Download Voltage adding circuit

CIRCUITS and
SYSTEMS – part II
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie
Lecture 14
Operational amplifier circuits
Ideal operational amplifier
U 2  AU  U



at
A
•Infinite gain A
•Zero output impedance
•Infinite input impedance
•These features valid for frequency from 0 to infinity.
3
2-port model of ideal op-amp
Hybrid description
 I 1   0 0 U 1 
U    A 0  I 
 2 
 2 
Voltage adding circuit
Voltage adding circuit - equations
Kirchhoff’s equations
G2 U 2  U    G0U 
G1 U 1  U    G0U   G f U   U 0 
U U
After simplification we get
G2
U  
U2
G0  G 2


G2 G0  G1  G f
G1
U0 
U2 
U1

Gf
G0  G 2
Gf
U 0  k1U 1  k 2U 2
Voltage adding circuit - gains
G1
k1  
Gf

G
G2 0  G1  G f
k2 
Gf
G0  G2
G2
k2 
Gf
if
G0  G1  G f  G0  G2
Integrator
Transfer function
U 1  RI
1
U2  
I
sC
U2
1
H (s) 

U1
sRC
Differentiator
Transfer function
1
U1 
I
sC
U 2   RI
U2
H ( s) 
  sRC
U1
Phase shifter
Kirchhoff’s equations
1 

U1   R 
I 2
sC 

U 1  2R f I1  U 2
U 2  R f I1 
1
I2
sC
Phase shifter (cont.)
The currents
U1
R  1 / sC
U U2
I1  1
2R f
I2 
Output voltage
U 2  R f
U1 U 2
U1
1
 s  1 / RC


U1
2R f
sC R  1 / sC
s  1 / RC
Transfer function
U 2  s  1 / RC
H ( s) 

U1
s  1 / RC
Negative impedance converter (NIC)
Kirchhoff’s equations
U1  U 2
R1 I 1  R2 I 2
Chain matrix description
1
0  U
U 1  
R2   2 

 I  0     I 
 1 
R1   2 
Gyrator
Kirchhoff’s equations
I 1  G z U 1  U 1  U 2   G zU 2
I 2  G z U 2  U 2  U 1   G zU 1
Admittance matrix description
 I1   0
 I    G
 2  z
Gz  U1 
0  U 2 
Mason signal flow graph (SFG)
Basic notions:
• Node – the point of graph associated with variable x
• Branch – the directed arch joining 2 nodes
• Gain – the transfer function describing branch
• Loop – the sequence of identically directed branches forming
closed loop
• Gain of the loop – the product of gains of branches of the loop
• Source node – the node from which the branches can only start
• Cascade – the sequence of identically directed branches from the
source node to the output node.
Example of SFG
Set of linear equations
Transformation to Mason form
Mason SFG
a11x1  a12 x2  F1
a21x1  a22 x2  F2
x1  (a11  1) x1  a12 x2  F1
x2  a21x1  (a22  1) x2  F2
Mason gain formula
Transfer function
T
 Tk  k
k

Δ - main determinant of SFG
  1   Gi   GiG j   GiG j Gk  ...
i
Gains of all loops
i, j
i, j ,k
Gains of non-touching loops
Tk – gain of kth cascade from source to output node
Δk - determinant of graph after eliminating kth cascade from SFG
Example
Graph
Transfer function
T
aej (1  l )  bgk (1  h)  adgk (1  h)  adfj(1  l )  bcej (1  l )  bfj(1  l )
1  cd  h  l   cdh  cdl  hl   cdhl
Direct construction of SFG for passive
elements connection
n
Ysk  Y0k   Yi
i 1
Circuit
Its SFG
Direct construction of SFG for op-amp
Op-amp
Its SFG
Vo  AV1  AV2
Example
SFG of the circuit
T
U wy
U we
Y3 Y1
Ys 2 Ys1
Y
YY
A 5 A 4 3
Ys 2
Ys1Ys 2
A

Y32
1
Ys1Ys 2
After simplification at
A   we finally get
 Y1Y3
T 
Y5 Y1  Y2  Y3  Y4   Y3Y4