Operational Amplifiers

Introduction to Operational Amplifiers

What is an Operational Amplifier?

  • An operational amplifier (op-amp) is a high-gain electronic voltage amplifier with a differential input and a single-ended output.

  • Originally used for mathematical operations (e.g., addition, subtraction, integration, differentiation).

  • Modern op-amps are linear integrated circuits (ICs):

    • Operate with low DC supply voltages.

    • Reliable and cost-effective.

  • Applications: signal amplification, filtering, analog computing, and more.

Op-Amp Symbol and Packages

  • Two inputs: Inverting (\(-\)) and Noninverting (\(+\)) and One output.

    Standard op-amp symbol.
    Standard op-amp symbol.

  • Typically powered by dual DC supplies (\(+V_{\text{CC}}\), \(-V_{\text{CC}}\)) or single supply.

  • Common IC packages: dual in-line (DIP), surface-mount technology (SMT).

    Typical op-amp packages.
    Typical op-amp packages.

Historical Context

  • 1947: Concept of op-amps proposed as analog building blocks.

  • Early op-amps used vacuum tubes (high voltages).

  • 1964: First IC op-amp developed by Fairchild Semiconductor designated as 702.

  • Later Developments: Fairchild 709 and 741 (industry standard).

  • Modern op-amps benefit from IC technology: compact, efficient, and versatile.

Ideal vs. Practical Op-Amps

Ideal Op-Amp Characteristics

  • Infinite voltage gain: \(A_{\text{ol}}\to \infty\).

  • Infinite bandwidth: Amplifies all frequencies equally.

  • Infinite input impedance: \(Z_{\text{in}}\to \infty\) (no input current).

  • Zero output impedance: \(Z_{\text{out}}= 0\) (ideal voltage source).

  • Zero noise: No unwanted signals.

Ideal and practical op-amp models.
Ideal and practical op-amp models.

Practical Op-Amp Characteristics

  • Very high voltage gain: \(A_{\text{ol}}\approx 10^5 - 10^6\).

  • Very high input impedance: \(Z_{\text{in}}\approx 10^6 - 10^{12} \Omega\).

  • Very low output impedance: \(Z_{\text{out}}\approx 50 - 100 \Omega\).

  • Limited bandwidth: Gain decreases at high frequencies.

  • Non-zero noise: Internal noise affects signal quality.

  • Voltage and current limitations:

    • Output voltage limited to \(\pm V_{\text{CC}}\) (slightly less).

    • Current limited by power dissipation and component ratings.

Internal Structure of an Op-Amp

  • Composed of three main stages:

    1. Differential Amplifier: Input stage, amplifies difference between inputs.

    2. Voltage Amplifier: Provides additional gain (Class A).

    3. Push-Pull Amplifier: Output stage (Class B) for efficient power delivery.

Block diagram of an op-amp.
Block diagram of an op-amp.

Input Modes and Parameters

Input Signal Modes and Configurations

  • Differential Mode:

    • Single-Ended: One input grounded, signal on the other.

    • Double-Ended: Two out-of-phase signals applied.

    • Output amplifies the difference between inputs.

Inverting input signal.
Inverting input signal.
Noninverting input signal.
Noninverting input signal.
Single-ended differential mode.
Double-ended differential mode.
Double-ended differential mode.

  • Common Mode:

    • Same signal (phase, frequency, amplitude) on both inputs.

    • Ideally, output is zero due to common-mode rejection.

Common-mode operation.
Common-mode operation.

Key Op-Amp Parameters

  • Common-Mode Rejection Ratio (CMRR): Measures ability to reject common-mode signals.

    • \[\text{CMRR}= \dfrac{A_{\text{ol}}}{A_{\text{cm}}}, \quad \text{CMRR}_{\text{dB}} = 20 \log \left( \dfrac{A_{\text{ol}}}{A_{\text{cm}}} \right)\]

  • Open-Loop Voltage Gain (\(A_{\text{ol}}\)): Typically \(10^5 - 10^6\).

  • Maximum Output Voltage Swing (\(V_{\text{O(p-p)}}\)): Limited to \(\pm V_{\text{CC}}\) (slightly less).

  • Input Offset Voltage (\(V_{\text{os}}\)): Small DC voltage needed to zero the output (typically \(< 2 \, \text{mV}\)).

  • Input impedance: Two Basic Ways to Specify

    • Differential Input Impedance:

      • Resistance between inverting and noninverting inputs.

      • Measured by change in bias current vs. differential input voltage.

    • Common-Mode Input Impedance:

      • Resistance between each input and ground.

      • Measured by change in bias current vs. common-mode input voltage.

    Differential and common-mode input-impedance
    Differential and common-mode input-impedance

  • Output impedance: resistance viewed from the output terminal of the op-amp.

    Output-impedance
    Output-impedance

  • Input Bias Current (\(I_{\text{BIAS}}\)):

    • \[I_{\text{BIAS}} = \dfrac{I_1 + I_2}{2}\]

  • Input Offset Current (\(I_{\text{os}}\)):

    Input Bias Current
    Input Bias Current
    Effect of input offset current
    Effect of input offset current

    • The offset voltage developed by the input offset current is

    • \[\begin{aligned} V_{os} & = (I_1-I_2)R_{in} = I_{os}R_{in} \\ V_{\text{OUT(error)}} & = A_v I_{os}R_{in} \end{aligned}\]
    \[I_{\text{os}}= |I_1 - I_2|\]

  • Slew Rate (SR): Maximum rate of change of output voltage in response to a step input measured in V/\(\mu\)s.

    • \[\text{Slew Rate (SR)} = \dfrac{\Delta V_{out}}{\Delta t} = \dfrac{+V_{max}-(-V_{max})}{\Delta t}\]
Slew-rate measurement
Slew-rate measurement

Noise in Op-Amps

  • Noise: Unwanted signals affecting signal quality.

  • Types:

    • 1/f Noise (Pink Noise): Dominant at low frequencies.

    • White Noise: Constant across higher frequencies.

  • Measured as noise voltage density (nV/\(\sqrt{\text{Hz}}\)).

  • Example: At 1 kHz, a low-noise op-amp may have 1.1 nV/\(\sqrt{\text{Hz}}\).

Noise vs. frequency for a typical op-amp.
Noise vs. frequency for a typical op-amp.

Negative Feedback

What is Negative Feedback?

  • Negative Feedback: A portion of the output is fed back to the inverting input, out of phase with the input.

  • Reduces gain but improves:

    • Stability.

    • Bandwidth.

    • Input/output impedance control.

Negative feedback configuration.
Negative feedback configuration.

Why Use Negative Feedback?

  • Without Feedback:

    • High \(A_{\text{ol}}\) (\(>100,000\)) drives op-amp into saturation.

    • Limited to comparator applications.

  • With Feedback:

    • Controlled, stable gain (\(A_{\text{cl}}\)).

    • Increased bandwidth.

    • Adjustable input/output impedances.

Effects of Negative Feedback
Gain Input \(Z_{\text{in}}\) Output \(Z_{\text{out}}\) Bandwidth
Open-Loop High (\(A_{\text{ol}}\)) High Low Narrow
Closed-Loop Controlled (\(A_{\text{cl}}\)) Adjustable Low Wider

Op-Amp Configurations

Noninverting Amplifier

  • Input applied to noninverting (\(+\)) input.

  • Feedback through resistors \(R_i\) and \(R_f\).

  • Closed-loop gain:

    • \[\begin{aligned} A_{\text{cl}}(\text{NI}) & = 1 + \dfrac{R_f}{R_i} \qquad (\text{Note:} ~ V_f = \left(\dfrac{R_i}{R_i+R_f}\right)V_{out}) \\ \text{Attenuation,}~B & = \left(\dfrac{R_i}{R_i+R_f}\right) \\ \text{Differential input,}~V_{diff} & = V_{in}-V_{f} \end{aligned}\]
Noninverting amplifier.
Noninverting amplifier.

Voltage Follower

  • Special case of noninverting amplifier.

  • All output fed back to inverting input (\(B=1\)).

  • Gain: \(A_{\text{cl}}(\text{VF}) = 1\).

  • Features:

    • Very high input impedance.

    • Very low output impedance.

    • Ideal buffer for impedance matching.

Voltage follower.
Voltage follower.

Inverting Amplifier

  • Input applied to inverting (\(-\)) input via \(R_i\).

  • Noninverting input grounded.

  • Feedback through \(R_f\).

  • Closed-loop gain:

    • \[A_{\text{cl}}(\text{I}) = -\dfrac{R_f}{R_i}\]

  • Negative sign indicates phase inversion.

Inverting amplifier.
Inverting amplifier.

Impedance Effects

Noninverting Amplifier Impedances

  • Input Impedance: Greatly increased by feedback.

    • \[Z_{\text{in}}(\text{NI}) = (1 + A_{\text{ol}}B) Z_{\text{in}}\]

  • Output Impedance: Significantly reduced.

    • \[Z_{\text{out}}(\text{NI}) = \dfrac{Z_{\text{out}}}{1 + A_{\text{ol}}B}\]

  • \(B = \dfrac{R_i}{R_i + R_f}\) (feedback attenuation).

Input and output impedance.
Input and output impedance.

Voltage Follower Impedances

  • Input Impedance: Even higher than noninverting amplifier (\(B=1\)).

    • \[Z_{\text{in}}(\text{VF}) = (1 + A_{\text{ol}}) Z_{\text{in}}\]

  • Output Impedance: Extremely low, ideal for buffering.

    • \[Z_{\text{out}}(\text{VF}) = \dfrac{Z_{\text{out}}}{1 + A_{\text{ol}}}\]

Inverting Amplifier Impedances

  • Input Impedance: Determined by input resistor due to virtual ground.

    • \[Z_{\text{in}}(\text{I}) \approx R_i\]

  • Output Impedance: Similar to noninverting, very low.

    • \[Z_{\text{out}}(\text{I}) = \dfrac{Z_{\text{out}}}{1 + A_{\text{ol}}B}\]
Inverting amplifier.
Inverting amplifier.

Bias Current and Offset Voltage

Effect of Input Bias Current

  • Small input bias currents (\(I_1\), \(I_2\)) cause output error voltages.

  • Example (Inverting Amplifier):

    • \[V_{\text{OUT(error)}} = I_1 R_f\]

  • Example (Voltage Follower):

    • \[V_{\text{OUT(error)}} = -I_1 R_s\]
Error due to bias current in inverting amplifier and voltage follower.
Error due to bias current in inverting amplifier and voltage follower.

Bias Current Compensation

  • Voltage Follower: Add resistor \(R = R_s\) in feedback path.

    • \[V_{\text{OUT(error)}} = I_{\text{os}}R_s\]

  • Noninverting/Inverting: Add compensating resistor \(R_c = R_i \parallel R_f\).

Bias current compensation.
Bias current compensation.

Input Offset Voltage

  • Small DC output error due to internal transistor mismatches.

  • Output error:

    • \[V_{\text{OUT(error)}} = A_{\text{cl}}V_{\text{IO}}\]

  • Compensation: Use external potentiometer (e.g., 741 op-amp offset null pins).

Offset voltage in voltage follower.
Offset voltage in voltage follower.

Frequency and Phase Responses

Open-Loop Frequency Response

  • Midrange Gain: Constant from DC to critical frequency \(f_{\text{c}}(\text{ol})\).

  • Roll-Off: -20 dB/decade above \(f_{\text{c}}(\text{ol})\).

  • Unity-Gain Bandwidth (\(f_{\text{T}}\)): Frequency where \(A_{\text{ol}}= 1\).

Open-loop gain vs. frequency.
Open-loop gain vs. frequency.

Phase Response

  • RC lag circuits cause phase shift:

    • \[\theta = -\tan^{-1}\left(\dfrac{f}{f_{\text{c}}}\right)\]

  • Phase lag increases with frequency, approaches \(-90^\circ\) per stage.

  • Multi-stage op-amps: Total phase lag sums contributions from each stage.

Output voltage lags input voltage.
Output voltage lags input voltage.

Closed-Loop Frequency Response

  • Negative feedback increases bandwidth:

    • \[BW_{\text{cl}} = BW_{\text{ol}} (1 + B A_{\text{ol}}(\text{mid}))\]

  • Gain-Bandwidth Product:

    • \[A_{\text{cl}}f_{\text{c}}(\text{cl}) = A_{\text{ol}}f_{\text{c}}(\text{ol}) = f_{\text{T}}\]

  • Higher gain reduces bandwidth, and vice versa.

Open-loop vs. closed-loop response.
Open-loop vs. closed-loop response.

Practical Considerations

Op-Amp Selection

  • Choose based on key parameters:

    • Gain, bandwidth, CMRR, noise.

    • Input/output impedance.

    • Slew rate, offset voltage/current.

  • Features: Short-circuit protection, no latch-up, offset nulling.

  • Consult datasheets for specific values.

Applications of Op-Amps

  • Amplifiers: Inverting, noninverting, voltage follower.

  • Filters: Active low-pass, high-pass, band-pass.

  • Oscillators: Sine wave, square wave generators.

  • Analog Computing: Integrators, differentiators.

  • Signal Conditioning: Buffering, impedance matching.

Conclusion

Summary

  • Op-amps are versatile, high-gain amplifiers with differential inputs.

  • Ideal vs. practical characteristics guide circuit design.

  • Negative feedback stabilizes gain, increases bandwidth, and controls impedances.

  • Configurations: Noninverting, inverting, voltage follower.

  • Key parameters: CMRR, slew rate, noise, offset voltage/current.

  • Frequency response and phase shift critical for high-frequency applications.

  • Wide range of applications in analog electronics.