 0.01 5.02ua
It should be noted that the above is a raw, best case calculation. Other
considerations may prohibit the higher currents from being realized in
practice.
Conclusion
It has been shown that there is an inherent power, accuracy and
gain-bandwidth limitation for any specific bipolar transistor fabrication
process. It is anticipated that the relations derived in this paper may result
in more optimal design of bipolar amplifiers.
Appendix A
Derivation of :
i
d
=
V
i
I
V
t
- 6
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGKbaabeaakiabg2da9iaadAfadaWgaaWcbaGaamyAaaqabaGcdaWcaaqaaiaadMeaaeaacaWGwbWaaSbaaSqaaiaadshaaeqaaaaakiaadccacaqGGaGaaeiiaiaabccacaqGTaGaaeiiaiaabAdacaqGGaGaaeiiaaaa@43BE@
It is well known that a reasonable accurate large signal design
equation for bipolar transistors in the saturation region is given by:
i
c
=
i
o
(
e
vbe
V
t
−1)≅
i
o
e
vbe
V
t
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaadMgadaWgaaWcbaGaam4BaaqabaGccaGGOaGaamyzamaaCaaaleqabaWaaSaaaeaacaWG2bGaamOyaiaadwgaaeaacaWGwbWaaSbaaWqaaiaadshaaeqaaaaaaaGccqGHsislcaaIXaGaaiykaiabgwKiajaadMgadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAhacaWGIbGaamyzaaqaaiaadAfadaWgaaadbaGaamiDaaqabaaaaaaaaaa@4D4C@
hence:
d
i
c
d(vbe)
=
i
o
1
V
t
e
vbe
V
t
=
1
V
t
i
c
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaamyAamaaBaaaleaacaWGJbaabeaaaOqaaiaadsgacaGGOaGaamODaiaadkgacaWGLbGaaiykaaaacqGH9aqpcaWGPbWaaSbaaSqaaiaad+gaaeqaaOWaaSaaaeaacaaIXaaabaGaamOvamaaBaaaleaacaWG0baabeaaaaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAhacaWGIbGaamyzaaqaaiaadAfadaWgaaadbaGaamiDaaqabaaaaaaakiabg2da9maalaaabaGaaGymaaqaaiaadAfadaWgaaWcbaGaamiDaaqabaaaaOGaamyAamaaBaaaleaacaWGJbaabeaaaaa@4FC5@
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@
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