Analog Design
Kevin Aylward B.Sc.
Bipolar Amplifier Design
Part 1
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to Contents
Abstract
This paper forms an introduction to the fundamental
aspects of the design of bipolar transistor amplifiers. The intent is that reader
should be able to gain a big picture view, such that many circuits can be
reasonably appreciated by inspection.
This paper addresses the key points that one
requires to effectively design analog circuits, and as such will not address
much of the underlying physics of transistor operation, i.e. the
"why". What is much important is "what" a transistor does,
i.e. how it behaves, irrespective of why.
It is assumed that the reader has some basic math,
including calculus, and has at least a cursory knowledge of circuits and
components. However, for those without such pre-requisites, other then the
actual derivations of some equations,
there should be little else to impede the understanding of the material
It is noted that detailed calculations of transistor
circuits are rarely required due to the availability of inexpensive Spice, e.g SuperSpice, simulation
software. However, this software does not replace the analogue design engineers
requirement to understand what he is doing! GIGO as they say.
In is also noted that the equations developed here
are based on inspection for the most part. The generalized small signal
equivalent circuit approach sometimes used for more complicated designs, will
be addressed in another paper.
Transistor Basics
A bipolar transistor is a device with 3 terminals,
collector, base and emitter. There are two types, NPN and PNP. This paper will
address mainly NPN types, as simply reversing all of the current flows and
terminal voltages will automatically explain PNP types.
Despite much literature that implies other wise, a
transistor is a voltage controlled device
not a current controlled device. This is a fundamental point that must
be taken completely to heart if one desires to effectively design transistor
level circuits. Applying a voltage to the base emitter terminals of a
transistor, when there is a voltage across the collector emitter, will result
in the flow of current through the collector to the emitter essentially
independent of the collector emitter voltage. Therefor the transistor should be
thought of as a voltage controlled
current source. The key point is
that the base emitter junction V/I must satisfy the basic diode equation. It is
the total diode current that matters. What leaks out of the base (re
combination) is essentially, irrelevant. The base terminal merely provides a
convenient method of impressing a voltage at the base emitter junction.
There is of course, a corresponding base to emitter
current (Ib ) that flows when a voltage is applied to the base
emitter (Vbe), but this current might be said to flap in the wind.
The ratio of collector current to base current is named hfe or
beta. This ratio is typically 50 to 500,
but is rather indeterminate having variations over the collector current range,
and from device to device. The emitter current is clearly (1+ hfe).Ib
, since it includes the base current as well. Most transistor circuits are
designed such that the base current does not unduly effect circuit operation.
The basic current equations for a transistor
amplifier are therefore:
I
c
≡
h
fe
I
b
-1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGJbaabeaakiabggMi6kaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaOGaamysamaaBaaaleaacaWGIbaabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabgdaaaa@430F@
I
e
=
I
c
+
I
b
=
h
fe
I
b
+
I
b
=(1+
h
fe
)
I
b
-2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4yaaqabaGccqGHRaWkcaWGjbWaaSbaaSqaaiaadkgaaeqaaOGaeyypa0JaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaWGjbWaaSbaaSqaaiaadkgaaeqaaOGaey4kaSIaamysamaaBaaaleaacaWGIbaabeaakiabg2da9iaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaGaamysamaaBaaaleaacaWGIbaabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGYaaaaa@545D@
or from (1) and (2)
I
c
=
I
e
h
fe
(1+
h
fe
)
-3
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGJbaabeaakiabg2da9maalaaabaGaamysamaaBaaaleaacaWGLbaabeaakiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaOGaaeiiaaqaaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaaaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabodaaaa@48F2@
or
α≡
I
c
I
e
=
h
fe
(1+
h
fe
)
-4
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyyyIO7aaSaaaeaacaWGjbWaaSbaaSqaaiaadogaaeqaaaGcbaGaamysamaaBaaaleaacaWGLbaabeaaaaGccqGH9aqpdaWcaaqaaiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaOGaaeiiaaqaaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaaaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabsdaaaa@4C6B@
Where it should be noted that as hfe is
of the order of at least 100, so that α is of the order of 0.99, hence the
collector current is essentially equal to the emitter current. For most 1st
order design calculations, they are usually taken to be equal.
Transistor Bias
A transistor can only pass current in one direction
when supplied by a single supply at its collector. This leads to the
introduction of the concept of bias.
With reference to Fig.1, if Vbe is
increased from 0 to some value, current will flow through the collector. The
initial voltage at the collector is equal to the supply voltage, Vs, as no
current is drooped in Rc. The increase in current will cause an increase in the
voltage dropped across Rc, hence the voltage at the collector will also drop.
This means that the output goes in a negative direction when the input goes
positive. This is usually referred to as phase inversion. However, if the input
goes negative, the output can not raise any further then the supply voltage.
The solution to this problem is to permanently "bias" the base
emitter voltage with some fixed dc voltage, and then add the signal effectively
in series with this voltage. This results in a fixed DC current always flowing
through the collector, and also results in the collector being at some intermediate
voltage between the power supply and zero.
Now, for example, if the fixed DC
voltage at the collector is say 5 volts, with a supply of 10 volts, the
collector voltage can now swing above and below this DC "operating"
condition.
It is noted that the use of the word
"phase" in describing the inverted action of a basic transistor
amplifier can be misleading, as it is not a phase shift in the exact sense. It
is a voltage polarity inversion.
However, for small signal, sine wave conditions (explained below), a
polarity inversion is identical to a phase shift of 180 degrees. If the signal
had say, a positive triangle and negative square shape, a 180 degrees phase
shift would not be the same as the polarity inversion of a transistor
amplifier.
Fundamental
Transistor Equation
The fundamental equation describing bipolar
operation is obtained from the classic diode equation, and is valid for diodes
and the diode junctions of transistors.
I
e
=
I
o
(
e
q
V
be
KT
−1) -5
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4BaaqabaGccaGGOaGaamyzamaaCaaaleqabaWaaSaaaeaacaWGXbGaamOvamaaBaaameaacaWGIbGaamyzaaqabaaaleaacaWGlbGaamivaaaaaaGccqGHsislcaaIXaGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeynaaaa@4879@
where K is boltzman's constant, q is the electronic
charge and T is the absolute temperature of the device.
The quantity Vt = KT/q is also usually
defined in order to simplify the notation. At room temperature Vt is
approximately 25mV.
Io
is named the reverse saturation current, (sometimes also called named Is) and is a function of many internal
parameters, including temperature. It is the current that will flow when the
diode is reversed biased. This will be examined in a paper on the design of
band gap voltage references.
With the above substation for Vt, the
diode equation becomes:
I
e
=
I
o
(
e
V
be
V
t
−1) -6
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4BaaqabaGccaGGOaGaamyzamaaCaaaleqabaWaaSaaaeaacaWGwbWaaSbaaWqaaiaadkgacaWGLbaabeaaaSqaaiaadAfadaWgaaadbaGaamiDaaqabaaaaaaakiabgkHiTiaaigdacaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqG2aaaaa@47DC@
The -1 is usually dropped as it is insignificant for
Vbe > 100mV, except when leakage currents are of interest.
I
e
=
I
o
e
V
be
V
t
-7
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAfadaWgaaadbaGaamOyaiaadwgaaeqaaaWcbaGaamOvamaaBaaameaacaWG0baabeaaaaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqG3aaaaa@44DC@
This equation is used in two ways.
1 Large signal conditions.
2 Small signal conditions
Large signal conditions are conditions such that the
circuit is analyzed using complete F(V, I) equations or equations that are
approximately correct for large voltage/current changes. Small signal
conditions deal with the effects of small signal changes of current and voltage
around some fixed large signal bias voltage or current. An example should make this a bit clearer.
Large signal conditions
Consider a 10 volt supply connected to a series
resistor and diode. It can be deduced from the diode/transistor equation above
that if the diode voltage changes by 60mv a 10:1 change will occur in the diode
current. Thus the diode voltage don't change much!. Lets pretend then that if
the diode has 0.75 volts across it at 100ma, then at 1A the diode voltage will
therefor be 0.81. So it can be said that if the current in the circuit is
somewhere between 100ma and 1A, the voltage across the resistor will be
somewhere between (10-0.75) and (10-0.81) i.e. around 9.22 +/- a little bit. If
we desire say, 500ma, letting R=9.22/.5 = 46ohms will set the current quite
accurately, even though the diode voltage is not known precisely. i.e. it must
be somewhere between 9.25/46 and 9.19/46. As an approximation therefore, it is
often sufficient to consider that a diode produces a constant DC voltage drop
irrespective of current.
Small signal conditions
Consider a small signal voltage i.e. say a signal of
10mV.Sin(wt) is added in series with the above circuit. The voltage that
appears across the diode due to this signal will be reduced by the series
resistor, and will not be constant in
the context of small signals. The next section shows how this signal is
calculat
Transistor GM
The gm, or mutual conductance, of the transistor may
be said to be the most fundamental design parameter of a transistor amplifier.
It is expressed by:
I
e
=gm.
V
i
-8
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadEgacaWGTbGaaiOlaiaadAfadaWgaaWcbaGaamyAaaqabaGccaqGGaGaaeiiaiaabccacaqGTaGaaeioaaaa@40BB@
That is, a small signal voltage at the base emitter
such as 10mV.Sin(wt) will result in a small signal emitter current of I=
gm.10mV.Sin(wt).
Another way of expressing this statement is by the
reciprocal of gm which is named "re". That is, re=1/gm. Then:
I
e
=
V
i
r
e
-9
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGPbaabeaaaOqaaiaadkhadaWgaaWcbaGaamyzaaqabaaaaOGaaeiiaiaabccacaqGGaGaaeylaiaabMdaaaa@4053@
This small signal effective resistance, re, is a
very useful concept. It states what the small signal voltage will be, if the
small signal current through re is known. For the example above, this allows an
easy calculation of the small signal voltage across the diode as Vd
= Vi (re/(R + re)) i.e. by the use of conventional formulas.
Derivation of
Transistor GM
Since gm is the small change in current produced by
a small change in voltage it is possible to approximate the gm by calculating a
derivative from a general mathematical result:
δI≅
dI
dV
δV -10
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaamysaiabgwKianaalaaabaGaamizaiaadMeaaeaacaWGKbGaamOvaaaacqaH0oazcaqGwbGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeymaiaabcdaaaa@44D9@
or
gm≡
dI
dV
≅
δI
δV
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGHHjIUdaWcaaqaaiaadsgacaWGjbaabaGaamizaiaadAfaaaGaaeiiaiabgwKianaalaaabaGaeqiTdqMaamysaaqaaiabes7aKjaabAfaaaaaaa@43ED@
From:
I
e
=
I
o
e
V
be
V
t
-11
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAfadaWgaaadbaGaamOyaiaadwgaaeqaaaWcbaGaamOvamaaBaaameaacaWG0baabeaaaaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaaeymaaaa@458A@
gm=
d
I
c
d
V
be
=
d
d
V
be
(
I
o
e
V
be
V
t
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpdaWcaaqaaiaadsgacaWGjbWaaSbaaSqaaiaadogaaeqaaaGcbaGaamizaiaadAfadaWgaaWcbaGaamOyaiaadwgaaeqaaaaakiabg2da9maalaaabaGaamizaaqaaiaadsgacaWGwbWaaSbaaSqaaiaadkgacaWGLbaabeaaaaGccaGGOaGaamysamaaBaaaleaacaWGVbaabeaakiaadwgadaahaaWcbeqaamaalaaabaGaamOvamaaBaaameaacaWGIbGaamyzaaqabaaaleaacaWGwbWaaSbaaWqaaiaadshaaeqaaaaaaaGccaqGPaaaaa@4EA8@
gm=
I
o
1
V
t
e
V
be
V
t
=(
I
o
e
V
be
V
t
)
1
V
t
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaWGjbWaaSbaaSqaaiaad+gaaeqaaOWaaSaaaeaacaaIXaaabaGaamOvamaaBaaaleaacaWG0baabeaaaaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAfadaWgaaadbaGaamOyaiaadwgaaeqaaaWcbaGaamOvamaaBaaameaacaWG0baabeaaaaaaaOGaeyypa0JaaiikaiaadMeadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAfadaWgaaadbaGaamOyaiaadwgaaeqaaaWcbaGaamOvamaaBaaameaacaWG0baabeaaaaaaaOGaaiykamaalaaabaGaaGymaaqaaiaadAfadaWgaaWcbaGaamiDaaqabaaaaaaa@50E3@
gm=
I
o
1
V
t
e
V
be
V
t
=
I
e
V
t
=
I
e
q
KT
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaWGjbWaaSbaaSqaaiaad+gaaeqaaOWaaSaaaeaacaaIXaaabaGaamOvamaaBaaaleaacaWG0baabeaaaaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiaadAfadaWgaaadbaGaamOyaiaadwgaaeqaaaWcbaGaamOvamaaBaaameaacaWG0baabeaaaaaaaOGaeyypa0ZaaSaaaeaacaWGjbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaamOvamaaBaaaleaacaWG0baabeaaaaGccqGH9aqpcaWGjbWaaSbaaSqaaiaadwgaaeqaaOWaaSaaaeaacaWGXbaabaGaam4saiaadsfaaaaaaa@4E5C@
gm=
I
e
q
KT
-12
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaWGjbWaaSbaaSqaaiaadwgaaeqaaOWaaSaaaeaacaWGXbaabaGaam4saiaadsfaaaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaaeOmaaaa@420A@
Hence the small signal gm is a function of the DC
bias current. Too obtain a large gm you need large current, which of course means
that a larger base current is required.
Plugging in the numbers for room temperature one
gets:
gm=40
I
e
-13
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaaI0aGaaGimaiaadMeadaWgaaWcbaGaamyzaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaae4maaaa@4177@
and
re=
25mV
I
e
-14
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaadwgacqGH9aqpdaWcaaqaaiaaikdacaaI1aGaamyBaiaadAfaaeaacaWGjbWaaSbaaSqaaiaadwgaaeqaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeymaiaabsdaaaa@42B8@
So at 1ma, re =25ohms
Transistor
Voltage Gain
The small signal collector current as a function of
small signal input voltage (Vbe) has now been found to be:
i
c
=gm
V
i
h
fe
(1+
h
fe
)
-15
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaadEgacaWGTbGaamOvamaaBaaaleaacaWGPbaabeaakmaalaaabaGaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaqGGaaabaGaaiikaiaaigdacqGHRaWkcaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaakiaacMcaaaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaaeynaaaa@4B14@
When equation (3) is used to obtain the collector
current from the emitter current
For simplicity, the hfe multiplier can be dropped by
the assumption of large hfe, such that:
i
c
=gm
V
i
-16
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGJbaabeaakiabg2da9iaadEgacaWGTbGaamOvamaaBaaaleaacaWGPbaabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabgdacaqG2aaaaa@421F@
There will be an output voltage developed across the
collector resistor by V=-IR:
V
c
=−gm
R
c
V
i
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGJbaabeaakiabg2da9iabgkHiTiaadEgacaWGTbGaamOuamaaBaaaleaacaWGJbaabeaakiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@3F98@
or
A
v
=
V
c
V
i
=−gm
R
c
-17
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGJbaabeaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0JaeyOeI0Iaam4zaiaad2gacaWGsbWaaSbaaSqaaiaadogaaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaae4naaaa@4759@
Which is often more usefully expressed as:
A
v
=−
R
c
r
e
-18
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9iabgkHiTmaalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaadkhacaWGLbWaaSbaaSqaaaqabaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGXaGaaeioaaaa@4295@
If an emitter resistor is added, the voltage to the
base emitter will be reduced by Ic.Re, which is equivalent to having re
increased by Re, since Ie flows through re, therefore:
A
v
=−
R
c
(
R
e
+re)
-19
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9iabgkHiTmaalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaacIcacaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOCaiaadwgacaGGPaaaaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeymaiaabMdaaaa@4692@
So the way to think about transistor gain is by way
of the load resister divided by effective emitter resistance.
It was noted above that at 1ma, re = 25 ohms so if
the added emitter resistance is say 1k, re drops out.
Mensa Test
The voltage across the collector resistor of a
transistor amplifier is 2V. What is the gain of the stage?
What bias current?. It don’t matter!
From (8) and (6)
A
v
=
V
c
V
i
=gm
R
c
-20
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGJbaabeaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0Jaam4zaiaad2gacaWGsbWaaSbaaSqaaiaadogaaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGYaGaaeimaaaa@4666@
gm=40
I
c
-21
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaaI0aGaaGimaiaadMeadaWgaaWcbaGaam4yaaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGYaGaaeymaaaa@4174@
So:
A
v
=40
I
c
R
c
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9iaaisdacaaIWaGaamysamaaBaaaleaacaWGJbaabeaakiaadkfadaWgaaWcbaGaam4yaaqabaaaaa@3E30@
But Ic.Rc is the DC voltage dropped across Rc so,
A
v
=40
V
cdc
-22
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9iaaisdacaaIWaGaamOvamaaBaaaleaacaWGJbGaamizaiaadogaaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGYaGaaeOmaaaa@42C9@
The gain is 80 then!
Second Order
Effects rbb' and Ro
So far base current and non-ideal current output has
been ignored. These effects of these will now be investigated.
Ro
It was stated that the transistor is a voltage
controlled current source. The first correction to this is that there is a
finite equivalent ac resistance across the collector emitter. This is noted by
the fact that increasing the collector emitter voltage results in a small
gradual increase of collector current. The change in Vce to the change in Ic is
called it output resistance ro, or in the literature often described by 1/hoe.
The effect of this resistance in a simple amplifier is to simply put it in
parallel with the load resistor.
This output resistance is specified by a term named
the "Early Voltage", or Va. It can be shown that the following holds
in general:
r
o
=
Va+Vce
I
c
-23
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGVbaabeaakiabg2da9maalaaabaGaamOvaiaadggacqGHRaWkcaWGwbGaam4yaiaadwgaaeaacaWGjbWaaSbaaSqaaiaabogaaeqaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeOmaiaabodaaaa@44FC@
That is, the small signal output resistance is a
direct function of a constant and inversely related to the DC bias current.
Typical Early voltages are from 30V to 200 volts.
If output characteristic plots are made, that is,
one of Vce (x-axis) against Ic (y-axis),
for various fixed base currents, all of the slopes of the Vce against Ic plots
will intersect at the same negative Vce voltage. This magnitude of this voltage
is the early voltage, hence why the ro slope is given by (21)
Often, the collector emitter voltage term is
drooped, as the early voltage might be say, 100V so that with a collector
emitter voltage of say, 5V, there is only 5% error.
r
o
=
Va
I
c
-24
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGVbaabeaakiabg2da9maalaaabaGaamOvaiaadggaaeaacaWGjbWaaSbaaSqaaiaabogaaeqaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeOmaiaabsdaaaa@416E@
This Early voltage leads to a maximum gain from a
simple amplifier stage as follows.
Consider an ideal case where a perfect current
source is the load on the transistor amplifier.
From (20)
A
v
=gm
R
c
=
I
c
V
t
r
o
=
I
c
V
t
V
a
I
c
=
V
a
V
t
-25
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@59C9@
For an Early voltage of 100V and a Vt of 25mV, then
the maximum voltage gain is Av=4000.
Ways to bypass this limit with cascodes will be
addressed in another paper.
rbb'
There is base resistance from the base terminal to
the real base of the transistor. This resistance is called the base spreading resistor. It varies
slightly with collector current, but this effect will be ignored for now.
Typically values for rbb' are from 10 ohms to 500 ohms. This
resistance can significantly effect noise performance.
The effect of this resistor is seen from inspection.
Since the real base emitter voltage is reduced by ib.rbb' which
equals Ic.rbb/hfe, this value can simply be added in to
our voltage gain equation:
A
v
=
R
c
(
R
e
+re+
rbb'
h
fe
)
-26
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaacIcacaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOCaiaadwgacqGHRaWkdaWcaaqaaiaadkhacaWGIbGaamOyaiaacEcaaeaacaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaaaaGccaGGPaaaaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeOmaiaabAdaaaa@4CFD@
Other then for noise, and high frequency response, rbb'
can generally be ignored in many cases. For example, if Re was only 100, and hfe
was 200, a 50 base resistance might only effect the gain by (100 + 50/200)/100
= 1.0025.
Input Resistance
Given an input voltage it is obviously desirable to
know what the resulting base current is. This is now calculated:
Equation (19) has to be corrected slightly as it was
derived assuming that Ic was equal to Ie (equation (15). It is easy to see that
Re and re need the prior dropped (1+hfe)/ hfe term as a
correction factor. Go back and check!
A
v
=
R
c
(
(
R
e
+re)(1+
h
fe
)
h
fe
+
rbb'
h
fe
)
-27
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaacIcadaWcaaqaaiaacIcacaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOCaiaadwgacaGGPaGaaiikaiaaigdacqGHRaWkcaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaakiaacMcaaeaacaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaaaaGccqGHRaWkdaWcaaqaaiaadkhacaWGIbGaamOyaiaacEcaaeaacaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaaaaGccaGGPaaaaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeOmaiaabEdaaaa@574D@
V
o
V
i
=
R
c
(
(
R
e
+re)(1+
h
fe
)
h
fe
+
rbb'
h
fe
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGwbWaaSbaaSqaaiaad+gaaeqaaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaaaaGccqGH9aqpdaWcaaqaaiaadkfadaWgaaWcbaGaam4yaaqabaaakeaacaGGOaWaaSaaaeaacaGGOaGaamOuamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkhacaWGLbGaaiykaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaaabaGaamiAamaaBaaaleaacaWGMbGaamyzaaqabaaaaOGaey4kaSYaaSaaaeaacaWGYbGaamOyaiaadkgacaGGNaaabaGaamiAamaaBaaaleaacaWGMbGaamyzaaqabaaaaOGaaiykaaaaaaa@54BF@
Rearranging
V
i
=
V
o
R
c
(
(
R
e
+re)(1+
h
fe
)
h
fe
+
rbb'
h
fe
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGVbaabeaaaOqaaiaadkfadaWgaaWcbaGaam4yaaqabaaaaOGaaiikamaalaaabaGaaiikaiaadkfadaWgaaWcbaGaamyzaaqabaGccqGHRaWkcaWGYbGaamyzaiaacMcacaGGOaGaaGymaiabgUcaRiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaOGaaiykaaqaaiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaaaakiabgUcaRmaalaaabaGaamOCaiaadkgacaWGIbGaai4jaaqaaiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaaaakiaacMcaaaa@54AF@
V
i
=
I
c
(
(
R
e
+re)(1+
h
fe
)
h
fe
+
rbb'
h
fe
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4yaaqabaGccaGGOaWaaSaaaeaacaGGOaGaamOuamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkhacaWGLbGaaiykaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaaabaGaamiAamaaBaaaleaacaWGMbGaamyzaaqabaaaaOGaey4kaSYaaSaaaeaacaWGYbGaamOyaiaadkgacaGGNaaabaGaamiAamaaBaaaleaacaWGMbGaamyzaaqabaaaaOGaaiykaaaa@5291@
V
i
=
I
b
h
fe
(
(
R
e
+re)(1+
h
fe
)
h
fe
+
rbb'
h
fe
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadMeadaWgaaWcbaGaamOyaaqabaGccaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaakiaacIcadaWcaaqaaiaacIcacaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOCaiaadwgacaGGPaGaaiikaiaaigdacqGHRaWkcaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaakiaacMcaaeaacaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaaaaGccqGHRaWkdaWcaaqaaiaadkhacaWGIbGaamOyaiaacEcaaeaacaWGObWaaSbaaSqaaiaadAgacaWGLbaabeaaaaGccaGGPaaaaa@5588@
V
i
=
I
b
((
R
e
+re)(1+
h
fe
)+rbb')
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadMeadaWgaaWcbaGaamOyaaqabaGccaGGOaGaaiikaiaadkfadaWgaaWcbaGaamyzaaqabaGccqGHRaWkcaWGYbGaamyzaiaacMcacaGGOaGaaGymaiabgUcaRiaadIgadaWgaaWcbaGaamOzaiaadwgaaeqaaOGaaiykaiabgUcaRiaadkhacaWGIbGaamOyaiaacEcacaGGPaaaaa@4C80@
or
R
i
=
V
i
I
b
=(rbb'+(
R
e
+re)(1+
h
fe
)) -28
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbaabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGPbaabeaaaOqaaiaadMeadaWgaaWcbaGaamOyaaqabaaaaOGaeyypa0JaaiikaiaadkhacaWGIbGaamOyaiaacEcacqGHRaWkcaGGOaGaamOuamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkhacaWGLbGaaiykaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaaeOmaiaabIdaaaa@543D@
And approximately Ri =hfe.Re'
Where Re' is the sum of Re and re
It is also shown later that Ri with Re=0, is often
called hie
Emitter Follower
Connection
This has the collector connected to the supply and
the output taken from the emitter and emitter resistor junction. The following
small signal equation can be seen from inspection.
V
i
=
V
be
+
I
e
R
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAfadaWgaaWcbaGaamOyaiaadwgaaeqaaOGaey4kaSIaamysamaaBaaaleaacaWGLbaabeaakiaadkfadaWgaaWcbaGaamyzaaqabaaaaa@4088@
But V can be expressed by the gm or re equation
previously derived, therefore
V
i
=
I
e
r
e
+
I
e
R
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadMeadaWgaaWcbaGaamyzaaqabaGccaWGYbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamysamaaBaaaleaacaWGLbaabeaakiaadkfadaWgaaWcbaGaamyzaaqabaaaaa@41AB@
V
i
=
I
e
(
r
e
+
R
e
) -29
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGPbaabeaakiabg2da9iaadMeadaWgaaWcbaGaamyzaaqabaGccaGGOaGaamOCamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkfadaWgaaWcbaGaamyzaaqabaGccaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGYaGaaeyoaaaa@45CD@
But
V
o
=
I
e
R
e
-30
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGVbaabeaakiabg2da9iaadMeadaWgaaWcbaGaamyzaaqabaGccaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaae4maiaabcdaaaa@421C@
and dividing (30 by (29)
A
v
=
V
oi
V
i
=
I
e
R
e
I
e
(
r
e
+
R
e
)
=
R
e
(
r
e
+
R
e
)
-31
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGVbGaamyAaaqabaaakeaacaWGwbWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9maalaaabaGaamysamaaBaaaleaacaWGLbaabeaakiaadkfadaWgaaWcbaGaamyzaaqabaaakeaacaWGjbWaaSbaaSqaaiaadwgaaeqaaOGaaiikaiaadkhadaWgaaWcbaGaamyzaaqabaGccqGHRaWkcaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaaiykaiaabccaaaGaaeiiaiaabccacqGH9aqpcaqGGaGaaeiiaiaabccadaWcaaqaaiaadkfadaWgaaWcbaGaamyzaaqabaaakeaacaGGOaGaamOCamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkfadaWgaaWcbaGaamyzaaqabaGccaGGPaGaaeiiaaaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGTaGaae4maiaabgdaaaa@5ECC@
Hence the gain is near unity for large Re.
Writing (31) as
V
o
=
V
i
R
e
(
r
e
+
R
e
)
-32
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGVbaabeaakiabg2da9iaabccacaWGwbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacaWGsbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaaiikaiaadkhadaWgaaWcbaGaamyzaaqabaGccqGHRaWkcaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaaiykaiaabccaaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeylaiaabodacaqGYaaaaa@4A71@
Which is the same expression for a voltage source of
Vi with internal impedance re driving a load Re. Hence, the emitter
follower has an output impedance of re (in || with Re). So, although the
voltage gain is only about unity, it has a low output resistance.
Base grounded, or
Common Base Connection
In this configuration the input is at the emitter
and the output is still taken from the collector. Vi.gm must still give the
emitter current, as the signal is still applied between the same base and
emitter! The gain must therefore also be the same, Rc/re. The difference in
this connection is that the input resistance is only re, not hfe re
as the input signal now supplies all of the emitter current, not just a small
fraction of it. The advantage of this connection is that it is often faster
then the common emitter amplifier. This will be examined in another paper.
Summary of the
main formulas
I
e
=
I
o
(
e
q
V
be
KT
−1)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGLbaabeaakiabg2da9iaadMeadaWgaaWcbaGaam4BaaqabaGccaGGOaGaamyzamaaCaaaleqabaWaaSaaaeaacaWGXbGaamOvamaaBaaameaacaWGIbGaamyzaaqabaaaleaacaWGlbGaamivaaaaaaGccqGHsislcaaIXaGaaiykaaaa@4485@
I
c
≅
I
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWGJbaabeaakiabgwKiajaadMeadaWgaaWcbaGaamyzaaqabaaaaa@3AE7@
gm=
I
e
q
KT
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaWGjbWaaSbaaSqaaiaadwgaaeqaaOWaaSaaaeaacaWGXbaabaGaam4saiaadsfaaaaaaa@3D65@
gm=40
I
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaad2gacqGH9aqpcaaI0aGaaGimaiaadMeadaWgaaWcbaGaamyzaaqabaaaaa@3C24@
re=
25mV
I
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaadwgacqGH9aqpdaWcaaqaaiaaikdacaaI1aGaamyBaiaadAfaaeaacaWGjbWaaSbaaSqaaiaadwgaaeqaaaaaaaa@3E07@
i
e
=gm
V
i
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBaaaleaacaWGLbaabeaakiabg2da9iaadEgacaWGTbGaamOvamaaBaaaleaacaWGPbaabeaaaaa@3CCB@
r
o
=
V
E
I
c
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGVbaabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGfbaabeaaaOqaaiaadMeadaWgaaWcbaGaae4yaaqabaaaaaaa@3CD6@
r
e
=
1
g
m
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGLbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadEgadaWgaaWcbaGaaeyBaaqabaaaaaaa@3BD4@
A
vmax
=
V
E
V
t
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2bGaciyBaiaacggacaGG4baabeaakiabg2da9maalaaabaGaamOvamaaBaaaleaacaWGfbaabeaaaOqaaiaadAfadaWgaaWcbaGaamiDaaqabaaaaOGaaeiiaaaa@404D@
Common Emitter
A
v
=
R
c
(
R
e
+re)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaacIcacaWGsbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOCaiaadwgacaGGPaaaaiaabccaaaa@419C@
R
i
=rbb'+(
R
e
+re)(1+
h
fe
)≅
h
fe
(
R
e
+re)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbaabeaakiabg2da9iaadkhacaWGIbGaamOyaiaacEcacqGHRaWkcaGGOaGaamOuamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkhacaWGLbGaaiykaiaacIcacaaIXaGaey4kaSIaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGPaGaeyyrIaKaamiAamaaBaaaleaacaWGMbGaamyzaaqabaGccaGGOaGaamOuamaaBaaaleaacaWGLbaabeaakiabgUcaRiaadkhacaWGLbGaaiykaaaa@5376@
Emitter follower
A
v
=
R
e
(
r
e
+
R
e
)
-31
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOuamaaBaaaleaacaWGLbaabeaaaOqaaiaacIcacaWGYbWaaSbaaSqaaiaadwgaaeqaaOGaey4kaSIaamOuamaaBaaaleaacaWGLbaabeaakiaacMcacaqGGaaaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaab2cacaqGZaGaaeymaaaa@47C0@
R
0
=
r
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaaIWaaabeaakiabg2da9iaadkhadaWgaaWcbaGaamyzaaqabaaaaa@3ABE@
Common Base
R
i
=
r
e
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbaabeaakiabg2da9iaadkhadaWgaaWcbaGaamyzaaqabaaaaa@3AF2@
A
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=
R
c
re
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWG2baabeaakiabg2da9maalaaabaGaamOuamaaBaaaleaacaWGJbaabeaaaOqaaiaadkhacaWGLbaaaaaa@3CC7@
The above formulas should be assimilated to heart.
They are indispensable.
© 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