Op-amp Integrator and Differentiator PDF

Op-amp Integrator  INTRODUCTION 1. The operational amplifier integrator is an electronic integration circuit. 2. Integrator performs the mathematical operation of integration with respect to time; that is, its output voltage is proportional to the input voltage integrated over time. 3....

Op-amp Integrator  INTRODUCTION 1. The operational amplifier integrator is an electronic integration circuit. 2. Integrator performs the mathematical operation of integration with respect to time; that is, its output voltage is proportional to the input voltage integrated over time. 3. The integrator circuit is mostly used in analog computers, analog-to-digital converters and wave-shaping circuits.  Ideal circuit Integrator  Ideal circuit 1. The circuit operates by passing a current that charges or discharges the capacitor CFduring the time under consideration. 2. The input voltage passes a current through the resistor producing a compensating current flow through the series capacitor to maintain the virtual ground. This charges or discharges the capacitor over time. 3. Because the resistor and capacitor are connected to a virtual ground, the input current does not vary with capacitor charge and a linear integration of output is achieved.  Ideal circuit The circuit can be analyzed by applying Kircchoff’s voltage law at the node v2, in an ideal op-amp, the capacitor has a voltage-current relationship,  Ideal circuit Substituting the appropriate variables: in an ideal op-amp, Integrating both sides with respect to time If the initial value of vo is assumed to be 0 V Input and output waveforms for Integrator  Practical circuit The ideal circuit is not a practical integrator design for a number of reasons. 1. Finite open loop gain, 2. input offset voltage and 3. input bias currents Practical circuit  Practical circuit 1. To negate the effect of the input bias current, it is necessary to set: The error voltage then becomes: The input bias current thus causes the same voltage drops at both the positive and negative terminals.  Practical circuit 2. In a DC steady state, the capacitor acts as an open circuit. The DC gain of the ideal circuit is therefore infinite.To counter this, a large resistor Rf is inserted in parallel with the feedback capacitor, this limits the DC gain of the circuit to a finite value, and hence changes the output drift into a finite, preferably small, DC error. Frequency response  Frequency response For both circuits, the crossover frequency , at which the gain is 0 dB, is given by: The 3dB cut-off frequency of the practical circuit is given by: The practical integrator circuit is equivalent to an active first-order low pass filter. The gain is relatively constant up to the cutoff frequency and decreases by 20 dB per decade beyond it.  Applications of Op-amp Integrator 1. Op-amp integrating amplifiers are used to perform calculus operations in analogue computers. 2. Integrating circuits are most commonly used in analogue- to-digital converters, ramp generators and also in wave shaping applications. 3. A common wave-shaping use is as a charge amplifier and they are usually constructed using an operational amplifier though they can use high gain discrete transistor configurations.  Op-amp Differentiator  INTRODUCTION 1.A differentiator is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change (the time derivative) of the input. 2. A true differentiator cannot be physically realized, because it has infinite gain at infinite frequency. 3. A similar effect can be achieved, however, by limiting the gain above some frequency. 4. The differentiator circuit is essentially a high pass filter.  Op-amp Differentiator 1. An op-amp differentiator or a differentiator amplifier is a circuit configuration which is inverse of the integrator circuit. 2. An op-amp differentiator is an inverting amplifier, which uses a capacitor in series with the input voltage. 3. Differentiators have frequency limitations while operating on sine wave inputs; the circuit attenuates all low frequency signal components and allows only high frequency components at the output. In other words, the circuit behaves like a high-pass filter. Ideal Op-Amp Differentiator Circuit  Ideal Op-Amp Differentiator Circuit 1. For DC input, the input capacitor C1, after reaching its potential, cannot accept any charge and behaves like an open-circuit. 2. The non-inverting input terminal of the op-amp is connected to ground through a resistor Rcomp, which provides the input bias compensation, and the inverting input terminal is connected to the output through the feedback resistor Rf. 3. Thus, the circuit behaves like a voltage follower.  Ideal Op-Amp Differentiator Circuit 1. When the input is a positive-going voltage, a current I flows into the capacitor C1, as shown in the figure. Since the current flowing into the op-amp’s internal circuit is zero, effectively all of the current I flows through the resistor Rf. The output voltage is, Vout = – (I * Rf) 2. node ‘X’ is virtually grounded and node ‘Y’ is also at ground potential i.e., VX = VY = 0.  Ideal Op-Amp Differentiator Circuit 3. From the input side, the current I can be given as: I = C1 {d(Vin – VX) / dt} = C1 {d(Vin) / dt} 4. From the output side, the current I is given as: I = -{(Vout – VX) / Rf} = -{Vout / Rf}  Ideal Op-Amp Differentiator Circuit 5. Equating the above two equations of current we get: C1 {d(Vin) / dt} = -Vout / Rf Vout = -C1 Rf {d(Vin) / dt} Above equation indicates that the output is C1 Rf times the differentiation of the input voltage. The product C1 Rf is called as the RC time constant of the differentiator circuit. The negative sign indicates the output is out of phase by 1800 with respect to the input. Input and output waveforms for Differentiator  Frequency Response of Ideal Differentiator 1. The gain of an op-amp differentiator is directly dependent on the frequency of the input signal. 2. Hence, for DC inputs where f = 0, the output is also zero. As the frequency of the input signal increases, the output also increases.  Practical Op-amp Differentiator Circuit 1. For an ideal differentiator, the gain increases as frequency increases. 2. Thus, at some higher frequencies, the differentiator may become unstable and cause oscillations which results in noise. 3. These problems can be avoided or corrected in a practical differentiator circuit, which uses a resistor R1 in series with the input capacitor and a capacitor Cf in parallel with the feedback resistor. Practical Op-amp Differentiator Circuit  Practical Op-amp Differentiator Circuit 1. The output voltage of the practical op-amp differentiating amplifier circuit is given as, Vout = - C1 Rf {d(Vin) / dt} i.e., the output voltage is C1Rf times the differentiation of the input voltage. 2. The addition of resistor R1 and capacitor Cf stabilizes the circuit at higher frequencies, and also reduces the effect of noise on the circuit.  Frequency Response of Practical Differentiator 1. The gain of the practical differentiator increases with increasing frequency and at a particular frequency, f1, the gain becomes the unity (0 dB). 2. The gain continues to increase at a rate of 20dB per decade till the input frequency reaches a frequency, f2. 3. Beyond this frequency of the input signal, the gain of the differentiator starts to decrease at a rate of 20dB per decade. This effect is due to the addition of the resistor R1 and capacitor Cf. Frequency Response of Practical Differentiator  Frequency Response of Practical Differentiator From the above plot, it can be seen that: 1. when f