Before the digitization of the electronic world, the control system based on differential equation solving used analog calculation to solve the equation. Therefore, analog computers are quite common, because the solution of almost all differential equations requires the ability to integrate the signal.
Although most control systems have been digitized and numerical integration has replaced analog integration, analog integrator circuit is still needed in the operation of sensor, signal generation and filtering.
These applications use operational amplifier based integrators with capacitive elements in the feedback loop to provide the necessary signal processing for low-power applications. Although practicality is still important, many designers may easily ignore it.
This paper summarizes the integrator circuit, and takes several products of Texas Instruments as examples to provide guidance on correct design, component selection and best practice in order to achieve excellent performance.
1、 Basic inverse integrator
The classical analog integrator uses operational amplifier and capacitor as feedback element (Fig. 1).
Figure 1: the basic inverse analog integrator contains an operational amplifier and a capacitor on the feedback path.
The output voltage Vout of the integrator is a function of the input voltage Vin and can be calculated using formula 1.
The gain coefficient of the basic inverse integrator is – 1 / RC, which can be applied to the input voltage integration. In fact, the capacitors used in the integrator should have a tolerance of less than 5% and low temperature drift. Polyester capacitor is a good choice. Resistors with a tolerance of ± 0.1% shall be used at critical path locations.
This circuit has limitations because under DC, the capacitor represents an open circuit and the gain will be infinite. In the working circuit, according to the polarity of non-zero DC input, the output will be transmitted to the positive power rail or negative power rail. This can be corrected by limiting the DC gain of the integrator (Fig. 2).
Figure 2: a large resistor connected in parallel to the feedback capacitor can limit the DC gain, so as to obtain a practical integrator.
By paralleling a high resistance resistor (RF) on the feedback capacitor, the DC gain of the basic integrator can be limited to – RF / R value, so as to obtain a practical device. This addition method solves the problem of DC gain, but limits the working frequency range of the integrator. Observing real circuits helps to understand this limitation (Figure 3).
Figure 3: TINA-TI simulation of practical integrator using real components.
The circuit uses the LM324 operational amplifier of Texas Instruments. LM324 is an excellent general-purpose operational amplifier with low input bias current (typical value 45na), low offset voltage (typical value 2mV) and 1.2MHz gain bandwidth product. The circuit input is driven by the function generator of the simulator with a square wave of 500Hz. This is shown as an upper trace on the simulator oscilloscope. The circuit will integrate the square wave and output a 500Hz trigonometric function, as shown in the lower trace of the oscilloscope.
The DC gain is – 270k Ω / 75K Ω or – 3.6 or 11db; This can be seen from the transfer function of the circuit, as shown in the lower right grid of Fig. 3. From about 100Hz to about 250kHz, the frequency response is rolled down by – 20dB / ten octave. This is the useful frequency range of the integrator and is related to the gain bandwidth product of the operational amplifier.
The tlv9002 of Texas Instruments is a newly introduced operational amplifier. This 1MHz gain bandwidth amplifier has an input offset voltage of ± 0.4mv and an extremely low bias current of 5pa. As a CMOS amplifier, it is suitable for a variety of low-cost portable applications.
For designers, it is important to remember that the integrator is an accumulation device. Therefore, without proper compensation, the input bias current and input offset voltage will cause the capacitor voltage to increase or decrease over time. In this application, the input bias current and offset voltage are relatively low, and the input voltage will force the feedback capacitor to discharge regularly.
In applications using the accumulation function, such as when measuring charge, there must be a mechanism in the integrator to reset the voltage and establish the initial conditions. The acf2101bu of Texas Instruments has this mechanism. It is a dual switch integrator that integrates a built-in switch to discharge the feedback capacitor. Since the device is suitable for applications requiring charge accumulation, it has a very low bias current of 100fa and a typical bias voltage of ± 0.5mv.
The ivc102u of Texas Instruments is a similar switching integrator / transimpedance amplifier. The device has the same application range as acf2101bu, but the difference is that each package contains a single device. In addition, there are three internal feedback capacitors. It includes a switch for discharging the capacitor bank and connecting the input source, so the designer can control the integration cycle, including holding operation and discharging the voltage on the capacitor.
2、 Non inverting integrator
The basic integrator inverts the integral of the signal. Although a second inverting operational amplifier in series with the basic integrator can restore the original phase, a non inverting integrator can also be designed in a single stage (Fig. 4).
Figure 4: non inverting integrator based on differential amplifier operational amplifier configuration can ensure that the output phase matches the input phase.
The non inverting version of the integrator uses a differential integrator to keep the output in phase with the input signal. This design adds additional passive components, which should be matched to achieve the best performance. The relationship between input and output voltages is the same as that of the basic integrator, but the symbols are different, as shown in formula 2:
By using the traditional operational amplifier circuit, other adjustments to the basic integrator can be realized. For example, multiple voltage inputs (V1, V2, V3…) can be added as long as they are added to the non inverting input of the operational amplifier through their respective input resistors (i.e. R1, R2, R3…).
The output is the integral of the sum of the inputs.
3、 Some common integrator applications
In the past, integrators have been used to solve differential equations. For example, mechanical acceleration is the rate of change or derivative of its velocity. Velocity is the derivative of displacement. The integrator can be used to obtain the output of the accelerometer and integrate it once to read the speed. If the velocity signal is integrated, the output is displacement. This means that by using an integrator, the output of a single sensor can produce three different signals: acceleration, velocity and displacement (Fig. 5).
Figure 5: using dual integrators, designers can generate acceleration, velocity and displacement readings from accelerometers.
The input of the accelerometer is integrated and filtered to obtain the velocity. The displacement can be obtained by integrating and filtering the velocity. Note that all outputs are AC coupled. In this way, it is no longer necessary to deal with the initial conditions of each integrator.
4、 Function generator
The function generator can output a variety of waveforms and can be composed of multiple integrators (Fig. 6).
Figure 6: function generator designed using three LM324 stages. OP1 is a relaxation oscillator that generates a square wave; Op2 is an integrator that converts a square wave into a triangular wave; OP3 is another integrator used as a low-pass filter to eliminate the harmonics of triangular waves and generate sine waves.
The function generator is designed around LM324, which is the practical integrator discussed earlier. In this design, three LM324 operational amplifiers are used, as shown in TINA-TI simulation. The first stage OP1 acts as a relaxation oscillator and generates a square wave output at the frequency determined by C1 and potentiometer P1. The connected second stage op2 is an integrator that converts a square wave into a triangular wave. The last connected stage OP3 is an integrator, but is used as a low-pass filter. The filter removes all harmonics in the triangular wave and outputs the fundamental sine wave. The output of each stage is shown in the simulator oscilloscope at the bottom right of Figure 6.
5、 Roche coil
Rogowski coil is a kind of current sensor, which uses a flexible coil wound on the measured current carrying conductor to measure AC power supply. They are used to measure high-speed current transients, pulse currents, or 50 / 60Hz line power.
Rogowski coils perform functions similar to current transformers. The main difference is that the Rogowski coil uses an empty core instead of the magnetic core used in the current transformer. The hollow core has a low insertion impedance, so it has faster response and no saturation effect when measuring high current. The Rogowski coil is very easy to use (Figure 7).
Figure 7: simplified schematic diagram showing the installation of Rogowski coil on current carrying conductor (left) and the equivalent circuit of this setting (right).
Rogowski coil, such as art-b22-d300 of lemusa, is simply wound on the current carrying conductor, as shown on the left side of Figure 7. The equivalent circuit of Rogowski coil is shown in the right figure. Note that the output of the coil is proportional to the derivative of the measured current. The integrator can be used to extract the sensed current.
The reference design of Rogowski coil integrator is shown in Figure 8. This design is characterized by high-precision output in the range of 0.5 to 200A (accuracy of 0.5%) and fast establishment output in the same current range (accuracy within 1% in less than 15ms).
Figure 8: the reference design of this Rogowski coil integrator uses opa2188 of Texas Instruments as the main operational amplifier in the design integrator element.
This reference design uses the opa2188 of Texas Instruments as the main operational amplifier in the design integrator element. Opa2188 is a dual operational amplifier with proprietary automatic zeroing technology. The maximum offset voltage is 25 microvolts (µ V) and the time or temperature drift is close to zero. The gain bandwidth product is 2MHz and the typical input bias current is ± 160pa.
For this reference design, Texas Instruments chose opa2188 because of low offset and low offset drift. Moreover, the low bias current can minimize the load on the Rogowski coil.
6、 Integrator in filter
The integrator is used in the design of state variables and double second-order filters. These related filter types use double integrators to obtain second-order filter responses. The state variable filter is a more interesting filter because a single design produces low-pass, high pass and band-pass responses at the same time. The filter uses two integrators and an adder / subtractor stage, as shown in TINA-TI simulation (Fig. 9). The figure shows the filter response of the low-pass output.
Figure 9: the state variable filter uses two integrators and an adder / subtractor stage to produce low-pass, high pass and band-pass outputs from the same circuit.
The advantage of this filter topology is that all three filter parameters (gain, cut-off frequency and Q value) can be adjusted independently in the design process. In this example, the DC gain is 1.9 (5.6dB), the cut-off frequency is 1kHz, and Q is 10.
The design of high-order filter is realized by connecting multiple state variable filters in series. These filters are usually used for anti aliasing in front of analog-to-digital converters, which requires high dynamic range and low noise.
Although sometimes the world seems to be fully digitized, analog integrator is still a very useful and universal circuit element in signal processing, sensor regulation, signal generation and filtering.