“Before the Electronic world went digital, control systems based on differential equation solving used analog computation to solve equations. Therefore, analog computers are fairly common, since the ability to integrate signals is required to solve almost all differential equations. Although control systems are mostly digitized, and numerical integration has replaced analog integration, analog integrator circuits are still required for the operations of sensors, signal generation, and filtering. These applications use an op amp based integrator with capacitive elements in the feedback loop to provide the necessary signal processing for low power applications.

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Author: Art Pini

Before the electronic world went digital, control systems based on differential equation solving used analog computation to solve equations. Therefore, analog computers are fairly common, since the ability to integrate signals is required to solve almost all differential equations. Although control systems are mostly digitized, and numerical integration has replaced analog integration, analog integrator circuits are still required for the operations of sensors, signal generation, and filtering. These applications use an op amp based integrator with capacitive elements in the feedback loop to provide the necessary signal processing for low power applications.

While practicality is still important, it may be easily overlooked by many designers. This article provides an overview of integrator circuits and provides guidance on proper design, component selection, and best practices to achieve superior performance, using several Texas Instruments products as examples.

**Basic Inverting Integrator**

A classic analog integrator uses an op amp with a capacitor as the feedback element (Figure 1).

Figure 1: A basic inverting analog integrator contains an op amp with a capacitor in the feedback path. (Image credit: Digi-Key Electronics)

The integrator’s output voltage, VOUT, is a function of the input voltage, VIN, and can be calculated using Equation 1.

Equation 1

The gain factor of the basic inverting integrator is -1/RC, which can be applied to the input voltage integration. In practice, the capacitors used in the integrator should have less than 5% tolerance and low temperature drift. Polyester capacitors are a good choice. Resistors with a tolerance of ±0.1% should be used at critical path locations.

This circuit has limitations because at DC, the capacitor represents an open circuit and the gain would be infinite. In a working circuit, depending on the polarity of the non-zero DC input, the output will be delivered to the positive or negative supply rail. This can be corrected by limiting the DC gain of the integrator (Figure 2).

Figure 2: A large resistor in parallel with the feedback capacitor limits the DC gain, resulting in a practical integrator. (Image credit: Digi-Key Electronics)

A high-value resistor (RF) in parallel with the feedback capacitor limits the DC gain of the basic integrator to the -RF/R value, resulting in a practical device. This addition solves the DC gain problem, but limits the integrator’s operating frequency range. Looking at a real circuit helps to understand this limitation (Figure 3).

Figure 3: Practical integrator TINA-TI simulation using real components. (Image credit: Digi-Key Electronics)

This circuit uses an LM324 op amp from Texas Instruments. The LM324 is an excellent general-purpose op amp with low input bias current (45 nA typ), low offset voltage (2 mV typ), and a gain-bandwidth product of 1.2 MHz. The circuit inputs are driven with a 500 Hz square wave by the simulator’s function generator. This is shown as the upper trace on the emulator oscilloscope. The circuit integrates the square wave and outputs a trigonometric function at 500 Hz, as shown in the lower trace of the oscilloscope.

The DC gain is -270 kΩ/75 kΩ or -3.6 or 11 dB; this can be seen from the transfer function of the circuit, shown in the lower right grid of Figure 3. From about 100 Hz to about 250 kHz, the frequency response rolls off by -20 dB/decade. This is the useful frequency range in which the integrator operates and is related to the op amp gain-bandwidth product.

Texas Instruments’ TLV9002 is a recent op amp. This 1 MHz gain bandwidth amplifier features an input offset voltage of ±0.4 mV and a very low bias current of 5 pA. As a CMOS amplifier, it is suitable for a variety of low-cost portable applications.

For designers, it is important to remember that an integrator is an accumulating device. Therefore, without proper compensation, the input bias current and input offset voltage can 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 forces the feedback capacitor to discharge periodically.

In applications that use the accumulation function, such as when measuring charge, there must be a mechanism in the integrator to reset the voltage and establish initial conditions. Texas Instruments’ ACF2101BU has this mechanism. It is a two-switch integrator that integrates a built-in switch to discharge the feedback capacitor. Since the device is suitable for applications requiring charge accumulation, it features an extremely low bias current of 100 fA and a typical bias voltage of ±0.5 mV.

Texas Instruments’ IVC102U is a similar switching integrator/transimpedance amplifier. This device has the same range of applications as the ACF2101BU, except that each package contains a single device. In addition, there are three internal feedback capacitors. These include switches to discharge the capacitor bank and connect the input source, so the designer can control the integration period, including holdover, and discharge the voltage across the capacitors.

**non-inverting integrator**

The basic integrator inverts the integral of the signal. Although a second inverting op amp in series with the basic integrator restores the original phase, it is also possible to design a non-inverting integrator in a single stage (Figure 4).

Figure 4: A non-inverting integrator based on a differential amplifier op amp configuration ensures that the output phase matches the input phase. (Image credit: Digi-Key Electronics)

The non-inverting version of the integrator uses a differential integrator to keep the output in phase with the input signal. This design has additional passive components that should be matched for optimum performance. The relationship between the input and output voltages is the same as for the basic integrator, with a different sign, as shown in Equation 2:

Equation 2

Additional adjustments to the basic integrator can be achieved by using conventional op-amp circuits. For example, multiple voltage inputs (V1, V2, V3…) can be added as long as they are added to the non-inverting inputs of the op amp through their respective input resistors (ie R1, R2, R3…). The final output of this additive integrator is calculated using Equation 3:

Equation 3

If R1=R2=R3=R, use Equation 4 to calculate the output:

Equation 4

The output is the integral of the sum of the inputs.

**Some common integrator applications**

In the past, integrators have been used for differential equation solving. 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 take the output of the accelerometer and integrate it once to read the velocity. If the velocity signal is integrated, the output is the displacement. This means that by using an integrator, the output of a single sensor can produce three distinct signals: acceleration, velocity, and displacement (Figure 5).

Figure 5: Using dual integrators, designers can generate acceleration, velocity, and displacement readings from an accelerometer. (Image credit: Digi-Key Electronics)

The input to the accelerometer is integrated and filtered to obtain the velocity. The velocity is integrated and filtered to obtain the displacement. Note that all outputs are AC coupled. This eliminates the need to deal with the initial conditions of each integrator.

**function generator**

The function generator can output a variety of waveforms, which can be composed of multiple integrators (Figure 6).

Figure 6: Function generator designed using three LM324 stages. OP1 is a relaxation oscillator that produces a square wave; OP2 is an integrator that converts the square wave to a triangular wave; OP3 is another integrator that acts as a low pass filter to remove the harmonics of the triangular wave, resulting in a sine wave. (Image credit: Digi-Key Electronics)

The function generator is designed around the LM324, the practical integrator discussed earlier. In this design, three LM324 op amps are used, as shown in the TINA-TI simulation. The first stage, OP1, acts as a relaxation oscillator and produces a square wave output at a frequency determined by C1 and potentiometer P1. The connected second stage OP2 is an integrator that converts the square wave into a triangle wave. The last stage connected, OP3, is an integrator, but acts as a low pass filter. This filter removes all harmonics from the triangle wave and outputs a fundamental frequency sine wave. The output of each stage is shown in the emulator oscilloscope at the bottom right of Figure 6.

**Rogowski coil**

Rogowski coils are a class of current sensors that measure AC power using a flexible coil wrapped around the current-carrying conductor under test. They are used to measure high speed current transients, pulse currents or 50/60 Hz line power.

Rogowski coils perform functions similar to current transformers. The main difference is that Rogowski coils use an air core instead of the magnetic core used in current transformers. Air cores have lower insertion impedance, resulting in faster response and no saturation effects when measuring high currents. Rogowski coils are very easy to use (Figure 7).

Figure 7: Simplified schematic showing the installation of a Rogowski coil on a current-carrying conductor (left) and the equivalent circuit of this setup (right). (Image credit: LEM USA)

A Rogowski coil, such as the ART-B22-D300 from LEM USA, is simply wrapped around a current-carrying conductor, as shown on the left in Figure 7. The equivalent circuit of the Rogowski coil is shown on the right. Note that the output of the coil is proportional to the derivative of the current being measured. An integrator can be used to extract the sensed current.

The reference design of the Rogowski coil integrator is shown in Figure 8. This design features a high-accuracy output (0.5% accuracy) from 0.5 to 200 A, and a fast settling output (within 1% accuracy in less than 15 ms) for the same current range.

Figure 8: This Rogowski coil integrator reference design uses the OPA2188 from Texas Instruments as the main op amp in the design integrator element. (Image credit: Texas Instruments)

This reference design uses the OPA2188 from Texas Instruments as the main op amp in the design integrator element. The OPA2188 is a dual op amp that uses a proprietary auto-zero technique with a maximum offset voltage of 25 microvolts (µV) and near-zero drift over time or temperature. The gain-bandwidth product is 2 MHz, and the typical input bias current is ±160 pA.

For this reference design, Texas Instruments chose the OPA2188 for its low offset and low offset drift. Also, the low bias current minimizes the load on the Rogowski coil.

**integrator in filter**

Integrators are used in both state variable and biquad filter designs. These related filter types use double integrators to obtain second order filter responses. State-variable filters are a more interesting type of filter because a single design produces low-pass, high-pass, and band-pass responses simultaneously. The filter uses two integrators and an adder/subtracter stage, as shown in the TINA-TI simulation (Figure 9). The figure shows the filter response for the low-pass output.

Figure 9: A state variable filter uses two integrators and an adder/subtracter stage to produce low-pass, high-pass, and band-pass outputs from the same circuit. (Image credit: Digi-Key Electronics)

The advantage of this filter topology is that all three filter parameters (gain, cutoff frequency, and Q) can be adjusted independently during the design process. In this example, the DC gain is 1.9 (5.6 dB), the cutoff frequency is 1 kHz, and the Q is 10.

The design of higher-order filters is accomplished by connecting multiple state-variable filters in series. These filters are typically used for antialiasing before analog-to-digital converters, where high dynamic range and low noise are required.

**Epilogue**

Although the world sometimes seems to be fully digital, the examples discussed in this article demonstrate that analog integrators are still very useful and versatile circuit elements for signal processing, sensor conditioning, signal generation, and filtering.

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