## Detecting a small fraction of impedance changes with improved sensitivity

### Introduction

There are normally two types of impedance measurement. The first type is the dynamic impedance measurement where impedance can change rapidly over time. This is the case in electrical impedance spectroscopy (EIS) in microfluidics for cell counting and cell discrimination applications, for example. The second type is of course the static impedance measurement where the impedance being measured does not really change with time. For static impedance measurement, some users may be looking for absolute accuracy. Quite often, users are more interested to find out the relative variation in impedance of a material being analyzed. This can be the case in the evaluation of biological cells or industrial treatment processes where impedance change provides valuable information on the quality of physical, electrical or chemical manipulations. For example, one can determine the content of ice in a soil by measuring impedance as a function of temperature. Another example is the deep level transient spectroscopy (DLTS) which relies on the time-domain measurement of junction capacitance transient in a photodiode to estimate carrier trap concentration.

### Estimate Required Instrument Sensitivity

Very often, the impedance variation being measured represents only a very small fraction of the fixed impedance value. This means that an instrument may need to have a very high sensitivity. Take a capacitance measurement as an example. If one wants to detect 1 pF change out of 100 pF driven by a 100 mV, 1 MHz sine wave, then one needs to detect about changes of about 600 nA out of 60 μA of signal current, or about 1%. This ratio can go down to one tenth or even one hundredth of a percent if the expected delta capacitance is even smaller. It is obvious that ratio of ‘fixed’ part and the variable part of the impedance pretty much determines the minimum signal-to-noise ratio the measurement instrument should have. Ideally, one should have at least an order of magnitude higher so the variation does not get buried inside the noise floor.

### Reduce Required Sensitivity

The concept is quite simple. If one can reduce the fixed part of the impedance, then one needs to just have an instrument capable of detecting the delta. How can this be achieved? The concept is actually quite simple: differential compensation. Below is a simplified schematic of a measurement setup using a lock-in amplifier. The lock-in amplifier in question can also be replaced with an impedance spectroscope. Assume the device-under-test has a fixed capacitance of *C _{dut}* with a variable part of

*ΔC*. One can then insert in parallel a compensation capacitor

*C*to cancel out

_{comp}*C*. The trick is to have a function generation that can generate a differential exciation voltage:

_{dut}*V*and –

_{drive}*V*, simultaneously.

_{drive}To prove this, we can do some circuit anaysis. We know from KCL that *i _{x}* =

*i*+

_{dut }*i*. With

_{comp}*i*= (

_{dut}*v*–

_{drive }*v*) s(

_{x}*C*+

_{dut }*ΔC*) and

*i*= (-v

_{comp}*– v*

_{drive }*) s(C*

_{x}*), We get,*

_{comp}One can see now that if we choose such that *C _{comp }*=

*C*, we are left with only,

_{dut}There will be some filtering effect from the added *C _{comp}*. But that should not play a big role usually. The key here is really the access to a differential voltage source. In practice, we should be able also to adjust the amplitude and the phase of the negative source to be able to fine tune the compensation current

*i*for parastics in the setup. In fact, the HF2LI Lock-in Amplifier and the HF2IS Impedance Spectroscope both offer the necessary features to perform this kind of measurement.

_{comp }### Measurement Example

In this section, an example will be shown for a delta capacitance measurement. Here are some setup parameters.

*C*≈_{dut }*C*≈ 240 pF_{comp }*ΔC ≈*1.3 pF*v*= 100 mV_{drive}- f = 300 kHz

The HF2IS Impedance Spectroscope is used for this exercise due to the fact that it can directly display the measured impedance value. The fact that all HF2 devices have two independent output generators, one can generate an output ac voltage and its inverted counterpart. The screenshot below shows a 2-Term measurement with Signal Output 2 still turned off i.e. inverted voltage off so no current flowing through *C _{comp}*. Note that

*ΔC*is also zero at this point. The measured capacitance is about 240 pF as expected.

Then the Signal Output 2 is turned on. The signal output 2 voltage has a negative sign to represented an inversion of 180 degrees with respect to the Signal Output 1. The amplitude of Signal Output 2 can actually be adjusted independently to try to minimize the compensated capacitance value. This may be necessary since the propagation delay and gain of two paths will surely not be identical in real cases.

Here, the measured capacitance drops from 240 pF to a very low value, about 1.3 pF. The input range as well as the HF2TA Transimpedance Amplifier gain must then adjusted accordingly. Be aware that it is not necessary to make this value close to zero. Having a value too small is actually counter productive since the measured current will be very small which will be heavily influenced by noise.

Note: If the HF2LI Lock-in Amplifier with HF2LI-MF Multi-frequency option is used, then the inverted output can be generated by simply entering an 180 degrees phase shift.

Now if we insert *ΔC* of about 1.3pF, we immediately see the measured capacitance goes from 1.3 pF to about 2.6 pF. Without compensation, one would be trying to measure a 0.5% variation. Now, one just has to optimize the instrument to detect a 50% change which relaxes the SNR requirement of the measurement system. In other words, if we try to capture too small a variation, then the filter time constant will have to increased to try to get noise down below the variation value. This means also that the measurement time will be very long due to the required filter integration time. In this example, a time constant of 10 ms was used which is quite fast such that one can imagine applying this technique to an impedance transient type measurement.

### Conclusion

In this blog, a compensation method is demonstrated using the HF2 device to increase delta impedance measurement sensitivity. The goal is not to achieve absolute accuracy. Rather, the objective is to achieve a good signal-to-noise ratio while trying to capture a very small relative change in a device-under-test’s impedance. A more concrete example will be given soon on how this setup can benefit hotodiode characterization.