Summing Amplifier Revisited
• Superposition can be used to easily evaluate summing amplfiers with infinite
open loop gain
R4
V1
V2
V3
R1
R2
R3
Lecture 5-1
Summing Amplifier Revisited
• Can we solve for the summing amplifier gain the same way when the open
loop gain is finite?
R4
V1
V2
V3
R1
R2
R3
Lecture 5-2
Other Opamp Nonidealities
• The open loop gain is generally so large that we can neglect its impact on the
closed-loop circuit gain
• But a significant nonideality that we cannot ignore is the finite open-loop
gain as a function of frequency
• Parasitic capacitance causes the gain to fall off at high frequency (can’t make
ωH infinite for any amplifier!)
• The gain may be designed to change with frequency to enure stability ----
compensation
Lecture 5-3
741 Opamp
• One of the most popular, general purpose opamps are the 741-type
VC1
15V
+-
741OPAMP0
+
VMIN
+
0.000 pV
-
200dB
e-1
e0
e1
e2
e3
+
VIN
SIN
-
+
e4
e5
e6
VC2
-15V
VMOUT
+
3.836 V
-
e7
e-1 e0
e1
e2
e3
e4
e5
frequency
e6 e7
0º
100 dB
-100º
0dB
-100dB
DB(VMOUT/VMIN)
-200º
frequency
PH(VMOUT/VMIN)
Lecture 5-4
741 Opamp Example
• A gain of 10, or 20dB is possible at 10kHz:
R16
10E3Ω
VC8
-15V
+-
R15
1E3Ω
+
VMIN
+
0.000 pV
-
0
V
741OPAMP12
-
VIN
SIN
+
+
2
4
VC9
15V
6
VMOUT
+
1.009 mV
-
8
time
10 e-4s
1
0
-1
VMOUT
VMIN
Lecture 5-5
741 Opamp Example
• But not at 100kHz:
R16
10E3 Ω
VC8
-15V
+-
R15
1E3Ω
VMIN
+
0.000 pV
-
0.0
0.2
+
741OPAMP12
-
VIN
SIN
+
+
0.4
VC9
15V
0.6
VMOUT
+
1.009 mV
-
0.8
time
1.0e-4s
1
V
0
-1
VMOUT
VMIN
Lecture 5-6
741 Opamp Example
• This decrease in gain is evident from a frequency domain analysis
R16
10E3Ω
VC8
-15V
R15
1E3Ω
+
VMIN
+
0.000 pV
-
e-1 e0
e1
e2
e3
VIN
SIN
+-
741OPAMP12
+
+
e4
e5
VC9
15V
frequency
e6 e7
VMOUT
+
1.009 mV
-
e-1 e0
e1
e2
e3
e4
e5
frequency
e6 e7
200º
20
10
0
100º
-10
-20
-30
0º
-40
DB(VMOUT/VMIN)
PH(VMOUT/VMIN)
Lecture 5-7
Gain as a Function of Frequency
• Will the closed-loop gain (with feedback) will always be less than or equal to
the open loop gain as a function of frequency?
200dB
open
loop
gain
e-1
e0
e1
e2
e3
e4
e5
e6
e7
100 dB
0dB
-100dB
DB(VMOUT/VMIN)
frequency
Lecture 5-8
Gain as a Function of Frequency
• We can assume that the close-loop response will follow the open loop
characteristic beyond the frequency at which they intersect if the load
capacitance and input terminal capacitances are small
• Why are the other capacitors a factor?
741 Open Loop Characteristics
200dB
e-1
e0
e1
e2
e3
e4
e5
e6
e7
e-1 e0
e1
e2
e3
e4
e5
frequency
e6 e7
0º
100 dB
-100º
0dB
-100dB
DB(VMOUT/VMIN)
-200º
frequency
PH(VMOUT/VMIN)
Lecture 5-9
Gain as a Function of Frequency
• It is important that the open loop characteristic has a phase shift of less than
180º (a change in gain of less than 40dB per decade) at the point of
intersection with the closed-loop characteristic --- why?
741 Open Loop Characteristics
200dB
e-1
e0
e1
e2
e3
e4
e5
e6
e7
e-1 e0
e1
e2
e3
e4
e5
frequency
e6 e7
0º
100 dB
-100º
0dB
-100dB
DB(VMOUT/VMIN)
-200º
frequency
PH(VMOUT/VMIN)
Lecture 5-10
Stability
• If the open loop gain has a phase shift of 180º, and we connect the opamp
with negative feedback (which represents another phase shift of 180º), then
the total phase shift will be 360º which is like positive feedback
R2
R1
vin
v o ( s ) = – A ( s )v 1
Lecture 5-11
Feedback
• The feedback can further complicate matters if it is also a function of
frequency
input
VS
∑_
Vi
+
A(ω )
Vf
β(ω)
Vo
load
A(ω )
A f ( ω ) = ----------------------------------1 + β ( ω )A ( ω )
Lecture 5-12
Stability
• The circuit will be unstable with this positive feedback
• What causes the gain characteristic to exhibit these unwanted phase shifts?
gain
(dB)
Stable
Marginally Stable
closed-loop gain
open-loop gain
Unstable
(phase becomes 180 degrees
at some frequency in this range)
f (Hz)
Lecture 5-13
Internal Compensation
• Compensation capacitors are added to purposely place a low frequency
(dominant) pole so that the change in gain will be 20dB/decade up until the
unity gain frequency
• Why isn’t the phase a concern beyond this frequency?
gain
(dB)
20dB/decade
0
f (Hz)
• Sacrifices gain for stability
Lecture 5-14
Feedback
• Most feedback we’ve considered has no phase shift (resistors), so
compensation will ensure stability
• However, frequency dependent feedback can pose a problem even for
compensated opamps
input
VS
∑_
V
+i
A(ω )
Vo
load
Vf
β( ω)
A(ω)
Af ( ω ) = ----------------------------------1 + β ( ω )A ( ω )
Lecture 5-15
Modeling Gain as a Function of Frequency
• With compensation, the frequency response appears like a STC
741 Open Loop Characteristics
200dB
e-1
e0
e1
e2
e3
e4
e5
e6
e7
Ao
A ( s ) = ---------------------1 + s ⁄ ωb
100 dB
0dB
-100dB
DB(VMOUT/VMIN)
frequency
Lecture 5-16
Unity Gain Bandwidth
• We can easily solve for the frequency, ωt, at which the gain is 0dB
741 Open Loop Characteristics
200dB
e-1
e0
e1
e2
e3
e4
e5
e6
e7
100 dB
0dB
-100dB
DB(VMOUT/VMIN)
frequency
Lecture 5-17
Frequency Dependence of Gain
• Using these expressions for gain as a function of frequency we can
approximate the inverting amplifier circuit gain as a function of frequency
R2
R1
vin
Ao
A ( s ) = ---------------------1 + s ⁄ ωb
R2
– -----Vo
R1
------ = ------------------------------Vi
R 2

 1 + ------
R 1

1 + --------------------A( s)
R2
– ------ A o
R1
Vo
------ = ---------------------------------------------------------------------Vi
R 2
R 2

s 
Ao +  1 + ------ + -------  1 + ------
R 1 ω b 
R 1

Lecture 5-18
Frequency Dependence of Gain
• We can assume that the closed-loop gain is much smaller than the dc open
loop gain
R2
– ------ Ao
Vo
R1
------ = ---------------------------------------------------------------------Vi
R 2
R 2

s 
A o +  1 + ------ + -------  1 + ------
R 1 ω b 
R 1

Lecture 5-19
Frequency Dependence of Gain
• Does this result seem correct?
Lecture 5-20
Opamp Macromodels
• We can model the frequency dependence of the gain using a macromodel for
the opamp
• Internal structure of an opamp:
C (compensation capacitor)
_
vid
+
1
Gm
–µ
+1
Differential
Input Transconductance
Amplifier
Voltage
Amplifier
Unity-Gain
Buffer
2
+
vo
_
• We’ll build all of these transistor level components later in the course
Lecture 5-21
Opamp Macromodels
• We can model the first two stages (the last is trivial) using controlled sources,
R’s and C’s:
C
_
vid
+
+
1
2
Gmvid
Ro1
+
– µv i2 vo
v i2
Ri2
_
_
• Which we can simplify by combining the parallel R’s
C
_
1
Gmvid
vid
R
+
2
+
+
v i2
vo
_
– µv i2
_
Lecture 5-22
Opamp Macromodel
• This circuit has the same STC form as a compensated opamp
C
_
1
Gmvid
vid
R
+
2
+
+
v i2
vo
_
– µv i2
_
Lecture 5-23
Opamp Macromodel
1
_
vid
+
2
+
Gmvid
R
v i2
C(1+µ)
_
Lecture 5-24
Opamp Macromodel
• The parameters for a 741 macromodel are:
mA
G m = 0.19 ---------volt
µ = 529
R i2 = 4MΩ
R o1 = 6.7MΩ
741 Open Loop Characteristics
200dB
e-1
e0
e1
e2
e3
e4
e5
e6
Macromodel
e7
200dB
100 dB
100 dB
0dB
0dB
-100dB
DB(VMOUT/VMIN)
C = 30pF
frequency
e-1 e0
e1
e2
e3
-100dB
DB(VMOUT/VMIN)
e4
e5
e6
e7
frequency
Lecture 5-25