Electrical Systems II

(1)

Electrical Systems II

(2)

Operational Amplifiers p p

Golden Rules of OP Amp

1. The voltage difference between V- and V+ is Zero.

2. The inputs draw no current.

(3)

Operational Amplifiers

e

1

K ( )

p p

e

2

e

o

0 K ( 2 1 )

e  e  e

5 6

K  10 ~ 10

i) for dc signals and ac of less than 10 Hz frequency.

ii) for ac 1 MHz ~ 50MHz iii) Ideal CP amps

K  1

K  

-

- no current flow into the input terminals

- the output voltage is not affected by the load connected to the output

terminal

(4)

Inverting Amplifier

R

2

R i i) 1 i 2 o

e e e e

i    i   

g p

e o

e i i 1

R

1

i 2

e

_



1 2

i) i i

R R

i i) 0 e i e e e o

i i i    

1 2

i i) op 0, , e i e e e o

i i i

R R

  

2

1 2 1

(0 ), 1, 0

o

i o

o i

since e K e K K e

e e R

e e

R R R

 

      

    

iii)

R 2

G   R

1 2 1

R 1

(5)

Non-inverting Amplifier

e i



e o e i e i

i R R

  

g p

( )

e   e i  K  

e i

e 0 R

2

R

1

e R 2 R 1

1 2

o i

e R e

R

 

    

 

i

 R 2   R 1 

1

1 R

G R

 

   

 

(6)

Example of Operational Amplifier Circuit



 C e   0, i 1   i 2 i 3

ex)

i 3

i 2

R 2

i 1

e

^

1

1 i , i e

 R 1 2

( )

c o

dv de

dt c i dt

 

      

 e i

e 0

1

e

R

_

( v c     e  e o e o )

3

o o

e e e

i R R

    

2 2

R R

1 1

1 1 de

e dv de

2 1

1 1

o

o i

de e e

dt CR CR

   

1 2 2

1 1

i c o

o o

e dv de

C e C e

R dt R dt R

    

(7)

Examples of Operational Amplifiers

Laplace Transform : 1 1

( ) ( )

o i

E s s E s

CR CR

 

  

 

 CR 2  CR 1

 

2

1 1 2

( ) 1 1 1

. ( ) 1 1

o i

E s R

T F E s CR R R Cs

s

      

 

1 0

1  t 

1 1 2

2

i ( ) s

 CR

Step input response : 1 1 0

( ) , ( )

0 0

i i

E s e t t

t s

 

    

2 1

( ) R i ( )

E s    E s

T  R C

2 o

( )

e t

t

1 2

2 2

1 2 1 2

( ) ( )

1

1 1 1 1

1 (1/ )

o i

E s E s

R R Cs

R R

R R Cs s R s s R C



 

  1 2     1      ( 2 )     R

²

 ( 1/

²

) 

( ) 2 1 R C t

o

e t R e

R

  



R

1

(8)

Examples of Complex Impedance

Z

₂

(s )

p p p

E E (s)

Z

₁(

s) I(s)

- +

I(s)

E (s) + E

₀

E

_i

(s) E

₀

(s)

-

1 2

( ) ( ) ( ), ( ) ( ) ( )

i o

E s  Z s I s E s  Z s I s

2 1

( ) ( )

o i

E s Z s E s   Z s

1

i

(9)

Examples of Complex Impedance p p p

ex)

C Z s 1 ( )  R 1

1 R

R

1

R

2

i

( )

e s e s ( )

2 2

( ) 1

1 1

Z s R

R C s Cs R

 

 

i ( )

e s e s o ( )

2 2

1 1 2

( ) ( ) 1

o i

E s Z s R

E s   Z s   R R C s

2 

( ) ( ) 1 ( )

I s Cs E s E s

   R

(10)

Other Examples

Differentiator/High Pass Filter Integrator/Low Pass Filter

(11)

Other Examples

Summing Amplifier Instrumentation Amplifier