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JOURNAL

OF MAGNETIC

RESONANCE

93,27-33

( 199 1)

Efficiency Estimation for Single-Coil, Separate-Input, Double-Tuned NMR Probes ELHADI Centre

NAJIM

AND

JEAN-PHILIPPE

GRIVET

de Biophysique Molkculaire, C.N.R.S., and Dkpartment de Physique, IA Avenue de la Recherche Scientifique, 45071 Orlkans Ckdex Received

June

5, 1990; revised

November

Universitk 2, France

d’Orl6an.x

2 1, 1990

The signal-to-noise ratio for the NMR observation of a nucleus using a double-tuned probe depends on the power efficiency of the observe channel. In a similar manner, the minimum decoupling power is a function of the efficiency ofthe proton channel. Efficiencies are often determined in several steps, which involve measuring electrical characteristics of disassembled probe elements. In this work, we show how efficiencies can be simply computed from measurements of the quality factors of each port, at its resonant frequency. Thus, an index of the probe performance, especially its signal-to-noise ratio, is readily available for the complete. loaded device. o 1991 Academic PKSS,IK

Metabolic NMR experiments often use special-purpose probes in order to observe heteronuclei with convenience. The simultaneous requirements of easy sample access, large volume, efficient proton decoupling, and high sensitivity have led to the development of single-coil, double-tuned resonant circuits. The signal-to-noise ratio for the observation of given nucleus using one channel of a double-tuned probe is proportional to the square root of the efficiency 77of that channel. This quantity is defined as the ratio of the power delivered to the sample coil to the total dissipated power ( 1). The design of a double-tuned probe involves choosing one channel for maximal sensitivity. The efficiency is an indication of the sensitivity advantage conferred to the observed nucleus. Thus, a knowledge of v is important to the designer of NMR probes. For a single-frequency probe, the signal-to-noise ratio is well known to be proportional to the square root of the quality factor Q of the coil (2, 3). This parameter can easily be measured outside of the spectrometer but under conditions (loading by the sample, radiofrequency shielding) completely identical to those that are obtained during the actual NMR experiment. In contrast, in the case of double-frequency operation, only the efficiency can be computed. The following approximate relations (I, 4) have usually been applied for this purpose:

These formulas refer to a type of circuit, to be described in detail below, comprising a high-frequency port (efficiency ~]nr) and a low-frequency port (efficiency qrr). The sample coil has inductance LM and quality factor QM; the parallel tank circuit includes a coil of inductance LB and quality factor QB. In principle, the quality factors must 27

0022-2364/91

$3.00

Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

28

NAJIM

AND

GRIVET

LF

FIG.

1. The electrical

HF

circuit

of a double-tuned,

single-coil,

two-port

NMR

probe.

each be measured at the relevant frequencies. All of the above inductances and Q factors can be measured only on the separate circuits, but not on the complete probe. This model has the serious shortcoming of assuming that all losses reside in the coils. Some authors report that capacitor and lead losses in high-field probes are comparable to coil losses (5). Nonetheless, this model is still useful. The efficiency of a completed probe can also be determined from two NMR experiments. It is equal to the ratio of 90” pulse lengths for the singly tuned probe (the tank circuit being either short-circuited or removed) and the double-tuned probe. It is also possible to measure, for each port, quality factors for the complete, loaded, and doubly-tuned probe; these are designated as Qnr and QLF below. We show that nLr and r)nr are simple functions of QLF and Qur and several other circuit parameters. Since the two double-tuned quality factors are easily determined for the complete loaded probe (6), we believe that they should be preferred when one is testing or setting up a probe before an NMR experiment. In the following, we derive relations between efficiencies and Q factors for a doubly tuned circuit. These are tested on a homemade probe, using samples of varying conductivity. Experimental results are in good agreement with theory. THEORY

The high-frequency port. The circuit of a double-tuned, separate-port probe is shown in Fig. 1. Inductance LB and capacitance C, in parallel make up a tank circuit tuned to the high angular frequency ~nr ( I ). The effective quality factor Qnr at ~nr can be computed in the following manner. On the high-frequency side, point A is effectively grounded; the two inductances LM and LB can be considered to be in parallel (see Fig. 2). Their equivalent resistances & = QM LMtiHF and RB = QsLBmHF are also in parallel. The effective Q is the ratio of the total resistance to the total inductance times UHF; thus

FIG. 2. Equivalent

circuit

for the probe,

at the higher

frequency.

EFFICIENCY

OF

DOUBLE-TUNED

29

PROBES

(bl

(a) FIG. 3. Two stages in the construction of a circuit equivalent plus high-frequency port. as seen from the low-frequency input.

to the parallel tank (b) Circuit equivalent

circuit. (a) The tank to (a).

[21 The efficiency at UHF is now found from [ 1] and [ 21 as 'VHF

=

Q

LB

gLM

+

LB.

This equation can be used to compute VHF, but it contains a parameter, QM , which is sensitive to the nature of the sample and which must be measured on a single-tuned probe. It is then convenient to transform Eq. [ 31 such that only quantities independent of the sample are used. From Eqs. [l] and [ 31 we derive

Q

VHF=

LB

lyzLM+LB.

Except for QnF, Eq. [ 41 uses constant or almost constant quantities. When the sample is changed, LM will change due to the differences in magnetic losses in the sample. These changes are, however, negligible compared to changes in capacitive losses ( 7). Equation [ 41 therefore proves a convenient relation between the easily measured Qur and the useful VHF, assuming that QB can be measured and the model is representative, i.e., negligible lead inductance and capacitor losses. Another form of VHF, useful in many derivations, can be obtained from Eqs. [ 21 and [ 3 1, QHF ‘HF

FIG.

4. The complete

=

QM

probe-equivalent

- QB - QB

circuit,

’

at the lower

frequency

30

NAJIM

AND

GRIVET

provided QM = Qs. Since qnp < 1, it follows that Qnr always lies between QM and QB. If QM = QB, then QHF = QB. The low-frequency port. At the lower angular frequency WLr, the tank circuit in parallel to the high-frequency tuning capacitor c4 is as shown in the scheme in Fig. 3a. A series connection is found to be easier to deal with than the parallel connection of Fig. 3a; we therefore consider instead the circuit of Fig. 3b, where L,, and r,, are to be chosen such that Figs. 3a and 3b are equivalent. By using the two resonance conditions L&&w& = 1 and LMC4w&r = 1, setting x = LB/ LM = CM/C,, and using the approximations Qr, $ 1 and ( wLF/ WHF)’ = k < 1, we find L,, 2: 2 where D = 1 - k( 1 + x). The complete probe-equivalent

;

LBWLF -

Ye4 =

-

QB

1

161

D2’

circuit is shown in Fig. 4. The efficiency is then

?&=

LB&I

1

-&&B

D2

1+--

[71 ’

As can be seen in Fig. 4, the two inductances and the two resistances are now connected in series; the effective quality factor QLr at the lower frequency is then

TABLE

1

Experimental Data and Derived Homemade Double-Tuned Sample

QM High-frequency

None 50 mM 100 mM 150 mM 200 mM

212 191 164 154 142

QM

Q BF

129 117 113 109 103

for a

(?“P>

141

port

192 165 133 123 110

Low-frequency None 50 mM 100 mM 150 mM 200 mA4

Q “F

Values Probe

81.8 83.8 86.3 87.1 88.3 [ 111

(RP>

port 138 133 131 129 126

61.8 63.7 64.3 65.1 66

Note. The quality factors have been measured and the square roots of the efficiencies (%) have been computed according to formulas [4] and [I 1] of the text, for various salt concentrations in the sample. For this probe, L,, = 22.5 nH, LB = 34.5 nH, QB = 254 (high frequency), and Qa = 144 (low frequency).

EFFICIENCY

OF

DOUBLE-TUNED

31

PROBES

89 88 a7 86 85 84 a3 82 81 50

0

150

100

FIG. 5, Square root of the high-frequency efficiency computed using [ 11: inverted triangles. n computed according to [ 51. The curve has been drawn smoothly

as a function of NaCl concentration. Open circles. using [ 31; squares, n according to [4]: triangles, through the open circles.

LM QLF

=

200

+

Le,

+

req

[81

WLF yM

n n

’

Using Eq. [ 61, we finally can write 1 LF= Q

D DLM +LB

t91

.

Following a derivation similar to that leading to Eqs. [ 31 and [ 41, we find

62 61 0

50

100

150

6. Square root of the low-frequency efficiency as a function of NaCl computed using [ 71: triangles, n computed using [lo]: squares, r~ according computed with [ 121. The curve has been drawn through the open circles. FIG.

200 concentration. Open circles, to [I I] ; inverted triangles,

n n

32

NAJIM

AND

Q

GRIVET

DLM

[lOI

‘LF=zDLM+L, 77LF =

Q

1 -

LB

-.JE

DQB LB + DLM ’

[III

Here, again, we can derive an alternative form of qLF: llLF

=

QLF QM -

QB QB

’

1121

This expression can be used to find the upper and lower bounds of Qrr, by again using the fact that qLr < 1. QLr must lie between &a and QM, provided these two values are different; if QB = QM, then QLr = Qrvr. We assume that the low-frequency port is used to observe a heteronucleus, while the high-frequency port is used for proton decoupling. If a lossy sample is introduced in the sample coil, QM is lowered; QLF and Q nr also decrease (see Eq. [ 91). From relations [ 41 and [ 111, we deduce that TLr and nur increase. This conclusion does not imply that the signal-to-noise ratio of the observe channel also increases. Indeed, Doty et al. (I ) have shown that SIN

a

(?ILFQM)“~.

Using the value of qLr given by [lo], we get S/N a (kiyfR)“2. The signal-to-noise ratio thus decreases, even though the efficiency of the loaded probe increases. EXPERIMENTAL

The above formulas have been experimentally verified in the following manner. We have used a double-tuned, single-coil, homemade probe with working frequencies of 300 and 75 MHz. The main coil was (single) tuned at either WLr or wnr and its loaded quality factor QM was measured, using samples of varying NaCl concentration. The characteristics of the isolated tank circuit were then determined. Next, the complete probe was assembled, and the double-tuned quality factors of both channels were measured for each sample. Inductive coupling between the two coils was minimized by keeping these two components as far as possible from each other and shielded. Efficiencies were computed at both frequencies, using the formulas given above. Results are presented in Table 1. These results are also shown in graphical form in Figs. 5 and 6. The characteristic shapes of these curves have been explained (for single frequency operation) by Gadian and Robinson (8). DISCUSSION

AND

CONCLUSION

As can be seen from the figures, formulas [l] and [ 41, for the high-frequency channel, and [I] and [ 111, for the low-frequency channel, are fully equivalent as regards the

EFFICIENCY

OF

DOUBLE-TUNED

33

PROBES

computation of the efficiency factors. This means that the efficiency can be determined for any sample using only a measurement of the double-tuned quality factors. This is much more convenient than the tedious measurements of QM using a partially disassembled probe. It often happens that a sample (for instance, an animal) must be removed from the probe and then replaced. The proposed method affords a convenient means of verifying the reproducibility of the electrical parameters of the loaded probe. REFERENCES 1. F. D. DOTY, R. R. INNERS, AND P. D. ELLIS, J. Mugs. 2. A. ABRAGAM, “The Principles of Nuclear Magnetism,” 3. D. I. HOULT 4. D. I. HOULT, 5 F. D. DOTY,

J. Mugn. Reson. Spectrosc. 12, 41 (1978).

AND R. E. RICHARDS,

Progr.

NMR

T. J. CONNICK,

Reson.

43, 399 ( 1981 ).

Clarendon

24,71

X. Z. NI, AND M. N. CLINGAN,

Press,

Oxford,

I96 I.

( 1976).

J. Magn.

Reson.

6. P. DAUGAARD, H. J. JAKOBSEN, H. R. GARBER, AND P. D. ELLIS, J. Mugn. 7. M. D. HARPEN. Ph.w Med. Biol. 33, 329 (1988). 8. D. G. GADIAN AND F. N. H. ROBINSON, .I. Magn. Reson. 34, 449 (1979).

77, 536 (1988). Reson. 44,224 ( 1981).