Impedance-Frequency Behavior & Resonance

FREQUENCY

2.3 Impedance-Frequency Behavior & Resonance

Section 3.5) of dimension 4 x 10 mm and separated by 4 mm (the approximate diameter of a standard sample tube)

w oz

~ w

a..~

4. ENDORCOILS AND RELATED RADIOFREQUENCY CIRCUITS 161

electromagnetic wave length at radio- and microwave frequencies, and one must treat the circuit as a distributed network (cf. Murdoch, 1970). The idealized model of the ENDOR coil circuit that was described in the preceding paragraphs does not account for parasitic capacitances that arise between (a) the turns of the wire loop and (b) the loop and the wall of the microwave cavity resonator. These parasitic capacitances are a factor in the mathematical description of the ENDOR coil network, but, as the name suggests, these capacitances are not desired, and their manifestation in the ENDOR spectroscopy experiment is associated with artifacts.

The impedance of an inductor varies linearly as a function of the frequency according to Z

=

^R+iroL, but the parasitic capacitance associated with a wire solenoid requires that the device be described as a parallel RLC network whose Z:fplot resembles Figure 3 (solid line). This plot is linear in the low frequency region, but deviates from the ideal behavior as some critical frequency

Ie

is approached. The impedance of a parallel RLC network is (Bowick, 1982; Nilson, 1983):

z=

R+iwL

iaJCR

+

(1-w²Le) ⁽³⁾

As the frequency of the signal increases from, for example, DC to the rf or microwave region of the spectrum, the impedance, as defined by Equation (3), undergoes a change in the sense that the reactance (complex-valued term) is initially dominated by the inductance, but as the frequency increases the capacitance term becomes larger and the total impedance is then dominated by the capacitance terms. At some critical frequency fe the imaginary part of the impedance vanishes, and this point demarcates the inductance-dominated and capacitance-dominated region of the Z:f plot. A seriesRLC network behaves in the inverse manner; the impedance achieves a minimum value at

I;

This critical frequency where the device, in effect, becomes purely resistive denotes a resonance.

As the impedance of an inductor increases, the power that is transmitted by the device declines, and therefore the radiofrequency field generated by the coil will likewise decline with increasing frequency. In other words, the coil is behaving as a low-pass filter. At resonance, the coil again

transmits

Frequency (MHz)

Frequency(MHz) 410 430

I

^4.0dB

450 470 490

Figure 4. Power transmission profiles of a bandpass filter. On the left is a pass band modelled by a polynomial function. The center of the plateau region corresponds to the center of tuned frequency of the filter, and the pass band corresponds to the frequency region over which the power transmitted is :53 dB relative to the power transmitted at the center frequency. On the right is a power transmission profile through the resonance region ofan 18-turn ENDOR coil.

power and behaves as a bandpass filter in a frequency 'window' near

f

^c- The ENDOR experiment is performed by sweeping through a frequency range, and one would ideally wish that the power and radiofrequency field strength remain very nearly constant during the course of the sweep. Since the pass band of the coil as filter is too narrow (see below), one typically operates the ENDOR coil in the low impedance region of the Z:f plot, where the coil's impedance is determined by the inductanceL in Equation (3). In this region of the operating range of the coil, the slope of the Z:f plot is approximately proportional to the inductance, and it follows that low inductance coils are desired in order to flatten the response profile and extend the useful radiofrequency sweep range.

The ENDOR coil is not an isolated circuit element, and therefore, besides the self-capacitance, there is a second parallel capacitance that arises between the coil and the microwave cavity walls . The impedance-frequency profile of an ENDOR coil changes when it is placed inside a cavity, and direct measurements of the assembled ENDOR probe determined that an additional parallel capacitance is introduced by the interaction between the coil and the grounded cavity walls; it was found that a resonance condition existed in the IS-turn wire coil and a Bruker TM110cavity at approximately 30 MHz (Bender & Babcock, 1992).

The ENDOR coil will behave both as a bandpass filter, owing to its characteristics as a parallel RLCnetwork, and as a low pass filter. Typically,

4. ENDORCOILS AND RELATED RADIOFREQUENCY CIRCUITS 163 however, the coil is operated in the low frequency region of its operating range (as defined by the

Z-J

plot), that is, well below fe, because the pass band of an ENDOR coil at resonance will not correspond to an rfsweep range of a typical spectrum. The power transmission profile of an ideal bandpass filter approximates a rectangular response that may be modelled by a polynomial expansion, as illustrated in Figure 4 (left panel), and the degree to which the pass band mimics a rectangle depends on the order and type of the polynomial function. Filters are typically defined by the polynomial that best approximates their behavior, examples being Bessel, Chebyshev (cf Bowick, 1982; Rhea, 1994), and a pass band is designated by the center frequency plus and minus those frequencies that are attenuated by 3 dB or less.

The ideal bandpass filter power transmission profile may now be compared with that of a TM110cylindrical ENDOR cavity apparatus outfitted with an 18-tum silver wire coil of length 1" (Figure 4, right, and 5). A Hewlett-Packard network analyzer and test set provided the swept frequency signal to the probehead and records the transmitted and reflected power through the device under test (DUT, see Appendix 2), which in this case is the ENDOR cavity assembly. The coil behaves as a classic inductor, as indicated by the steady decline in transmitted power as the frequency increases. Nearly all the power is attenuated at 300 MHz, but near 450 MHz the coil again transmits and behaves as though it were a bandpass filter with its 3 dB band spanning only 1.4 MHz. This narrow region in which the coil behaves as a bandpass filter is unsuitable for swept frequency ENDOR experiments despite some control over tuning the center frequency; if one spoils the coil 's Qby altering the coil geometry(e.g. spacing between turns, height to diameter ratio, etc.) one will still not significantly broaden the pass band.'

The ENDOR coil is therefore operated in a frequency range well below resonance at frequencies where its impedance is low (optimally, near 500).

The power fluctuation across the sweep range may be reduced by lowering the inductance of the coil (designs are reviewed in Section 3), and the optimum impedance attained by matching the transmission line near the center of the frequency sweep range (Section 4). Increasing the amplifier gain offsets the magnetic field strength losses accompanied by loweringL.

For the standard commercial cavity apparatus, that is, a 20-tum silver wire coil that is inserted into a cylindrical TM₁₁₀ cavity, the power transmission

I Large coils denote ENDOR probeheads in which the wan of the microwave cavity resonator constitutes the rf coil. For example, wire-wound TE₁₀₂(Hyde, 1965) or spiral-cut TE_II2 (Gruber et al., 1974) cylindrical walls. Comparing the dimensions of these probeheads with coil design formulae suggest that therfcoil Qis high since the length:diameter ratio is nearly 1:1.

profile (Figure 5) shows that 5 to 15 MHz sweep is accompanied by only a 1

14.0dB I1 .0dB

50 150 250 350 450 12 19 26 33

Frequency (MHz) Frequency (MHz)

Figure 5. Power transmission profiles of a 20-turn silver wire helix that is inserted into a cylindrical TM₁₁₀ cavity. On the left is a broad frequency sweep that shows resonance, whereas on the right is a narrow range sweep that demonstrates how power losses may be significant even on sweeps commonly used in an ENDOR experiment.

dB decline in power, but a 5 to 40 MHz sweep is subject to a 6 dB decline, which can affect the quality of the spectrum. The 5 to 15 MHz sweep is suitable for many IH-ENDOR spectra, and the 1 dB decline in power is sufficiently small that impedance matching is not necessary/ Beyond 20 MHz the transmitted power declines further, but with nearly the same slope, and therefore a corrective impedance match at 25 MHz will ensure that the radiofrequency field is optimal. At 30 MHz and beyond the slope of the power transmission plot steepens and there is a weak resonance because of the parallel capacitance to the cavity wall (Bender & Babcock, 1992).

Beyond 30 MHz one should switch to a lower inductance coil or, if the frequency range of the sweep is small, use the ' same strategy as was employed between 20 and 30 MHz. This approach was used by Benderet al.

(1989) for the IH-ENDOR study of the tyrosyl radical of ribonucleotide reductase; the spectrum spanned 40 MHz, but because the affected protons consisted of both0.-and

f3-

types, the power requirements for observing the ENDOR effect (cf Hyde et al., 1968; Allendoerfer & Maki, 1970) varied and the spectrum was assembled from 10 MHz portions. A full 2-50 MHz sweep was possible with a very low inductance coil, such as a wire loop, but

2Impedance will vary with design; in this example the impedance was approximately 12n^at

15 MHz and may be easily matched by using 1:4 transformers, see Section 4

4. ENDORCOILS AND RELATED RAoIOFREQUENCY CIRCUITS 165 for a given radiofrequency power level one would not have been able to optimize signal intensity of thec-and l3-protons simultaneously.