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MAE140 Linear Circuits 76
Active Circuits: Life gets interesting Active cct elements – operational amplifiers (OP
AMPS) and transistors Devices which can inject power into the cct External power supply – normally comes from connection to
the voltage supply rails Capable of linear operation – amplifiers
and nonlinear operation – typically switches Triodes, pentodes, transistors
MAE140 Linear Circuits 77
Active Cct Elements
Amplifiers – linear & active Signal processors Stymied until 1927 and Harold Black
Negative Feedback Amplifier Control rescues communications Telephone relay stations manageable
against manufacturing variability Linearity
Output signal is proportional to the input signal Note distinction between signals and systems which
transform them Yes! Just like your stereo amplifier
Idea – controlled current and voltage sources
MAE140 Linear Circuits 78
A Brief Aside  Transistors
Bipolar Junction Transistors Semiconductors – doped silicon
ndoping: mobile electrons Si doped with Sb, P or As
pdoping: mobile holes Si doped with B, Ga, In
Two types npn and pnp Heavily doped Collector and Emitter Lightly doped Base and very thin Collector and Emitter thick and dopey
Need to bias the two junctions properly Then the base current modulates a strong C®E current
Amplification iC=biB
B
E
C
B
E
C
n p nE
B
C
p n pE
B
C
MAE140 Linear Circuits 79
Transistors
Common Emitter Amplifier Stage Biasing resistors R1 and R2
Keep transistor junctions biased in amplifying range
Blocking capacitors CB1 and CB2 Keep dc currents out
Feedback capacitor CE Grounds emitter at high
frequencies
B
E
C
+VCC
RL
RE
RC
R2
R1CB1
CB2
CE vin
vout +

+

Small changes in vin Produce large changes in vout
MAE140 Linear Circuits 80
Linear Dependent Sources
Active device models in linear mode Transistor takes an input voltage vi and produces an output
current i0=gvi where g is the gain This is a linear voltagecontrolled current source VCCS
+ 
i1 ri1
CCVS
i1 bi1
CCCS
VCVS
+ v1 µv1
+
 VCCS
v1 gv1
+

r transresistance
g transconductance
b current gain
µ voltage gain
MAE140 Linear Circuits 81
Linear dependent source (contd)
Linear dependent sources are parts of active cct models – they are not separate components But they allow us to extend our cct analysis techniques to
really useful applications This will become more critical as we get into dynamic ccts
Dependent elements change properties according to the values of other cct variables
+ 
i1 ri1=riS=v0iS
+

i1 ri1=v0=0iS=0
+

Source on Source off
MAE140 Linear Circuits 82
Cct Analysis with Dependent Sources
Golden rule – do not lose track of control variables Find iO, vO and PO for the 500W load
Current divider on LHS Current divider on RHS
Ohm s law
Power
iyix iO
iS 48ix
500W vO 300W25W
50W
+

A
Sixi 3 2
=
€
iO = 3 8(−48)ix =−18ix =−12iS
SiOiOv 6000500 −==
2000,72 SiOvOiOp ==
MAE140 Linear Circuits 83
Analysis with dependent sources
Power provided by ICS
Power delivered to load
Power gain
Where did the energy come from? External power supply
iyix iO
iS 48ix
500W vO 300W25W
50W
+

A
2 3 502)2550( SiSiSp ==
€
72000iS 2
€
G = pOpS
= 72000iS
2
50 3iS 2 = 4320
MAE140 Linear Circuits 84
Nodal Analysis with Dependent Source
KCL at node C KCL at node D CCCS element description
Substitute and solve
+_vS1 +_vS2
R1 R2
RB
RP
RE iB
biB
vDvCvB
vA
vO
+

0)()2(2)1(1 =−++−+− DvCvPGCvBGSvCvGSvCvG 0)( =−+− BiDvEGCvDvPG β
)( DvCvPGBi −=
[ ] 0)1()1( )( 221121
=++++−
+=−+++
DEPCP
SSDPCPB vGGvG
vGvGvGvGGGG ββ
MAE140 Linear Circuits 85
T&R, 5th ed, Example 43 p 148
+_ + +
vx µvx
R1
vOvS R2
R3
R4
+
 
iO Find vO in terms of vS What happens as µ®¥?
+ +
vx µvx
R1 vOR2
R3
R4
+
 
iO
1R Sv
vA vB Node A:
Node B:
Solution:
SvGBvGAvGGG 13)321( =−++
AvxvBv µµ −=−=
SvGGG G
AvBvOv ! " #
$ % &
+++
− =−==
3)1(21 1 µ
µ µ
For large gains µ: (1+µ)G3>>G1+G2
SvR R
SvG G
Ov 1 3
3)1( 1 −≈
+
− ≈ #$
% &' (
µ
µ
This is a model of an inverting opamp
MAE140 Linear Circuits 86
Mesh Current Analysis with Dependent Sources
Dual of Nodal Analysis with dependent sources Treat the dependent sources as independent and sort out
during the solution
+_
R1 R2
R3 R4vS vx
vO ++


Rin
iO
gvx
+ 
+_
R1 R2
R3 R4vS
vx vO ++


Rin
iO
gR3vx iA
iB
0)43()323( 3)32321(
=+++−
=−+++
BiRRAiRgRR
vBiRAiRgRRRR S
MAE140 Linear Circuits 87
T&R, 5th ed, Example 45 BJTransistor Needs a supermesh
Current source in two loops without R in parallel
Supermesh = entire outer loop
Supermesh equation
Current source constraint
Solution
+_RB RC
RE
B
C
EVg + 
VCC
iB
iC
iE
biB i1 i2
012 =++− CCBE VRiVRi γ
Biii β=− 21
EB
CC B RR
VV ii
)1(1 ++ −
=−= β
γ
MAE140 Linear Circuits 88
T&R, 5th ed, Example 46 Field Effect Transistor
Since cct is linear Solve via superposition
First vS1 on and vS2 off, then vS1 off and vS2 on This gives K1 and K2
+_ R1 +_R2R3 R4vS1 vS2
vx
gvx
vy
gvy rdsrds
++ 

+ vO
2211 SSO vKvKv +=
MAE140 Linear Circuits 89
Operational Amplifiers  OpAmps
Basic building block of linear analog circuits Package of transistors, capacitors, resistors, diodes in a chip
Five terminals – Positive power supply VCC – Negative power supply  VCC – Noninverting input vp – Inverting input vn – Output vO
Linear region of operation
Ideal behavior
Saturation at VCC/VCC limits range
1
2
3
4
8
7
6
5 + 
VCC
VCC
vn vp vO
vO VCC
VCC
vpvn
Slope A)( npO vvAv −=
85 1010
MAE140 Linear Circuits 90
Real OpAmp (u741)
MAE140 Linear Circuits 91
Ideal OpAmp
Equivalent linear circuit Dependent source model
Need to stay in linear range
Ideal conditions
vO VCC
VCC
vpvn
Slope A
+ 
RI RO
A(vpvn)
+

ip
in
iO vO
vn
vp
+
+
+
106 < RI
MAE140 Linear Circuits 92
Noninverting OpAmp  Feedback What happens now?
Voltage divider feedback
Operating condition vp=vS
Linear noninverting amplifier
Gain K=
+
+_
vp
vS
vO
R2
R1 vn
On vRR Rv
21
2 +
=
SO vR RRv 2
21 +=
2
21 R RR +
vO = RIA(R1 + R2 )+ R2RO
RI (AR2 + RO + R1 + R2 )+ R2 (R1 + RO ) vS
With dependent source model
MAE140 Linear Circuits 93
T&R, 5th ed, Example 413
Analyze this
Ideal OpAmp has zero output resistance RL does not affect vO
+ 
+_
vp
vS
vO
R4
R3 vn
RL
R1
R2 21
2
0
RR R
v v
K
i
S
p S
p
+ ==
=
4
43 AMP R
RR v vK p
O +==
ú û
ù ê ë
é + ú û
ù ê ë
é +
=== 4
43
21
2 AMPTotal R
RR RR
R v vKKK S
O S
MAE140 Linear Circuits 94
Voltage Follower  Buffer
Feedback path
Infinite input resistance
Ideal OpAmp
Loop gain is 1 Power is supplied from the Vcc/Vcc rails
+ 
+_
vp
vS
vO vn
RL
R1 iO in