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Instrumentation & Experimental

Complex Impedance

An alternating voltage, monochromatic sine waveform, applied to a circuit will create an alternating current (A.C.).  This will modify the resistive Ohm’s Law to an impedance, or

(eq. I01)

where Z is the complex, real and imaginary, impedance and replaces the purely real resistance R in Ohm’s Law in a static, non-time varying, direct current (D.C.) circuit.

Halliday [Halliday, 2001 #252], Serway [Serway, 2004 #253], and Feynman [Feynman, 1989 #254] discuss the reactance of a capacitor in an A.C. circuit, as shown

(fig. I01)

where the voltage source V drives the charge Q buildup and decay on the capacitor with capacitance C and the current I is measured with an appropriate A.C. ammeter.

The voltage across the capacitor is given by

(eq. I02)

and the current I is the given by the rate of change of the charge within the capacitor, or

(eq. I03)

and finally the rate of change of the sinusoidally varying voltage can be expressed by

(eq. I04)

where i = Ö-1 and is the angular frequency of the source or = 2 f.  Upon combining the three latter equations derives

(eq. I05)

and thus equating the latter with Eq. I01 gives

(eq. I06)

where the complex impedance for a capacitor is also known as the capacitive reactance Zc.  The purely imaginary capacitive reactance in an A.C. driven circuit implies a phase difference between the applied voltage and the measured current.  An ideal or perfect capacitor, therefore, will produce a current that leads by 90° as compared to the applied sine wave voltage, where leading implies a -90° phase difference, as shown in Eq. I06 by the -i.

 

A perfectly resistive A.C. circuit will exhibit a zero phase difference between the source voltage and the measured current thus leading to a purely real reactance of

(eq. I07)

in which is purely resistive A.C. circuit is not frequency dependent.

Thusfar, the discussion has only considered ideal capacitors and resistors.  A real capacitor will always exhibit some loss.  The loss of a capacitor has been traditionally treated as a resistive factor providing the impedance of a real capacitor as

(eq. I08)

A capacitor allows the propagation of an electric field through a dielectric medium.  The ability of the capacitor to hold that charge and maintain a constant electric field between the conductive plates is expressed in the latter imaginary term, hence an ideal capacitor.  However, a real capacitor is constructed with a dielectric material between the plates in which the induced and orientational polarizabilities of the material will couple with the applied electric field.  The coupling may cause the material to absorb some of the energy from the applied electric field and thus dissipate the absorbed energy into other forms of energy, usually thermal.

For example, a polar liquid is the dielectric material which will couple to the electric field through orientational polarization.  Such coupled molecules will rotate to maintain coherence with the field, and in so doing, loose energy to the bulk medium through intermolecular friction with its locally surrounding solvent molecules.  This is the conversion of the absorbed field energy with loss to thermal energy, in which the loss is detected as the real component of the complex impedance.  Vector voltmeters, phase discriminators, and modern lock-in amplifiers will simultaneously measure both the real and imaginary components of the complex impedance, therefore providing the relative degree of capacitance, or charge storage, and resistance, or energetic loss, for a particular dielectric medium and associated real capacitor.

Measuring the complex impedance or V/I provides a direct correlation to the resistance and capacitance, via the reactance, of the cell.  It is also a common practice to measure the complex admittance which is the measurement of I/V, or simply the inverse of the impedance.  This leads to the conductance and susceptance of the cell, in which both are inverses of resistance and the capacitive reactance, respectively.

Solving Eq. I08 for the complex capacitance C will yield

(eq. I09)

where Ix is the real component of the measured current, Iy is the respective imaginary component, and .  Each term in the numerator is a power term, where P = IV = I2R, and the denominator is the angular frequency, , times the square of the impedance, Z2.

Empirical results of the measured capacitance reveal that the measured capacitance C has a more complicated dependence on frequency then a simple 1/ relationship.  In fact, the response of the bulk material, and that of the underlying intra and intermolecular potentials, to the applied electric field are expressed through the measured complex current.  In fact

(eq. I10)

where the resistive component is implied through the real component of the current, I’.  Therefore, the direct measurement of the complex current will lend itself to the resistance and capacitance of a real capacitor/condenser.  The design of such a capacitor, or sample holder, with a liquid dielectric material between the conductive metal plates will measure the direct measurement of the relative permittivity at a particular imposed frequency or

(eq. I11)

or the relative permittivity, , is the measured capacitance C of the dielectric medium at a particular frequency is divided by the measured capacitance C0 of the same condenser/capacitor under vacuum.

Since the ultimate goal of the PDP is to measure the complex dielectric response of peptides and proteins, the complex permittivity is given by

(eq. I12)

where Zr is the impedance of the reference or blank cell, such as water, and the Zs is that of the sample, such as a protein in water.

Substituting Equation I01 into the latter will yield

(eq. I13)

where the sample and reference are measured at a single specific frequency .

The latter currents are measured relative to the applied voltages, therefore the voltages are expressed with only a real component and the expression of the currents will be complex.  This leads to the complex permittivity with separation of the components as

(eq. I14)

The measurement of the complex current of the sample, along with comparison to the reference, will lead to the complex permittivity of the sample.  This will ensure that the capacitor geometry and most fringe effects will be nulled from the permittivity of the sample.  The measured complex permittivity then leads to the anomalous dispersion and absorption as discussed in the Theoretical section, and finally to the orientational polarizability and solvent effects of the dipole within the same section.

Phase I Dielectric Spectrometer

Traditional impedance/dielectric spectrometers were based on bridge circuits, such as the deSauty and Schering bridge, to null out noise and measure the complex capacitance of a sample cell.  Although bridge circuits perform well, they can have a complicated circuit design and have a limited frequency range.  Bridge circuits require a balancing method to null the detected complex current across the bridge.  The method of balancing requires that the bridge be rebalanced for each change in frequency, hence the mass-produced auto-balancing bridge, as that of Agilent Technologies.

Traditional spectrometers have measured the permittivity using two techniques.  The first method introduced a pulse, of arbitrary shape, into the sample and performed a fast-fourier transform (FFT) on the signal versus the original pulse.  This technique is analogous to modern infrared (FT-IR) spectrometers and has the advantage of nearly instantaneous measurements and spectral acquisition.  Although this procedure has had promise, proper amplification and signal processing, as well as noise effects, have made it impractical for most peptide and protein studies.  In the future, we would like to reconsider this method for biopolymer dielectric measurements in our research endeavors.

The second technique imposed a single monochromatic sinuously-varying voltage and detected the response at the imposed frequency.  This technique has proven extremely successful because modern op.amps. can be configured for a specific and predetermined frequency range, thus allowing proper calibration, amplification, and characterization within the said frequency range.  Additionally, focusing the detector on a single frequency allows the digital-signal processor (DSP) circuits to properly filter any noise inherent to or introduced by the circuit.  Essentially, detectors are far more sensitive by focusing on a single sinusoidal frequency.

Overall, the latter method starts a scan at a predetermined frequency and incrementally steps through a range of frequencies.  The complex current is then measured at each incremental frequency.  A computer is used to accumulate the data and plot the respective dielectric spectrum.

Traditional bridge circuits are complicated, especially auto-balancing, and require rebalancing and recalibration at each frequency, therefore bridges will not be used in our research.  The first FFT technique, although promising, will also not be initially employed in our studies because of amplification and calibration issues.  The second incremental technique will be employed in our studies because modern electronic detectors exhibit the sensitivity to design and build an extremely simple circuit to measure the complex capacitive reactance, as in Fig. I01.

Modern electronic detectors, such as vector voltmeters, lock-in amplifiers, and phase discriminators, contain state-of-the-art solid-state instrumental and operational amplifiers (op.amps.) coupled to digital-signal processing (DSP).  Modern op.amps. have much broader frequency response ranges and well calibrated common-mode rejection ratios (CMRR).  The CMRR allows the op.amp. to increase the gain of a input signal without introducing noise into the output, thus allowing op.amps. far greater sensitivity and amplification.  Once the raw analog signal is amplified through modern op.amps., the detector can deploy digital circuits and calculations to pull from the preprocessed signal only the signal of interest, therefore, ignoring all noise in and around the true signal.  The DSP output can be directly fed to a computer for further acquisition, storage, and analysis.

The Ametek 7265 DSP lock-in amplifier will be used as the voltage source and simultaneous measure the complex current.  The 7265 will amplify the signal to better than 100 dB CMRR and then digitally process the amplified signal for a maximum current sensitivity down to 2 fA.  Our initial studies prove that our current range will be on the order of 3 to 300 pA, therefore, we are well within the detection range of the 7265 lock-in amplifier.

In other words, a simple capacitive circuit containing a voltage source of stable sinusoidal frequency, from the 7265, is fed into a custom capacitive sample holder.  The complex current is then detected through the sample by the 7265 lock-in amplifier yielding the complex impedance of the sample cell.  The measurement of the complex current or impedance, in comparison with a second identical vacuum or air capacitive holder, will yield the complex permittivity of the peptide or protein sample.  The molecular dipole and solvent effects of the peptide or protein response from the measured complex permittivity is outlined under the Theoretical and Biological Molecules section.

National Instruments provides LabView, an excellent development suite and programming language specifically designed for laboratory and instrument use.  LabView will be used to create a custom user interface and control panel within the computer to control all aspects of the physical instrument.  LabView will provide an interface to control the frequency range and voltage across the capacitive circuit of a spectrum and acquire the complex current data from the 7265 detector using a GPIB interface.  LabView acquires the real and imaginary current data and stores and processes it in real-time.  The data is processed through real-time statistical calculations and plots.  A screen shot of the user interface is below.

(fig. I02)

where the upper right is a four-decade spectrum of a mica capacitor at room temperature owing to a transition frequency of anomalous dispersion at 14.4 kHz and the bottom right shows a real-time strip chart of the raw data.

The left side of the user interface, split into two tabs, controls all aspects of the physical instrument, as shown in greater detail below.

(fig. I03)

The left tab, or “Lockin Control”, of the control interface manages the 7265 output voltage, amplification gain, sensitivity range, and frequency, as well as other instrument controls.  The bottom of the tab reports any GPIB communication, signal processing, and sensitivity errors from the 7265 with indicator lights.  It also reports current settings of the instrument and the complex current data, as well as its magnitude and units.

The right tab, or “Primary Control”, of the control interface manages the frequency run, either single frequency or an entire spectrum.  It allows for single or multiple decade ranges with a linear or logarithmic mode in which all of the indicators and graphical spectra follow suit.  The interface also allows for control over the steps or resolution of a spectrum, the number of samples for averaging per frequency, and dynamic settling.  Dynamic settling is for the frequency range less than 10 Hz where the circuit and instrument must settle into a stable complex current reading.

The raw data can be fed into a strip chart, as shown below, to allow for a signal vs. noise comparison in real-time.  The numbers above the strip chart report the average and standard deviation for both the real (X in white) and the imaginary (Y in pink) current, as well as the units.

(fig. I04)

LabView also allows the raw data to be fed into a real-time histogram, which is also expressed in complex X, real and white, and Y, imaginary and pink, form, as shown below.  These two visualizations are invaluable for gauging the signal to noise ratios and the responses of various samples and circuit designs.

(fig. I05)

And finally, the user interface allows for the full acquisition and storage of all raw and statistically processed data into in-memory arrays.  The arrays are used for storage onto computer disk and allow for the real-time graphical display of a spectral run, as shown below

.

(fig. I06)

The above complex current-based spectrum shows the distinctive anomalous dispersion in pink and the absorption in white.


As a cool aside, the source code for LabView is completely graphical.  An example of the PDP’s source code is below.

(fig. I07)

The sample holder of the Phase I Dielectric Spectrometer (P1DS) consists of a plexiglass holder with a bored out 1” diameter cylinder, Figure I08.  Both open ends of the holder have cylindrically based bronze inserts to act as the capacitive electrodes, in which the minimum separation of the parallel plate electrodes is less than 1/32”, and adjustable using cylindrical-disk shims.  The sample holder, therefore, will exhibit a volume of approximately 4/10 ml.  O-rings are used to ensure a liquid tight seal and an ultrasonic vibrator is used to shake air bubbles from the holder.

(fig. I08)

The last few months have produced encouraging results from the Phase I Dielectric Spectrometer (P1DS).  Also, many new phenomena have been observed.  The first natural study is on a vacuum or air sample.  The full-range complex admittance is below.

(fig. I09)

As can be seen in Figure I09, both the real and imaginary components of the admittance are well-defined mathematical functions with the expected anomalous dispersion characteristics.  The error is also reported in the above plot, revealing the extreme sensitivity and reproducibility of the P1DS.  The dispersion detected at 62 kHz, however, should not be observed and is probably due to stray or parasitic effects within sample cell, cabling, or amplifiers.  Future spectrometers will require better cabling and specially designed circuits to encompass better guarding and shielding.  A lack of effective guarding was the major cause of the above stray effect.

Most dielectric studies of peptides and proteins will be performed in aqueous solution.  Therefore, the next natural study is the admittance of pure water, as shown below.

(fig. I10)

As can be seen, the admittance magnitudes of water can be three orders of magnitude greater than that of air.  The expected result should be a constant eighty times greater within this frequency range.  The dielectric constant of water is a constant eighty until around 1 GHz.  The peaks reported in Figure I10 are probably due to electrode polarization effects, for the dispersion occurs in the frequency range reported by previous studies.  Electrode polarization will need to be decreased or negated in future spectrometers and techniques.

Hemoglobin was then studied in aqueous solution.  The complex admittance is below in Figure I11.

(fig. I11)

As can be seen the stray admittance of the air sample is far lower than discernable in either the pure water to the aqueous hemoglobin.  The aqueous hemoglobin has a similar dispersion due to electrode polarization than that of pure water.  However, the dispersion is shifted to a lower frequency and the peak is broadened.  This is to be expected for electrode polarization effects for aqueous hemoglobin.

Dividing the complex admittance of the aqueous hemoglobin over that of pure water, or using Equations I14 at each frequency point, the following complex dielectric spectrum is derived for aqueous hemoglobin.

(fig. I12)

where the large dispersion due to electrode polarization is centered at 100 Hz and is still prevalent compared to the admittance spectrum.  An interesting phenomena, however, appears at less than 20 mHz.  A rather obvious absorption peak occurs which is shown in the imaginary permittivity and centered at 4 mHz.  It is characterized by normal dispersion as revealed by its real permittivity.  Normal dispersion has only been generally observed in the infrared, optical, or ultra-violet regions of the electromagnetic spectrum.  Traditionally, normal dispersion is attributed to resonant processes, whereas anomalous dispersion is attributed to relaxation processes.

The data is clear in revealing that the 4 mHz absorption peak is a primary peak and harmonics can be distinguished at approximately 8, 12, and 16 mHz.  This is additionally confirmed in the real dielectric spectrum.  Therefore, the data apparently reveals normal dispersion and harmonics in the region from 4 to 16 mHz, with the primary peak at 4 mHz.  Initial interpretation can attribute the observed data to resonance effects due to intramolecular dielectric responses of structural motifs or to dielectric responses of hydration layers within or directly surrounding the protein.

The ultra-low frequency dielectric spectrum of pure dry flakes of hemoglobin was performed to confirm the above results.  The data is shown in Figure I13.

(fig. I13)

The absorption peak at 4 mHz is apparent, albeit the harmonics are not distinguishable.  The dry studies of hemoglobin would negate any electrode polarization effects, for these are due to mobile charge or ion carriers, such as that found in solution.  Also, the great majority of the immediately external and surrounding hydration layers would not exist, thus the 4 mHz response must be attributed to either a dielectric response of the intramolecular structural motifs or to the response of protein-embedded water.  Therefore, it can concluded that the 4 mHz dielectric response is due to a intramolecular dielectric response of hemoglobin.  Since the frequency resolution of the P1DS is only 1 mHz, the exact location of the primary and harmonic peaks can not be accurately determined.  Further study is required in this frequency region.

Proposed Phase II and III Dielectric Spectrometers

The proposed Phase II and III Dielectric Spectrometers (P2DS and P3DS) will be designed to increase the overall frequency range, and resolution, from 10 uHz to 32 MHz.  The proposal will employ modern electronics and custom circuitry to increase the sensitivity of the current capabilities by an order of at least two.  The proposed spectrometers will be the most sensitive and broadest ultra-low frequency instruments available.

Much of the deleterious effects observed in the P1DS will be reduced or removed in the P2DS and P3DS.  Better mechanical and electronic techniques will be used to properly shield and guard the sample cells to greatly reduce or fully negate stray, parasitic, or transient electronic effects, those usually produced from electronic components.  This alone will reduce noise and thus increase sensitivity.  Better impedance matching techniques will be employed to also reduce noise and acquire a precise impedance signal.

And finally, sample cell design and chemical techniques will be developed to provide far greater control and reproducibility of the protein samples themselves.  Accurate temperature control will be employed by using a combination of fluid and thermoelectric refrigerators.  And, chemical techniques will be employed to greatly reduce electrode polarization and rotationally and translationally suspend the proteins in aqueous solution.

The proposed spectrometers will push the limits of technology and science to detect and characterize the intramolecular dielectric responses of peptides and proteins.  This understanding will propel our biophysical and biochemical understanding of peptides and proteins, and living systems, in general.


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