Modern Control Engineering

Section 3–3 / Mathematical Modeling of Electrical Systems 79

The equation for this circuit can be obtained as follows: Define

Since only a negligible current flows into the amplifier, the current i 1 must be equal to

currenti 2 .Thus

Since and e¿must be almost zero, or Hence we have

or

Thus the circuit shown is an inverting amplifier. If R 1 =R 2 ,then the op-amp circuit

shown acts as a sign inverter.

Noninverting Amplifier. Figure 3–15(a) shows a noninverting amplifier. A circuit

equivalent to this one is shown in Figure 3–15(b). For the circuit of Figure 3–15(b), we

have

whereKis the differential gain of the amplifier. From this last equation, we get

Since if then

This equation gives the output voltage eo.Sinceeoandeihave the same signs, the op-amp

circuit shown in Figure 3–15(a) is noninverting.

eo= a 1 +

R 2

R 1

bei

K1, R 1 AR 1 +R 2 B 1 K,

ei=a

R 1

R 1 +R 2

+

1

K

beo

eo=Kaei-

R 1

R 1 +R 2

eob

eo=-

R 2

R 1

ei

ei

R 1

=

- eo

R 2

K(0-e¿)=e 0 K  1, e¿0.

ei-e¿

R 1

=

e¿-eo

R 2

i 1 =

ei-e¿

R 1

, i 2 =

e¿-eo

R 2

eo

ei

R 2

R 1

+

• eo

ei

R 2

R 1

• 
  

+

(a) (b)

Figure 3–15
(a) Noninverting
operational
amplifier;
(b) equivalent
circuit.