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Questions and Answers
What is the main function of an operational amplifier integrator?
The operational amplifier integrator performs the mathematical operation of integration with respect to time.
Where are integrator circuits mostly used?
The input current to an integrator circuit varies with capacitor charge.
False
What effect does input bias current have in a practical integrator circuit?
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What happens to the capacitor in a DC steady state in a practical integrator circuit?
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How is the cutoff frequency of a practical integrator circuit characterized?
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Which of the following is a typical application of op-amp integrating amplifiers?
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A true differentiator can be physically realized.
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What type of filter does the differentiator circuit essentially act as?
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In the ideal op-amp differentiator circuit, how is the input capacitor connected?
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In an ideal op-amp differentiator circuit, the non-inverting input terminal is connected to ground.
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Study Notes
Op-amp Integrator
- An operational amplifier integrator is an electronic circuit for integration.
- It performs mathematical integration with respect to time. Output voltage is proportional to the input voltage integrated over time.
- Typically used in analog computers, analog-to-digital converters, and wave-shaping circuits.
Op-amp Integrator - Ideal Circuit
- The circuit passes current that charges or discharges the capacitor during the time period considered.
- Input voltage creates a current through the resistor; this generates a compensating current through the capacitor to maintain virtual ground.
- Capacitor charges/discharges over time.
- Input current doesn't change with capacitor charge; this results in a linear integration output.
- Ideal op-amp has zero input bias current (IB = 0).
- Capacitor follows voltage-current relationship (Ic = C dV/dt).
Op-amp Integrator - Analysis
- Apply Kirchhoff's voltage law at node v2.
- In an ideal op-amp, input bias current (IB) is zero, so i₁ = iF.
- Substituting variables into the equations and integrating both sides with respect to time yields the output voltage equation.
- Output voltage is expressed as (Vo = -1/(R₁C₁) *∫Vin dt) and considers the initial value of Vo to be 0.
Op-amp Integrator - Waveforms
- Input waveforms typically show square or rectangular shapes.
- Output waveforms typically show triangular shapes.
Op-amp Integrator - Practical Circuit
- Ideal circuit is not practical due to finite open-loop gain, input offset voltage, and input bias currents.
- To negate input bias current, set Rcom = R1||RF||RL.
- Error voltage is expressed as (VE = (RF / R1 + 1) VIOS).
- Parallel resistor (R) inserted with feedback capacitor (C) limits DC gain, reducing output drift to a finite, preferably small, DC error.
Op-amp Integrator - Frequency Response
- The crossover frequency (fb) where the gain is 0dB is expressed as (1/(2πR₁C₁)).
- 3dB cut-off frequency (fa) of the practical circuit is expressed as (1/(2πRFC₁)).
- The practical integrator behaves like a first-order low-pass filter; the gain remains relatively constant up to the cutoff frequency. The gain decreases by 20dB per decade beyond the cutoff frequency.
Op-amp Integrator - Applications
- Used to perform calculus operations in analog computers.
- Commonly used in analog-to-digital converters, ramp generators, and wave shaping applications.
- A common wave-shaping use is as a charge amplifier.
- Often constructed using op-amps, but can also use high-gain discrete transistors.
Op-amp Differentiator
- A differentiator circuit produces an output approximately proportional to the rate of change (time derivative) of the input signal.
- A true differentiator is not physically realizable due to infinite gain at infinite frequency.
- In a practical differentiator, the gain is limited above some frequency.
- The differentiator circuit acts essentially as a high-pass filter.
Op-amp Differentiator - Ideal Circuit
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Input capacitor (C₁) blocks DC current, acting as an open circuit for DC inputs.
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The non-inverting input is grounded via Rcomp, which accounts for input bias compensation.
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Inverting input connects to output via feedback resistor (RF).
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The circuit functions like a voltage follower in the DC steady state.
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A positive-going input voltage flows into capacitor (C₁), producing a current (I) that flows through RF.
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The output voltage is given by the following equation: Vout = -C₁RF (d(Vin)/dt)
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Current I can also be expressed as I = C₁ (d(Vin - VX) / dt , but given that Vx ≈0, I ≈ C₁ (d(Vin) / dt)
Op-amp Differentiator - Waveforms
- Input waveforms typically shown as square or rectangular wave;
- Output waveforms typically shown as pulses or spikes.
Op-amp Differentiator - Frequency Response
- The gain of an ideal differentiator depends on the input signal frequency.
- DC inputs (f=0) produce zero output.
- The output increases as the input frequency increases.
- The ideal differentiator is equivalent to a high pass filter.
- The gain of the practical differentiator increases, reaching unity (0dB) at a certain frequency (f1), then increases at 20dB per decade up to a certain frequency (f2). Beyond f2, the gain decreases at 20dB per decade.
Op-amp Differentiator - Practical Circuit
- A practical differentiator includes a resistor (R₁) in series with the input capacitor (C₁).
- A capacitor (Cf) is added in parallel to the feedback resistor (RF).
- These elements stabilize the circuit at higher frequencies and reduce noise.
- The output voltage of a practical circuit is still based on the following equation: Vout = - C₁ Rf (d(Vin)/dt)
Op-amp Differentiator - Applications
- Typically used for differentiating triangular and rectangular signals, and detecting high-frequency components in an input signal.
- Important part of analogue computers and analogue PID controllers.
- Used in frequency modulators as rate-of-change detectors.
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Description
Explore the principles and applications of the Op-amp integrator circuit in this quiz. Discover how these circuits perform mathematical integration and their relevance in various analog systems. Test your understanding of ideal circuits and the analysis techniques involved.