Inverting AmplifierUnder stable linear operation– AOL= ∞, Rin= ∞– Vo= AOL (Vin(+) – Vin(-))

– Vid = (Vin(+) – Vin(-)) = Vo/AOL = 0 V

– I1 = Vin/R1

– IB(+) = IB(-) = 0

– IF = -I1

– Vo= IFRF = -I1RF = -VinRF/R1

– Closed loop voltage gain of circuit ACL = Vo/Vin = -(Rf/Ri)

Vo

-

+

RF

R1

Vin

+ -IFI1

Vid

IB(+)

IB(-)

Virtual ground

Summing Amplifier• Circuit yields the

weighted sum of different input voltages

• Each input voltage is connected to the negative input terminal of the op-amp by an individual resistor

-

+

RF

R4

+

IFI4

Vo

R3

+I3

V3

V4

R2+

I2

V2

R1

+I1

V1

Summing Amplifier

• KCL current law: IF = I1 + I2 + I3 + I4

• I1 = V1/R1, I2 = V2/R2, I3 = V3/R3, I4 = V4/R4

• IF = -V0/RF

• -V0/RF = V1/R1+ V2/R2 + V3/R3 + V4/R4

• V0 = -RF (V1/R1+ V2/R2 + V3/R3 + V4/R4)

Noninverting AmplifierUnder stable linear operation– AOL= ∞, Rin= ∞; iin =0 and i2=0,

– Vid = 0; Vin = Vf

– Vf = Voβ = Vo [R1/(R1+RF)]

– Vin = Vo [R1/(R1+RF)] – Closed loop voltage gain of

circuit ACL = Vo/Vin = (R1+RF)/R1

Vo

-

+

RF

Vf

R1

Vin

Vid

i2

iin

Differential Amplifier• Op-amp should amplify V1 and V2

equally• Above is possible if R1 = R2 and R1F

= R2F• V3 = V2 {R2F/(R2 + R2F)}• I = (V1 – V3)/R1

= (V3 – V0)/R1F

• (V3 – V0) = (V1 – V3) R1F/R1

• V0 = V3 (1+ R1F/R1) - V1 R1F/R1

= V2 {R2F/(R2 + R2F)} {(R1+R1F)/R1} – (V1 R1F/R1)

=V2 (R1F/R1) – (V1 R1F/R1)• V0 = (V2 – V1) (R1F/R1)

Vo

-

+

R1FR1 I

V1

V2

R2

R2F

V3

I

Comparators• Circuit that compares

the input voltage with a reference voltage

• If (Vin + Vref) > 0 –V0 = -13 V

• Else V0 = +13 V

Vo

-

+

R2

R1

Vin

Vref

R1

Rectifiers• Full-wave rectifier circuit

using one op-amp• If Vin < 0, circuit behaves like

an inverting amplifier rectifier with gain = 0.5

• If Vin > 0, the op-amp disconnects and passive resistor chain yields a gain = 0.5

Vo

-

+

RF = 1KR1 =2K

Vin

RL = 3K

Effect of Negative Feedback on Output Resistance

• RoF = Ro/(1+βAOL)• For inverting amplifier–β = R1/RF

• For noninverting amplifier–β = R1/(R1+RF)

Typical Bode Plots

elexp.com, people.seas.harvard.edu/

Bandwidth Limitations• fc (corner frequency or break frequency or critical

frequency): frequency at which the gain of the op-amp deviates from the passband gain

• fc frequency at which gain of op-amp has dropped 3 dB from passband gain

• Midband: range of frequencies from 0 to fc • Bandwidth: range of frequencies for which gain of

op-amp is within 3 dB of maximum• Unity gain: 0 dB gain (numerical gain is 1)

Bandwidth Limitations

• funity gain = frequency at which AOL = 1• Gain-Bandwidth Product (GBW) =

numerical closed loop gain value X frequency

• For both inverting and noninverting amplifiers– fc = funity / {1+(RF/R1)}

Cascaded Amplifiers

• Total gain = ACL1 ACL2

• Overall bandwidth (BWT) = BWs √(21/n-1)– Where n is the number of stages cascaded

-

+

14.14R

R+

-

+

14.14R

R

Vin

Vo

A1A2