Technote 6
April 1990
Revised 11/22/02
Op Amp Definitions
Tim J. Sobering
SDE Consulting
sdeconsulting@pobox.com
© 1990 Tim J. Sobering. All rights reserved.
Op Amp Definitions
Page 2
Op Amp Definitions
This Technote summarizes the basic operational amplifier definitions and compares the
ideal Op Amp assumptions with the performance of real amplifiers.
Open Loop Gain:
Av
Va
Av =
Vo
Va
Vo
(1)
Figure 1. Op Amp in open-loop configuration
Closed Loop Gain (inverting):
Avcl (inv ) =
Vo
Avα
=−
Vi
1 + Av β
Zf
(2)
or:
Vi
Va
Zf
1
Avcl (inv ) = −
Zi 1 + 1
Zi
Av
Vo
(3)
Av β
Figure 2. Inverting Op Amp configuration
Inverting Amplifier block diagram:
α
Vi
Σ
Va
Av
Vo
β
Figure 3. Inverting Op Amp block diagram
Closed Loop Gain (non-inverting):
Avcl (non ) =
Vo
Av
=
Vi 1 + Av β
Zf
(4)
or:
Zi
Zf ⎞
⎛
1
⎟⎟
Avcl (non ) = ⎜⎜1 +
(5)
Zi ⎠ 1 + 1
⎝
Av β
Va
Av
Vo
Vi
Figure 4. Non-Inverting Op Amp configuration
Copyright © 1990 Tim J. Sobering. All rights reserved.
Op Amp Definitions
Page 3
Non-inverting Amplifier block diagram:
Vi
Σ
Va
Av
Vo
β
Figure 5. Non-inverting Op Amp block diagram
Loop Gain:
Zf
Avl =
Vo
= − Av β
V1
V1
Zi
(6)
Va
Av
Vo
Figure 6. Definition of loop gain
Feedback Factor:
Zf
β=
Zi
V2
=
V1 Z i + Z f
α = (1 − β ) =
Zf
Zi + Z f
V1
(7)
Zi
(8)
V2
Figure 7. Definition of feedback factor
Derivation of Avcl, Inverting Amplifier
Referring to Figure 2 above and applying KCL at the inverting node:
Vi + Va Vo + Va
+
=0
Zi
Zf
(9)
Vi
Vo
V
Vo
+
+ o +
=0
Z i Av Z i Z f Av Z f
(10)
⎡ 1
1
1
+
+
Vo ⎢
⎢⎣ Av Z i Z f Av Z f
Vo
1
= Avcl (inv ) = −
Vi
Zi
⎤
Vi
⎥=−
Zi
⎥⎦
1
1
1
1
+
+
Av Z i Z f Av Z f
Copyright © 1990 Tim J. Sobering. All rights reserved.
(11)
(12)
Revised 11/19/02
Page 4
Avcl (inv ) = −
Av Z f
(13)
Z i + Z f + Av Z i
⎛ Zf ⎞
⎟
Av ⎜⎜
⎟
+
Z
Z
i
f
⎝
⎠
Avcl (inv ) = −
⎛ Zi
1 + Av ⎜
⎜Z + Z
f
⎝ i
(14)
⎞
⎟
⎟
⎠
Recognizing that
β =
Zf
Zi
and α =
Zi + Z f
Zi + Z f
(15)
Avα
1 + Av β
(16)
Z
α
=− f
β
Zi
(17)
Avcl (inv ) = −
As Av → ∞
Avcl (inv ) = −
Note that Equation (16) can also be expressed as:
Avcl (inv ) = −
Zf
1
Zi 1 + 1
(18)
Av β
Derivation of Avcl, Non-inverting Amplifier
− Vi + Va Vo − Vi + Va
+
=0
Zi
Zf
⎛1
⎛ 1
1 ⎞⎟
1
1
⎜
− Vi ⎜⎜
+
+
+
+
V
o
⎟
⎜
⎝ Zi Z f ⎠
⎝ Av Z i Z f Av Z f
(19)
⎞
⎟=0
⎟
⎠
1
1
+
Zi Z f
Vo = Vi
1
1
1
+
+
Av Z i Z f Av Z f
Avcl (non ) =
Avcl (non ) =
(21)
Av (Z i + Z f )
(22)
Av Z i + Z i + Z f
Av
⎛ Zi
1 + Av ⎜
⎜Z +Z
f
⎝ i
(20)
⎞
⎟
⎟
⎠
Copyright © 1990 Tim J. Sobering. All rights reserved.
(23)
Revised 11/19/02
Page 5
Recognizing that
β =
Zi
Zi + Z f
Avcl (non ) =
(24)
Av
1 + Av β
(25)
As Av → ∞
Avcl (non ) =
1
β
=
Zi + Z f
Zi
=1+
Zf
(26)
Zi
Equation (25) can also be expressed as:
Zf ⎞
⎛
1
⎟⎟
Avcl (non ) = ⎜⎜1 +
Zi ⎠ 1 + 1
⎝
Av β
(27)
Equations (18) and (27) are very useful in examining the gain error as a function of
frequency as the error term appears in the same form in both equations. This shows why
high open loop gain is important and also why it is often necessary to select a an Op Amp
with high open-loop gain and/or a higher then expected bandwidth.
Gain _ Error = 1 −
1
1+ 1
(28)
Av β
Settling time is a related, but separate, parameter. Due to the delays in propagation of a
signal through the internal opamp circuitry, the output takes time to reach it’s final value.
This is a particular concern for step inputs when slew rate, overshoot and ringing may
dominate the output response.
Gain (dB)
OPEN LOOP GAIN
Avβ
fCL = CLOSED-LOOP BANDWIDTH
CLOSED LOOP
GAIN
fCL
LOG f
Figure 8. Open Loop and Closed Loop Magnitude Responses.
Copyright © 1990 Tim J. Sobering. All rights reserved.
Revised 11/19/02
Page 6
Ideal Op Amp assumptions:
Va
Ri
AvVa
Ro
Vo
Figure 9. Op Amp Equivalent Circuit
Ideal Characteristic
No current flows in the inputs
Voltage gain is infinite
No output voltage for zero input
Output impedance is zero
Frequency response is flat and infinite
Slew Rate is infinite
CMRR is infinite
PSRR is infinite
Noise free
Circuit Effect
Ibias(+,−) = 0 or Zin = ∞
Av = ∞, Va = 0
Vos = 0
Zout = 0
BW = ∞ or Delay = 0 and GBW = ∞
BW = ∞ or Delay = 0 and GBW = ∞
Unlimited input voltage range
Immune to PS offset and variations
Output due to signal only
Table 1. Ideal Op Amp Characteristics
Realities:
1. Bias currents cause offsets and drift.
2. Finite open loop gain and finite bandwidth result in gain errors – examine loop gain
term, Avβ, in Equations (16) and (25).
3. Input resistance (impedance) introduces errors – appears in parallel with Zi.
4. Output resistance (impedance) introduces errors – appears as a voltage divider with Zo
and the load.
5. Offset voltage appears at the output multiplied by the noise gain. Offset voltage also
drifts over time due to temperature changes.
6. Bias currents cause offsets and are not matched between the inputs. They also drift
over time due to temperature changes.
7. Power supply mismatch causes offsets – appears as a common mode voltage at the
input.
8. CMRR is not infinite so the input voltage range is limited and nonlinearities appear in
the oputput as a result of common mode voltages, partuicularly in the non-inverting
configuration.
9. Real Op Amps have noise.
10. Op Amps have group delay, i.e. not all frequencies experience the same time delay
through the circuit. This results in distortion.
11. Finite slew rate can result in output distortion.
Copyright © 1990 Tim J. Sobering. All rights reserved.