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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
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