Formula Sheets: Operational Amplifier | Analog and Digital Electronics - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Operational Ampli?ers (Analog
and Digital Electronics) – GATE
1. Basic Concepts
• Operational Ampli?er (Op-Amp): High-gain di?erential ampli?er with high
input impedance and low output impedance.
• Ideal Op-Amp Characteristics:
– In?nite open-loop gain (A?8).
– In?nite input impedance (Z
in
?8).
– Zero output impedance (Z
out
? 0).
– In?nite bandwidth.
– Zero o?set voltage and input bias current.
• Real Op-Amp: Finite gain (A˜ 10
5
-10
6
), ?nite bandwidth, non-zero o?set.
2. Ideal Op-Amp Rules
• Virtual Short: V
+
=V
-
(voltages at inverting and non-inverting inputs are equal
due to high gain).
• No Input Current: I
+
=I
-
= 0 (in?nite input impedance).
3. Op-Amp Con?gurations
3.1 Inverting Ampli?er
• Voltage Gain:
A
v
=-
R
f
R
1
where R
f
: Feedback resistor, R
1
: Input resistor.
• Output Voltage:
V
out
=-
R
f
R
1
V
in
• Input Impedance: Z
in
˜R
1
.
3.2 Non-Inverting Ampli?er
• Voltage Gain:
A
v
= 1+
R
f
R
1
• Output Voltage:
V
out
=
(
1+
R
f
R
1
)
V
in
1
Page 2


Formula Sheet for Operational Ampli?ers (Analog
and Digital Electronics) – GATE
1. Basic Concepts
• Operational Ampli?er (Op-Amp): High-gain di?erential ampli?er with high
input impedance and low output impedance.
• Ideal Op-Amp Characteristics:
– In?nite open-loop gain (A?8).
– In?nite input impedance (Z
in
?8).
– Zero output impedance (Z
out
? 0).
– In?nite bandwidth.
– Zero o?set voltage and input bias current.
• Real Op-Amp: Finite gain (A˜ 10
5
-10
6
), ?nite bandwidth, non-zero o?set.
2. Ideal Op-Amp Rules
• Virtual Short: V
+
=V
-
(voltages at inverting and non-inverting inputs are equal
due to high gain).
• No Input Current: I
+
=I
-
= 0 (in?nite input impedance).
3. Op-Amp Con?gurations
3.1 Inverting Ampli?er
• Voltage Gain:
A
v
=-
R
f
R
1
where R
f
: Feedback resistor, R
1
: Input resistor.
• Output Voltage:
V
out
=-
R
f
R
1
V
in
• Input Impedance: Z
in
˜R
1
.
3.2 Non-Inverting Ampli?er
• Voltage Gain:
A
v
= 1+
R
f
R
1
• Output Voltage:
V
out
=
(
1+
R
f
R
1
)
V
in
1
• Input Impedance: Z
in
˜8.
3.3 Voltage Follower (Bu?er)
• Voltage Gain:
A
v
= 1
• Output Voltage: V
out
=V
in
.
• Characteristics: High input impedance, low output impedance.
4. Summing Ampli?er
• Inverting Summing Ampli?er:
V
out
=-
(
R
f
R
1
V
1
+
R
f
R
2
V
2
+···+
R
f
R
n
V
n
)
• Non-Inverting Summing Ampli?er: Uses additional inverting stage.
5. Di?erence Ampli?er
• Output Voltage:
V
out
=
R
2
R
1
(V
2
-V
1
)
where R
1
/R
2
=R
3
/R
4
.
• Common-Mode Rejection Ratio (CMRR):
CMRR (dB) = 20log
10
(
A
d
A
c
)
where A
d
: Di?erential gain, A
c
: Common-mode gain.
6. Integrator and Di?erentiator
• Inverting Integrator:
V
out
=-
1
RC
?
V
in
dt
• Inverting Di?erentiator:
V
out
=-RC
dV
in
dt
7. Frequency Response
• Open-Loop Gain:
A(f) =
A
0
1+j
f
f
b
where A
0
: DC open-loop gain, f
b
: Break frequency.
2
Page 3


Formula Sheet for Operational Ampli?ers (Analog
and Digital Electronics) – GATE
1. Basic Concepts
• Operational Ampli?er (Op-Amp): High-gain di?erential ampli?er with high
input impedance and low output impedance.
• Ideal Op-Amp Characteristics:
– In?nite open-loop gain (A?8).
– In?nite input impedance (Z
in
?8).
– Zero output impedance (Z
out
? 0).
– In?nite bandwidth.
– Zero o?set voltage and input bias current.
• Real Op-Amp: Finite gain (A˜ 10
5
-10
6
), ?nite bandwidth, non-zero o?set.
2. Ideal Op-Amp Rules
• Virtual Short: V
+
=V
-
(voltages at inverting and non-inverting inputs are equal
due to high gain).
• No Input Current: I
+
=I
-
= 0 (in?nite input impedance).
3. Op-Amp Con?gurations
3.1 Inverting Ampli?er
• Voltage Gain:
A
v
=-
R
f
R
1
where R
f
: Feedback resistor, R
1
: Input resistor.
• Output Voltage:
V
out
=-
R
f
R
1
V
in
• Input Impedance: Z
in
˜R
1
.
3.2 Non-Inverting Ampli?er
• Voltage Gain:
A
v
= 1+
R
f
R
1
• Output Voltage:
V
out
=
(
1+
R
f
R
1
)
V
in
1
• Input Impedance: Z
in
˜8.
3.3 Voltage Follower (Bu?er)
• Voltage Gain:
A
v
= 1
• Output Voltage: V
out
=V
in
.
• Characteristics: High input impedance, low output impedance.
4. Summing Ampli?er
• Inverting Summing Ampli?er:
V
out
=-
(
R
f
R
1
V
1
+
R
f
R
2
V
2
+···+
R
f
R
n
V
n
)
• Non-Inverting Summing Ampli?er: Uses additional inverting stage.
5. Di?erence Ampli?er
• Output Voltage:
V
out
=
R
2
R
1
(V
2
-V
1
)
where R
1
/R
2
=R
3
/R
4
.
• Common-Mode Rejection Ratio (CMRR):
CMRR (dB) = 20log
10
(
A
d
A
c
)
where A
d
: Di?erential gain, A
c
: Common-mode gain.
6. Integrator and Di?erentiator
• Inverting Integrator:
V
out
=-
1
RC
?
V
in
dt
• Inverting Di?erentiator:
V
out
=-RC
dV
in
dt
7. Frequency Response
• Open-Loop Gain:
A(f) =
A
0
1+j
f
f
b
where A
0
: DC open-loop gain, f
b
: Break frequency.
2
• Gain-Bandwidth Product (GBW):
GBW =A
v
·f
3dB
• Closed-Loop Bandwidth:
f
3dB
=
GBW
A
v
8. Op-Amp Parameters
• Slew Rate:
SR =
? V
out
? t
(V/µ s)
• Full-Power Bandwidth:
f
FP
=
SR
2pV
pk
where V
pk
: Peak output voltage.
• Input O?set Voltage: V
OS
˜ 1-10mV (non-ideal).
• Input Bias Current: I
B
˜ 10-100nA (non-ideal).
9. Comparator
• Output:
V
out
=
?
?
?
V
sat+
if V
+
>V
-
V
sat-
if V
+
<V
-
• Applications: Zero-crossing detector, Schmitt trigger.
10. Design Considerations
• Stability: Ensure phase margin > 45
?
to avoid oscillations.
• Power Supply: V
out
=V
CC
-1V (typical rail limit).
• Applications: Ampli?ers, ?lters, oscillators, comparators, ADC/DAC interfaces.
• Feedback: Negative feedback reduces gain, improves linearity and bandwidth.
3