Common Emitter Amp
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Lab-Report
Analogue Electronics
The Common-Emitter Amplifier
Name:
Course:
Group:
Student No.:
Date:
Dirk Becker
BEng 2
A
9801351
30/10/1998
1. Contents
1.
2.
3.
a)
b)
c)
d)
4.
a)
b)
c)
d)
e)
5.
a)
b)
c)
6.
7.
a)
b)
c)
8.
a)
b)
c)
d)
9.
a)
b)
c)
CONTENTS ............................................................................................................................................ 2
INTRODUCTION .................................................................................................................................. 3
DC BIAS OF COMMON-EMITTER AMPLIFIER ........................................................................... 3
Measurment of the “true” resistor values.................................................................................................. 3
DC-bias precalculation with given resistors ............................................................................................. 4
Measurement of DC-Voltages (DC-bias) ................................................................................................. 5
Calculation of IB, IC and β with measured values ..................................................................................... 5
i. IB with IE and IC (RE and RC)..................................................................................................................... 5
ii.
IB with I1 and I2 (R1 and R2) ................................................................................................................. 5
AC AND SMALL SIGNAL CHARACTERISTICS............................................................................ 6
BJT small signal behavioural model......................................................................................................... 6
Definitions of the small signal parameters................................................................................................ 6
Measured small signal values ................................................................................................................... 7
Calculation and derivation of the small signal parameters ....................................................................... 7
i. Amplifier input resistance Ri ..................................................................................................................... 7
ii.
Amplifier output resistance Ro ............................................................................................................. 8
iii. Voltage amplification without load Avo ................................................................................................ 9
iv. Voltage amplification with load Av ...................................................................................................... 9
v.
Current amplification Ai .................................................................................................................... 10
vi. Amplifier power gain Ap ..................................................................................................................... 10
Comparison between measured and calculated values ........................................................................... 11
i. DC-Bias .................................................................................................................................................. 11
ii.
Small signal parameters..................................................................................................................... 11
FREQUENCY RESPONSE OF THE CE-AMPLIFIER................................................................... 12
Measured frequency response of Av ....................................................................................................... 12
Determination of the theoretical cut-off frequency................................................................................. 13
Frequency response of Ai........................................................................................................................ 16
DISCONNECTION OF BYPASS CAPACITOR C3 ......................................................................... 18
DC-BIAS WITH DISCONNECTED C3 ............................................................................................. 18
Calculation of DC-Bias........................................................................................................................... 18
Measurement of DC-Voltages (DC-bias) ............................................................................................... 18
Calculation of IB, IC and β with measured values( C3 disconnected) ...................................................... 19
i. IB with IE and IC (RE and RC)................................................................................................................... 19
ii.
IB with I1 and I2 (R1 and R2) ............................................................................................................... 19
AC AND SMALL SIGNAL CHARACTERISTICS (WITHOUT C3) ............................................. 20
BJT small signal behavioural model....................................................................................................... 20
Measured small signal values without C3 ............................................................................................... 20
Calculation and derivation of the small signal parameters (mid freq., without C3) ................................ 21
i. Amplifier input resistance Ri ................................................................................................................... 21
ii.
Amplifier output resistance Ro ........................................................................................................... 22
iii. Voltage amplification at mid frequencies without load...................................................................... 22
iv. Voltage amplification at mid frequencies with load........................................................................... 23
v.
Current amplification Ai .................................................................................................................... 23
vi. Amplifier power gain Ap ..................................................................................................................... 24
Comparison between measured and calculated values ........................................................................... 24
FREQUENCY RESPONSE OF THE CE-AMPLIFIER WITHOUT C3 ......................................... 25
Frequency response of Av ....................................................................................................................... 25
Theoretical calculation of the lower cut-off frequency........................................................................... 27
Frequency response of Ai without C3...................................................................................................... 29
2
2. Introduction
The common-emitter amplifier is one of the most common discrete amplifier used in
conventional electronics. Its special characteristics were to be discovered in this lab session.
3. DC bias of common-emitter amplifier
figure 1
a) Measurment of the “true” resistor values
For calculating the correct bias values at first the real values of the used resistors and
capacitors have to be measured, because they all have got deviations from the printed values.
Also a measurement of the very small currents is not possible. So the voltage drop across the
different devices has to be determined and then the flowing current can be calculated via KCL
and Ohm’s law.
The following values were measured:
RS=3.26kΩ
R1=9.99kΩ
R2=55.7kΩ
RE=986Ω
RC=3.82kΩ
RL=6.72kΩ
C1=80.54µF
C2=8.949µF
C3=9.177µF
3
b) DC-bias precalculation with given resistors
β=200, VBE=0.7V
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RI =
R 1R 2
10kΩ ⋅ 56kΩ
=
= 8.485kΩ
R 1 + R 2 10kΩ + 56kΩ
Vo =
R1
10kΩ
VCC =
10V = 1.515V
R1 + R 2
10kΩ + 56kΩ
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KCL :
0V = I E R E − VO + R I I B
-
7
-
+
7
9
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:
8
0V = (I C + I B ) − VO + R I I B
0V = I B (βR E + R E + R 1 ) − VO
R1
10kΩ
VCC
10V
VO
R1 + R 2
10
k
56
k
Ω
+
Ω
IB =
=
=
= 7.232µA
R 1R 2
βR E + R E + R I
200 ⋅ 1000Ω + 1000Ω + 8.485kΩ
βR E + R E +
R1 + R 2
I C = βI B = 200 ⋅ 7.232µA = 1.446mA
I E = (β + 1)Ι Β
I E = 201 ⋅ 7.232µΑ = 1.4536mA
VRC = R C I C
VRC = 3.9kΩ ⋅ 1.446mA
VR 2 = VCC − VBE − VRE
VRC = 5.639V
VR 2 = 10V − 0.7 V − 1.454V
VR 2 = 7.846V
VRE = R E I E = R C [I B (β + 1)
VRE = 1000Ω(7.232µA ⋅ 201)
IR2 =
VRE = 1.454V
VR 2 7.846V
=
R2
56kΩ
I R 2 = 0.140mA
VCE = VCC − VRE − VRE
VCE = 10V − 5.639V − 1.454V
VC = VRE + VCE = 1.454V + 2.907V
VCE = 2.907 V
VC = 4.361V
I R1 =
VR1 = VCC − VR 2
VR1 2.54V
=
= 0.215mA
R 1 10kΩ
VR1 = 10V − 7.846V = 2.154V
4
c) Measurement of DC-Voltages (DC-bias)
Next step was to measure the voltages in order to check the real bias and to calculate based on
this values the DC current amplification and the other occurring currents and voltages.
VR1=1.477V
VR2=8.51V
VRE=0.863V
VRC=3.33V
VC=6.66
VBE=0.618
VCE=5.80
d) Calculation of IB, IC and β with measured values
i. IB with IE and IC (RE and RC)
IE =
VRE 0.863V
=
= 0.875mA
RE
986Ω
IC =
VRC
3.33V
=
= 0.8713mA
R C 3.82kΩ
IB = IE − IC
I B = 0.875mA − 0.8713mA
I B = 3.7µA
β = h fe =
I C 0.8713mA
=
= 235
IB
3.7µA
ii. IB with I1 and I2 (R1 and R2)
I R1 =
VR1 1.477V
=
= 0.14785mA
R 1 9.99kΩ
IR2 =
VR 2
8.51V
=
= 0.15278mA
R2
55.7kΩ
I B = I R 2 − I R1 = 0.15278mA − 0.14785mA = 4.93µΑ
β=
I C 0.8713mA
=
= 177
IB
4.93µA
5
4. AC and small signal characteristics
Next part of the lab was to determine the small signal characteristics of the common-emitter
amplifier. Small signal characteristics are the essential values for the amplification function of
the circuit.
For calculation of these characteristics a special model is used, the small signal model of the
BJT. These models are derived from the characteristic curves of the real devices and, when
properly devised, can be used to predict accurately the behaviour of semiconductor devices in
practical applications such as voltage and current amplification.
By computer simulation more complex models can be used.
a) BJT small signal behavioural model
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figure 2
In the small signal model for mid frequencies the capacitors are short circuited because their
impedance is negligble to the rest of the circuit.
b) Definitions of the small signal parameters
voc:
vo:
vs:
vi:
vrs:
Ri:
Ro:
Avo:
Av:
Ai:
Ap:
open circuit output voltage
output voltage with load
signal voltage
ampifier input voltage
voltage drop across sensor resistor
amplifier input resistance
amplifier output resistance
amplifier voltage gain without load
amplifier voltage gain with load
amplifier current gain
amplifier power gain
6
[
c) Measured small signal values
(A generator with f=1kHz works as voltage source V1)
vs=20mV
voc=1.242V
vo=0.799V
vi=10.8mV
vrs=9.1mV
d) Calculation and derivation of the small signal parameters
ALL THEORETICAL VALUES WITH INDEX ‘ (i.e. R’)
i. Amplifier input resistance Ri
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figure 3
Measured value of R i :
Theoretical Values for R i :
R 'i =
R 'i = R 1 || R 2 || h fe re
R 'i = R 1 || R 2 || h fe
with re =
26mV
IE
vi
v
10.8mV
= i =
= 3.87kΩ
v rs
9.1mV
ii
3.26kΩ
rs
26mV
IE
R 'i = 10kΩ || 56kΩ || 200
R 'i =
Ri =
vi
= R 1 || R 2 || h ie
ii
26mV
1.453mA
1
1
1
1
+
+
10kΩ 56kΩ 3.578kΩ
= 2.52kΩ
The theoretical value of Ri fits good with the determination due the measured values. The
measurement of vi and ii was relative uncertain because of the small values and not using the
whole range of the measuring instrument (~20mV to 200mV Range).
7
ii. Amplifier output resistance Ro
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v
w
x
v
{
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w
}
figure 4
Theoretical value of R o :
Measured value of R o :
R 'o = R C
Ro =
with h OE = 0S
R ' o = 3.9kΩ
Ro =
v oc − v o
vo
RL
1.242V − 0.799V
= 3.72kΩ
0.799V
6.72kΩ
From these in the lab measured values hOE can be easily determined:
1
1
1
=
+
R O h OE R C
1
h OE
=
h OE =
1
1
−
RO RC
1
1
1
−
RO RC
=
1
1
1
−
3.72kΩ 3.82kΩ
h OE = 142.1kΩ
It can be seen, that hOE can be let out at calculation of BJTs, because of its large value to the
rest of the circuit.
8
iii. Voltage amplification without load Avo
Theoretical values of A vo :
A' vo =
v oc
vi
A' vo =
− icR C
h i R
= − fe B C
h ie i B
h ie i B
A' vo = −
A' vo =
A vo =
R L →∞
h fe R C
=−
h ie
Measured value of A vo :
h fe R C
26mV
h fe
IE
v oc
vi
1.242V
10.8mV
A vo = 115
A vo =
3.9kΩ
= 218
26mV
1.4536mA
iv. Voltage amplification with load Av
Measured value of A v :
Theoretical value of A v :
A' v =
A' v =
vo
vi
Av =
− i c (R C R L )
A' v = −
h fe (R c R L )
h ie
0.799V
10.8mV
A v = 74
Av =
h ie i B
=
h fe (R c R L )
26mV
h fe
IE
RC ⋅RL
RC RL
R + RL
A' v =
= C
26mV
26mV
IE
IE
A' v =
vo
vi
2.479kΩ
= 138
26mV
1.4536mA
9
v. Current amplification Ai
Theoretical value of A i :
A'i =
iC
iI
A'i =
h fe i B
iI
with i B =
Measured value of A i :
R1 R 2
h ie + R 1 R 2
v oc
iC
Rc
Ai =
=
v rs
iI
Rs
iI
R1 R 2
R1 R 2
A'i = h fe
= h fe
h ie + R 1 R 2
h fe 26mV + R 1 R 2
IE
8.485kΩ
A'i = 200
3.58kΩ + 8.485kΩ
A'i = 140
Ai =
1.242V
9.1mV
A i = 117
3.82kΩ
3.26kΩ
vi. Amplifier power gain Ap
Theeoretic values of A p :
Measured value of A p :
A' p = A i ⋅ A v
A 'p = A i ⋅ A v
A' p = h fe
A' p = h fe
h fe (R C R L )
RI
h ie + R I
h ie
A'p = 74 ⋅117
A'p = 8658
RC RL
RI
h ie + R I 26mV
IE
8.458kΩ
2.479kΩ
⋅
3.58kΩ + 8.458kΩ 26mV
1.1436mA
A' p = 19494
A' p = 200
10
e) Comparison between measured and calculated values
i. DC-Bias
Measurand
measured
VRC
3.33V
VRE
0.863V
VCE
5.80V
VC
6.66V
VR1
2.154V
VR2
7.846V
calculated
5.639V
1.454V
2.907V
4.361V
2.154V
7.846V
hfe
237/
177
200
ii. Small signal parameters
Measurand
measured
calculated
Rin
3.87 kΩ
2.52 kΩ
Rout
3.72kΩ
3.9kΩ
Avo
115
218
Av
74
138
Ai
117
140
Ap
8685
19494
The measured and the calculated values are partly very different. Major problem of the
determination of the Bias and the small signal parameters were the uncertain measurements.
Especially the small values like IB and the input current of the small signal equivalent circuit
were very difficult to determine exactly.
11
5. Frequency response of the CE-amplifier
a) Measured frequency response of Av
The voltage amplification over a range from 10Hz to 1kHz was to measure to determine the
cut-off frequency of the CE-Amplifier.
f/Hz
vi/mV
vo/mV
vo/vi
10
13.1
183.4
14
20
12.8
0.324
25.31
50
11.4
555
48.7
70
11.1
618
55.7
100
10.7
667
62.3
120
11.4
730
64
200
10.2
714
70
500
10
734
73.4
700
10
736
73.6
1000
10
738
73.8
Voltage Amplification vs frequency
Av
100
− 3dB =
1
A v max
2
10
10
100
1000
f
fc=58Hz
figure 5
The cut-off frequency is defined as the frequency, when the amplification is dropped down to
70.7% (-3dB or ½√2).
Below this frequency there is still amplification, but the frequency range of amplifiers is
commonly defined by the 3dB cut-off freuency.
12
b) Determination of the theoretical cut-off frequency
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figure 6
Figure 6 shows the equivalent circuit for low and mid frequencies.
To determine the cut-off frequency the input and output circuit has to be separated.
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figure 7
Input
Current source
ic=constant
Output
Í
The cut-off frequency of the CE-Amplifier is above all depending on the emitter bypass
capacitor CE=C3. The impedance of the two coupling capacitors is about 1500Ω at 10Hz each
and can be neglected because of these small values in comparison to the rest of the circuit.
For the input circuit:
IB =
vi
h ie + R In + Z EIn
where Z EIn = (β + 1)Z E =
β +1
1
+ jω C E
RE
=
1
C
1
+ jω E
β +1
(β + 1)R E
13
The current through (β+1)RE can be neglected because of it’s large resistance
v in
iB =
β +1
ωC E
h ie + R In − j
The current IB will drop to 70.7% of it’s maximum value when the voltage drop at (hie+RIn)
and ZE are of the same magnitude.
1
= R In + h ie
ωC E
β +1
β +1
ωcutoff =
where R in = R 1 R 2 R S = 2.38kΩ
C E (R in + h ie )
ωc =
fc =
201
1
= 337
100µF(2.38kΩ + 3.577 kΩ)
s
ωc
= 54Hz
2πf
The calculated cut-off frequency is with 54Hz very close at the measured value of 58 Hz.
The exact calculation of the cut-off frequency is very complex and is done in most cases with
a computer and a simulation software like (P) or (H)spice.
On the following pages (P)spice printouts of the simulated CE-Amplifier can be found.
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figure 8
figure 8 shows the BIAS calculated by a (P)spice simulation.
14
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Spice printout of frequency response AV
15
c) Frequency response of Ai
f/Hz
vrs/mV
vo/mV
v rs
µA
Rs
10
4.5
197
1.38
20
6.2
351
1.90
50
8.0
600
2.45
70
8.3
668
2.54
100
8.7
720
2.67
120
8.9
738
2.73
200
9.1
771
2.79
500
9.2
792
2.82
700
9.2
795
2.82
1000
9.3
797
2.85
vo
µA
RL
Ai
29.31
52.23
89.29 99.4
107
110
115
118
118
119
21
27.5
36
40
40,5
41
42
42
42
39
Current Amplification vs frequency
Ai
100
− 3dB =
1
A i max
2
10
10
100
fc=23Hz
figure 9
The cut-off frequency of the current amplification is read out from the diagram 23Hz.
This frequency is above all determined by the capacitor CE.
16
1000
f
Spice printout of the frequency response of the current amplification Ai vs frequency
17
6. Disconnection of bypass capacitor C3
Next step of the Lab was to disconnect the Emitter bypass capacitor to proof the importance
of the bypass capacitor especially on the voltage and current amplification, the main
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figure 10
Figure 10 shows the schematics of the CE-Amplifier with disconnected C3.
7. DC-Bias with disconnected C3
a) Calculation of DC-Bias
The calculated DC-Bias with disconnected capacitor C3 can’t change, because C3 has no
influence to any DC-values. C3 is not used for calculation of the DC-Bias and so it’s not able
to change it.
û
see DC-Bias values at DC-bias precalculation with given resistors
b) Measurement of DC-Voltages (DC-bias)
The same DC-bias measurements were to make with disconnected C3 like in the first part of
the Lab.
VR1=1.479V
VR2=8.47V
VRE=0.849V
VRC=3.27V
VC=6.65
VBE=0.625
VCE=5.81
18
c) Calculation of IB, IC and β with measured values( C3 disconnected)
i. IB with IE and IC (RE and RC)
IE =
VRE 0.849V
=
= 0.861mA
RE
986Ω
IC =
VRC
3.27 V
=
= 0.856mA
R C 3.82kΩ
IB = IE − IC
I B = 0.861mA − 0.856mA
I B = 5µA
β = h fe =
I C 0.856mA
=
= 171.2
IB
5µA
ii. IB with I1 and I2 (R1 and R2)
I R1 =
VR1 1.479V
=
= 0.14805mA
R 1 9.99kΩ
IR2 =
VR 2
8.47V
=
= 0.15206mA
R2
55.7kΩ
I B = I R 2 − I R1 = 0.15206mA − 0.14805mA = 4.01µΑ
β=
I C 0.856mA
=
= 213
IB
4.01µA
The different values of β=hfe can be led back to the uncertainty of measurement, because the
measured values were very small and at the lower range of the measuring instrument.
19
8. AC and small signal characteristics (without C3)
Next part of the lab was to determine the small signal characteristics of the common-emitter
amplifier.
By computer simulation more complex models can be used.
a) BJT small signal behavioural model
With the disconnected Capacitor C3 another model of the circuit has to be used now, because
the Emitter Resistor is now not short circuited at mid-frequencies and generates an important
current feedback of the AC-signals through IE, which reduces the amplification of the CEAmplifier considerably.
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figure 11
All the definitions can be taken from Definitions of the small signal parameters
b) Measured small signal values without C3
The input voltages are increased by a factor 10 to obtain better measurands at the low
amplifications without bypass capacitor.
(A generator with f=1kHz works as voltage source V1)
vs=200mV
voc=533mV
vo=340mV
vi=143mV
vrs=57.3mV
20
c) Calculation and derivation of the small signal parameters (mid freq., without C3)
ALL THEORETICAL VALUES WITH INDEX ‘ (i.e. R’)
i. Amplifier input resistance Ri
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+
,
-
9
(
0
1
2
3
4
5
6
)
3
.
)
8
+
8
+
;
<
=
7
#
:
%
>
+
.
/
8
7
/
*
8
;
7
+
,
@
7
<
A
7
%
?
(
8
B
%
6
&
'
8
B
D
-
E
C
figure 12
The
Figure 12 shows, that the Emitter resistor is divided into two parts for separating input and
output circuit.
The connection between both is given through the voltage drop at the emitter resistor RE.
Measured Value of R in :
Theoretical value of R in :
R 'in = R 2 R 1 (h ie + R EIn ) where R EIn =
R 'in = R 2 R 1 [h ie + (h fe + 1)R E ]
v RE (h fe + 1)R E i B
=
iB
iB
vi
vi
=
i i v rs
Rs
143mV
57.3mV
3.26kΩ
R in = 8.14kΩ
R in =
R 'in ≈ R 2 R 1 (h ie + h fe R E )
R 'in ≈
R in =
1
1
1
1
+
+
56kΩ 10kΩ 3.577kΩ + 200 × 1kΩ
R 'in ≈ 8.15kΩ
The calculated value of Rin is very close to the measured value. The approximation of REin can
be done because there’s a negligible difference between hfe and (hfe+1).
21
ii. Amplifier output resistance Ro
I
H
L
R
J
F
G
H
F
K
L
N
G
M
O
P
Q
figure 13
It’s the same small signal model than in the first part. The series connection of the current
source and the emitter resistor can be reduced to the current source only.
Theoretical value of R o :
Measured value of R o :
R 'o = R C
Ro =
with h OE = 0S
R ' o = 3.9kΩ
Ro =
v oc − v o
vo
RL
533mV − 340mV
= 3.81kΩ
340mV
6.72kΩ
The calculated and the measured value are again very close together.
iii. Voltage amplification at mid frequencies without load
Theoretica l value of A vo :
Measured value of A vo :
A ' vo =
− h fe i B R C
v oc
=
vi
i B ( h ie + R Ein )
A vo =
A vo =
− h fe R C
( h ie + R Ein )
A vo =
A vo = −
A vo =
h ie
a vo
h fe R C
h fe R C
≈
+ ( h fe + 1) R E
h ie + h fe R E
v oc
vi
533mV
143mV
= 3.3
200 ⋅ 3 . 9 k Ω
= 3 . 83
3 . 577 k Ω + 200 ⋅ 1k Ω
The Voltage amplification is very low, because the emitter resistor without bypass capacitor
causes a current feedback which results in a low Amplification at all frequencies (see
frequency response).
22
iv. Voltage amplification at mid frequencies with load
Theoretical value of A vo :
A' v =
A' v =
Measured value of A vo :
v o − h fe i B (R C R L )
=
vi
i B (h ie + R Ein )
A vo =
− h fe (R C R L )
A' v = −
h ie + (h fe + 1)R E
340mV
143mV
= 2.38
A vo =
(h ie + R Ein )
h fe (R C R L )
v oc
vi
≈
A vo
h fe (R C R L )
h ie + h fe R E
3.9kΩ ⋅ 6.8kΩ
3.9kΩ + 6.8kΩ = 2.45
A' v =
3.577 kΩ + 200 ⋅ 1kΩ
200 ⋅
The amplification without bypass capacitor is again very low, but the theoretical value and the
measured value are very close together.
v. Current amplification Ai
Theoretical value of A i :
A'i =
iC
iI
A'i =
h fe i B
iI
A'i = h fe
with i B =
R1 R 2
h ie + (h fe + 1)R E + R 1 R 2
R1 R 2
h ie + R E (h fe + 1) + R 1 R 2
A'i = 200
= h fe
R1 R 2
h fe 26mV
8.485kΩ
3.58kΩ + 1kΩ(201) + 8.485kΩ
A'i ≈ 8
Measured value of A i :
v oc
iC
Rc
Ai =
=
v rs
iI
Rs
533mV
3.82kΩ
57.3mV
3.26kΩ
A i = 7.93
Ai =
iI
23
IE
+ R E (h fe + 1) + R 1 R 2
vi. Amplifier power gain Ap
Theeoretic values of A p :
A' p = A i ⋅ A v
A' p = h fe
R1 R 2
h ie + R E (β + 1) + R 1 R 2
A' p = 200
×
RC RL
h ie
h fe
+ RE
8.48kΩ
2.479kΩ
×
3.577 kΩ + 1kΩ ⋅ 201 + 8.48kΩ 3.577 kΩ
+ 1kΩ
200
A' p = 19
Measured value of A p :
A' p = A i ⋅ A v
A' p = 7.93 ⋅ 2.38
A' p = 19
d) Comparison between measured and calculated values
Measurand
measured
calculated
hfe
171/
213
200
Rin
8.14kΩ
Rout
3.81kΩ
Avo
3.3
Av
2.38
Ai Ap
8 19
8.15kΩ
3.9kΩ
3.83
2.45
8
24
19
9. Frequency response of the CE-amplifier without C3
a) Frequency response of Av
f/Hz
vi/mV
vo/mV
vo/vi
10
134
305
2.3
20
131.8
309
2.3
50
131.4
311
2.4
70
131.2
310
2.3
100
131.3
311
2.3
200
130.9
310
2.3
500
130.9
310
2.3
700
730.9
310
2.3
1000
130.9
310
2.3
Voltage Amplification vs frequency
Av
10
1
10
100
1000
f
figure 14
fc is not
readable
and so the cut-off frequency is much lower than 10Hz. The current amplification is linear over
the low and mid frequency band.
25
(P)spice printout of the Voltage Amplification without bypass capacitor
26
b) Theoretical calculation of the lower cut-off frequency
At this circuit only the coupling capacitors have an influence to the lower cut -off frequency
because the bypass capacitor is removed.
S
S
S
\
T
\
^
k
Y
]
^
S
X
V
U
_
W
`
a
^
_
`
l
[
\
d
e
f
g
h
i
f
\
a
k
^
k
k
q
p
j
j
b
]
k
c
^
^
n
o
V
m
n
j
^
s
_
j
o
X
t
j
U
X
b
X
r
[
k
u
X
^
Y
Z
j
figure 15
figure 15 shows the small signal equivalent circuit for the CE-Ampflifier with disconnected
bypass capacitor.
S
S
S
\
T
\
^
S
^
k
X
Y
V
U
_
]
W
`
^
a
_
`
l
[
\
d
e
f
g
h
i
f
a
\
k
^
k
k
q
p
j
j
c
b
]
k
^
U
X
^
n
o
V
m
j
^
_
s
j
o
t
j
X
r
b
X
n
[
v
w
x
y
v
w
z
{
|
X
Y
Z
figure 16
In figure 16 the emitter resistor is separated into its input and output circuit part.
For the input circuit:
ii =
iB =
iB =
vi
R in +
1
jω C
R1 R 2
R 1 R 2 + h ie + R Ein
ii =
RB
R B + h ie + R Ein
vi
R in +
1
jω C 1
1
v i where R in = R B (h ie + R Ein )
1
(h ie + R Ein )1 +
jωR in C1
27
For the output circuit:
io =
RC
1
RC + RL +
jω C 2
v o = −R L i o = −
vo =
ic =
RC
1
RC + RL +
jω C 2
R L R C h fe
RL + RC +
1
jω C 2
h fe i B
i b where R C R L =
R CR L
RC + RL
(R C R L )h fe
iB
1
1+
jω(R L + R C )C 2
Combining the I/P and O/P circuit
Av =
vo
=−
vi
(R C R L )h fe
h fe (R C R L )
1
where A vmid = −
1
1
h ie + (1 + h fe )R E
1+
1+
(h ie + R Ein )
jω(R L + R C )C 2
jωR In C1
τ1 = R In C1 = [R B (h ie + (1 + h fe )R E )]C1
τ 2 = ( R C + R L )C 2
A V (s) =
vo
τ1 τ 2 s 2
= A vmid
vi
(1 + τ1s)(1 + τ 2 s)
Using the calculated values:
τ1 = (8.48kΩ 8.14kΩ)10µF
τ1 = 0.081s
ω1 =
ω
1
= 12.28 ⇒ f c1 = 1 = 1.95Hz
2π
τ1
τ 2 = (3,9kΩ + 6.8kΩ)10µF
τ 2 = 0.107s
ω2 =
ω
1
= 9.34Hz ⇒ f c1 = 2 = 1.48Hz
τ2
2π
ω1 ≈ ω 2
⇒
A(s)
s2
ω2
=
=
A mid
(s + ω c ) 2
ω 2 + ωc2
ω L = 1.55ωC
ωc ≈ 2Hz
This low frequency was not measurable in the lab, because the generator and the measuring
instruments were not able to determine such a low frequency correctly.
The amplification of the CE-Amplifier without bypass capacitor reaches nearly into the DCregion.
28
c) Frequency response of Ai without C3
f/Hz
vrs/mV
vo/mV
v rs
µA
Rs
10
56
333
17.17
20
57
337
17.5
50
57
339
17.5
70
57
339
17.5
100
57
340
17.5
200
57
339
17.5
500
57
339
17.5
700
57
339
17.5
1000
57
339
17.5
vo
µA
RL
Ai
49.55
50
50.5
50.5
50.5
50.5
50.5
50.5
50.5
2.88
2.87
2.88
2.88
2.88
2.88
2.88
2.88
2.88
Current Amplification vs frequency
Ai
10
1
10
100
1000
f
figure 17
fc is not readable and so the cut-off frequency is much lower than 10Hz.
The current and voltage amplification of the CE-amplifier is very low, but constant during the
lower frequencies, therefor the CE-Amplifier without bypass capacitor is not often used.
For low frequency amplification usually direct coupled amplifiers with bypass capacitors are
used. If the circuit is well designed the lower cut-off frequency is mostly determined by the
emitter capacitor and NOT by the coupling capacitors.
29
(P)spice printout of the current amplification vs frequency
30
