Recently I was asked to investigate an analog issue with a optical sensor channel. The problem was not related to the sensor but was thought to be with an integrator circuit used to integrate a series of pulses. The circuit was using what is called a Howland current source. This was not the problem and this is not your usual analog integrator. It was developed in 1962 by Prof. Bradford Howland of MIT. See Figure 1. I'm writing about it here because of its novelty.
Figure 1 - Howland Integrator
Note the output is taken from the op-amp output and not the integrator capacitor. Also there is both positive and negative feedback to the op-amp input. For a dual supply op-amp the integrator output can be either polarity and is ground referenced. Often there might be a small compensating capacitor paralleling R2 for HF stability. As shown the output is inverting however the input can be moved to R4 with R1 grounded and it becomes a non-inverting integrator.
As reference the more common topology for an integrator is the textbook op-amp feedback capacitor as shown in Figure 2. Given a dual supply op-amp this circuit will integrate both positive or negative with inverting gain.
Figure 2 - Op-amp Miller Integrator
The op-amp circuit in Figure 2 has a transfer function defined by equation 1 where A(s) is the op-amp transfer function in the s-domain. Ideally there is infinite impedance paralleling the feedback capacitor. In practice there could be PCB level contamination that provides a finite impedance. Normally this is not an issue.
Equation 1:Â Â Â
When the op-amp loop gain, Ao, is large the pole in the transfer function is always at s = 0 and the gain is G(s) = -1/RCs
Another approach is the single ended design shown in Figure 3 using transistor current mirrors for a non-inverting positive only output.
Figure 3 - Current Mirror Integrator
Back to the Howland current source, this is a dual polarity integrator if needed. The presence of both the positive and negative feedback is unique and presents its own challenges. The small signal transfer function from input to the capacitor output is given in equation 2 for the non-inverting configuration.
Equation 2:
The effective impedance paralleling the integrator capacitor is given by the time constant term in equation 2. Note that for matching resistors the pole is at s=0 but for real world tolerance the pole can be negative or positive. In order to achieve accurate results it is common to need very tight tolerances on the resistors R1, R2, R3 and R4. Any mismatch from ideal provides a leakage current to the integrator and degrades accuracy. One solution is to use four matched resistors in a single package, however that is an expensive solution.
As example, if R1~R4 are all 1.00k 0.1% the effective leakage impedance can be as small as 249.75k. What's more, depending on the tolerance build-up the polarity of the leakage impedance can be positive or negative. This places the pole on the real axis in either a stable (<0) or unstable position.
For the special case of matched resistors, 
Equation 3:Â
An advantage of both the Howland and the Current Mirror integrator is that they are both single ended and a reset switch can be paralleled across the capacitor to ground for easy resetting. The disadvantage of the Howland integrator is the reliance on tight tolerance (expensive) resistors to minimize the effective leakage impedance. Also, for non-ideal resistors the pole of the system can be slightly positive or negative.
The Howland integrator was reviewed as a historical topology from 1962.