Op-amp Integrator Circuit Analysis

Study Notes

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.

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).

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.

Ideal circuit is not practical due to finite open-loop gain, input offset voltage, and input bias currents.
To negate input bias current, set R_com = R₁||R_F||R_L.
Error voltage is expressed as (V_E = (R_F / R₁ + 1) V_IOS).
Parallel resistor (R) inserted with feedback capacitor (C) limits DC gain, reducing output drift to a finite, preferably small, DC error.

The crossover frequency (f_b) where the gain is 0dB is expressed as (1/(2πR₁C₁)).
3dB cut-off frequency (f_a) of the practical circuit is expressed as (1/(2πR_FC₁)).
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.

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.

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.

Input capacitor (C₁) blocks DC current, acting as an open circuit for DC inputs.
The non-inverting input is grounded via R_comp, which accounts for input bias compensation.
Inverting input connects to output via feedback resistor (R_F).
The circuit functions like a voltage follower in the DC steady state.
A positive-going input voltage flows into capacitor (C₁), producing a current (I) that flows through R_F.
The output voltage is given by the following equation: V_out = -C₁R_F (d(V_in)/dt)
Current I can also be expressed as I = C₁ (d(V_in - V_X) / dt , but given that V_x ≈0, I ≈ C₁ (d(V_in) / dt)

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.

A practical differentiator includes a resistor (R₁) in series with the input capacitor (C₁).
A capacitor (C_f) is added in parallel to the feedback resistor (R_F).
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: V_out = - C₁ Rf (d(Vin)/dt)

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.