Operational Amplifiers: Theory and Design

Lesson 1 of 5

Module 1: Ideal vs. Practical Op-Amps

Moving beyond the ideal model to understand real-world limitations essential for precision design.

Core90 min

Ideal vs. Practical Models

Ideal Op-Amp Assumptions

1. Infinite Open-Loop Gain: $A_{OL} = \infty$ 2. Infinite Input Impedance: $R_{in} = \infty$ (no current flows into inputs) 3. Zero Output Impedance: $R_{out} = 0$ (can drive any load) 4. Infinite Bandwidth: Works at all frequencies 5. Zero Offset Voltage: $V_{os} = 0$

The Real World Error Sources

Engineering design requires accounting for non-ideal parameters:

1. Finite Open-Loop Gain ( $A_{OL}$ )

Typical values: 10,000 to 1,000,000 (80-120 dB)

The actual closed-loop gain ( $A_{CL}$ ) differs from the ideal gain ( $A_{ideal}$ ): $A_{CL} = \frac{A_{ideal}}{1 + \frac{1}{A_{OL}\beta}}$

Where $\beta$ is the feedback factor. For high precision, $A_{OL}\beta$ (Loop Gain) must be very large (>1000).

Example: For a non-inverting amplifier with ideal gain of 10:

If $A_{OL} = 100,000$ : Actual gain = 9.999
If $A_{OL} = 1,000$ : Actual gain = 9.90 (1% error)

2. Input Offset Voltage ( $V_{os}$ )

Typical values: 0.1mV to 10mV

Real op-amps have a small DC voltage between inputs even when grounded, causing output DC error. $V_{out(error)} = V_{os} \times (1 + \frac{R_f}{R_{in}})$

Mitigation: Use offset nulling pins or AC coupling.

3. Input Bias Current ( $I_B$ ) and Offset Current ( $I_{os}$ )

BJT inputs: 10nA to 500nA JFET/CMOS inputs: 1pA to 100pA

Base/Gate currents required to bias input transistors flow through external resistors, creating voltage errors.

Error Voltage: $V_{error} = I_B \times R_{equivalent}$

Mitigation: Use a compensation resistor $R_{comp} = R_{in} || R_f$ on the non-inverting terminal to balance voltage drops.

4. Finite Input Impedance

BJT: 1MΩ - 10MΩ JFET: 10^{12}Ω

5. Non-Zero Output Impedance

Typical: 50Ω - 500Ω Affects load driving capability and accuracy.

Activity: Datasheet Comparison

20 min · read

Op-Amp Datasheet Comparison

Parameter	LM741 (BJT)	TL081 (JFET)	OP07 (Precision)	LM358 (Low Power)
$V_{os}$	1-6 mV	3-15 mV	25-150 μV	2-7 mV
$I_B$	80 nA	30 pA	2 nA	45 nA
$A_{OL}$	200,000	200,000	400,000	100,000
GBP	1 MHz	3 MHz	0.6 MHz	1 MHz
SR	0.5 V/μs	13 V/μs	0.3 V/μs	0.6 V/μs
Supply	±15V	±15V	±15V	+3V to +32V
$R_{in}$	2 MΩ	10^{12} Ω	4 MΩ	10 MΩ

Key Observations:

1. Precision apps (strain gauges, thermocouples): Use OP07 for low $V_{os}$ 2. High-impedance sources (pH probes, piezo sensors): Use TL081 for low $I_B$ 3. General purpose: LM741 (educational standard, discontinued for new designs) 4. Single supply/battery: LM358 rail-to-rail operation

Application Selection Guide:

Audio preamplifiers → Low noise (TL07x, NE5532)
Instrumentation → Low offset (OP07, AD620)
High speed → Fast SR and GBP (LM318, THS4001)
Low power → Micropower (LPV521, MCP6001)

Derivation: Finite Gain Error

25 min · read

Derive the exact gain expression for a non-inverting amplifier assuming finite $A_{OL}$ .

Derivation Steps

Given Circuit: Non-inverting amplifier with feedback resistors $R_1$ and $R_f$ .

Step 1: Write voltage at inverting terminal $V_- = V_{out}\frac{R_1}{R_1 + R_f}$

Step 2: Write fundamental op-amp equation $V_{out} = A_{OL}(V_+ - V_-)$ $V_{out} = A_{OL}\left(V_{in} - V_{out}\frac{R_1}{R_1 + R_f}\right)$

Step 3: Expand and collect $V_{out}$ terms $V_{out} = A_{OL}V_{in} - A_{OL}V_{out}\frac{R_1}{R_1 + R_f}$ $V_{out}\left(1 + A_{OL}\frac{R_1}{R_1 + R_f}\right) = A_{OL}V_{in}$

Step 4: Solve for gain $A_{CL} = \frac{V_{out}}{V_{in}} = \frac{A_{OL}}{1 + A_{OL}\frac{R_1}{R_1 + R_f}}$

Step 5: Identify feedback factor $\beta = \frac{R_1}{R_1 + R_f}$

$A_{CL} = \frac{A_{OL}}{1 + A_{OL}\beta}$

Step 6: Take limit as $A_{OL} \to \infty$ $A_{CL} = \frac{A_{OL}}{1 + A_{OL}\beta} \approx \frac{A_{OL}}{A_{OL}\beta} = \frac{1}{\beta} = \frac{R_1 + R_f}{R_1} = 1 + \frac{R_f}{R_1}$

This is the ideal gain formula! ✓

Error Analysis: Percent error from ideal: $\epsilon = \frac{A_{ideal} - A_{actual}}{A_{ideal}} \times 100\% = \frac{1}{A_{OL}\beta} \times 100\%$

For 1% accuracy, need $A_{OL}\beta > 100$ .

Next Lesson

Operational Amplifiers: Theory and Design

Lesson 1 of 5

Module 1: Ideal vs. Practical Op-Amps

Moving beyond the ideal model to understand real-world limitations essential for precision design.

Core90 min

Ideal vs. Practical Models

Ideal Op-Amp Assumptions

The Real World Error Sources

Engineering design requires accounting for non-ideal parameters:

1. Finite Open-Loop Gain ( $A_{OL}$ )

Typical values: 10,000 to 1,000,000 (80-120 dB)

The actual closed-loop gain ( $A_{CL}$ ) differs from the ideal gain ( $A_{ideal}$ ): $A_{CL} = \frac{A_{ideal}}{1 + \frac{1}{A_{OL}\beta}}$

Where $\beta$ is the feedback factor. For high precision, $A_{OL}\beta$ (Loop Gain) must be very large (>1000).

Example: For a non-inverting amplifier with ideal gain of 10:

If $A_{OL} = 100,000$ : Actual gain = 9.999
If $A_{OL} = 1,000$ : Actual gain = 9.90 (1% error)

2. Input Offset Voltage ( $V_{os}$ )

Typical values: 0.1mV to 10mV

Real op-amps have a small DC voltage between inputs even when grounded, causing output DC error. $V_{out(error)} = V_{os} \times (1 + \frac{R_f}{R_{in}})$

Mitigation: Use offset nulling pins or AC coupling.

3. Input Bias Current ( $I_B$ ) and Offset Current ( $I_{os}$ )

BJT inputs: 10nA to 500nA JFET/CMOS inputs: 1pA to 100pA

Base/Gate currents required to bias input transistors flow through external resistors, creating voltage errors.

Error Voltage: $V_{error} = I_B \times R_{equivalent}$

Mitigation: Use a compensation resistor $R_{comp} = R_{in} || R_f$ on the non-inverting terminal to balance voltage drops.

4. Finite Input Impedance

BJT: 1MΩ - 10MΩ JFET: 10^{12}Ω

5. Non-Zero Output Impedance

Typical: 50Ω - 500Ω Affects load driving capability and accuracy.

Activity: Datasheet Comparison

20 min · read

Op-Amp Datasheet Comparison

Parameter	LM741 (BJT)	TL081 (JFET)	OP07 (Precision)	LM358 (Low Power)
$V_{os}$	1-6 mV	3-15 mV	25-150 μV	2-7 mV
$I_B$	80 nA	30 pA	2 nA	45 nA
$A_{OL}$	200,000	200,000	400,000	100,000
GBP	1 MHz	3 MHz	0.6 MHz	1 MHz
SR	0.5 V/μs	13 V/μs	0.3 V/μs	0.6 V/μs
Supply	±15V	±15V	±15V	+3V to +32V
$R_{in}$	2 MΩ	10^{12} Ω	4 MΩ	10 MΩ

Key Observations:

Application Selection Guide:

Audio preamplifiers → Low noise (TL07x, NE5532)
Instrumentation → Low offset (OP07, AD620)
High speed → Fast SR and GBP (LM318, THS4001)
Low power → Micropower (LPV521, MCP6001)

Derivation: Finite Gain Error

25 min · read

Derive the exact gain expression for a non-inverting amplifier assuming finite $A_{OL}$ .

Derivation Steps

Given Circuit: Non-inverting amplifier with feedback resistors $R_1$ and $R_f$ .

Step 1: Write voltage at inverting terminal $V_- = V_{out}\frac{R_1}{R_1 + R_f}$

Step 2: Write fundamental op-amp equation $V_{out} = A_{OL}(V_+ - V_-)$ $V_{out} = A_{OL}\left(V_{in} - V_{out}\frac{R_1}{R_1 + R_f}\right)$

Step 3: Expand and collect $V_{out}$ terms $V_{out} = A_{OL}V_{in} - A_{OL}V_{out}\frac{R_1}{R_1 + R_f}$ $V_{out}\left(1 + A_{OL}\frac{R_1}{R_1 + R_f}\right) = A_{OL}V_{in}$

Step 4: Solve for gain $A_{CL} = \frac{V_{out}}{V_{in}} = \frac{A_{OL}}{1 + A_{OL}\frac{R_1}{R_1 + R_f}}$

Step 5: Identify feedback factor $\beta = \frac{R_1}{R_1 + R_f}$

$A_{CL} = \frac{A_{OL}}{1 + A_{OL}\beta}$

Step 6: Take limit as $A_{OL} \to \infty$ $A_{CL} = \frac{A_{OL}}{1 + A_{OL}\beta} \approx \frac{A_{OL}}{A_{OL}\beta} = \frac{1}{\beta} = \frac{R_1 + R_f}{R_1} = 1 + \frac{R_f}{R_1}$

This is the ideal gain formula! ✓

Error Analysis: Percent error from ideal: $\epsilon = \frac{A_{ideal} - A_{actual}}{A_{ideal}} \times 100\% = \frac{1}{A_{OL}\beta} \times 100\%$

For 1% accuracy, need $A_{OL}\beta > 100$ .

Next Lesson

Ideal vs. Practical Models

Ideal Op-Amp Assumptions

The Real World Error Sources

1. Finite Open-Loop Gain (AOLA_{OL}AOL​)

2. Input Offset Voltage (VosV_{os}Vos​)

3. Input Bias Current (IBI_BIB​) and Offset Current (IosI_{os}Ios​)

4. Finite Input Impedance

5. Non-Zero Output Impedance

Activity: Datasheet Comparison

Op-Amp Datasheet Comparison

Key Observations:

Application Selection Guide:

Derivation: Finite Gain Error

Derivation Steps

Ideal vs. Practical Models

Ideal Op-Amp Assumptions

The Real World Error Sources

1. Finite Open-Loop Gain (AOLA_{OL}AOL​)

2. Input Offset Voltage (VosV_{os}Vos​)

3. Input Bias Current (IBI_BIB​) and Offset Current (IosI_{os}Ios​)

4. Finite Input Impedance

5. Non-Zero Output Impedance

Activity: Datasheet Comparison

Op-Amp Datasheet Comparison

Key Observations:

Application Selection Guide:

Derivation: Finite Gain Error

Derivation Steps

1. Finite Open-Loop Gain ( $A_{OL}$ )

2. Input Offset Voltage ( $V_{os}$ )

3. Input Bias Current ( $I_B$ ) and Offset Current ( $I_{os}$ )

1. Finite Open-Loop Gain ( $A_{OL}$ )

2. Input Offset Voltage ( $V_{os}$ )

3. Input Bias Current ( $I_B$ ) and Offset Current ( $I_{os}$ )