Lecture 13•Review:

•Operational Amplifiers•Operational amplifier examples•Dependent Sources•Related educational materials:

–Chapter 5.1 - 5.4

Operational Amplifiers• Circuit Symbol:

• Assume: ip = 0, in = 0; vin = 0; V - < vout < V +

ip

in

+

-vin

Lecture 12, example 1

inin

fOUT V

R

RV

Op-amp circuit applications• Operational-amplifier based circuits are often used

to implement mathematical operations

• Example:• The circuit of Lecture 12, example 1 performs

multiplication by a negative constant

• Inverting voltage amplifier

ininin

fOUT KVV

R

RV

Saturation• The op-amp output is limited by V+ and V-

• The operational amplifier output is not purely linear• If the output reaches the supply “rails”, the output

“saturates”

• Lecture 12, Example 1:

inOUT KVV Þ

Op-amp circuit – Example 1• Find Vout

R+-V1 +

-VOUT

+-

V2

R

R

+

-

Op-amp circuit – example 2

• Find Vout

Op-amp circuit applications

• Operational-amplifier can be used to isolate sections of a circuit from one another

• The operational amplifier has a very high input resistance

• Example:• The circuit example 2 draws no power from Vin

Example 2 – revisited

• We have a 3k load which requires 6V, but we only have access to a 12V supply.

Dependent Sources

• Operational amplifier based circuits are often modeled as dependent sources

• Review: Four types of dependent sources:• Voltage controlled voltage source (VCVS)• Current controlled voltage source (CCVS)• Voltage controlled current source (VCCS)• Current controlled current source (CCCS)

Dependent Sources – circuit symbols• VCVS:

• VCCS:

• CCVS:

• CCCS:

Op-amp circuit as dependent source -- example

• Inverting voltage amplifier:

inin

fOUT V

R

RV

Analysis of circuits with dependent sources

• Can still use:– KVL– KCL– Ohm’s Law– Nodal analysis– Mesh analysis

Dependent source analysis – example 1

• Find the current, i

Dependent source analysis – example 2

• Find the voltage, v