The Q-factor and peak frequency of resonant phenomena give useful information about the propagation and storage of energy in an electronic system and therefore its electromagnetic compatibility performance. However, the calculation of Q by linear interpolation of a discrete frequency response to obtain the half-power bandwidth can give inaccurate results, particularly if the data are noisy or the frequency resolution is low. We describe a more accurate method that makes use of the Lorentzian shape of the resonant peaks and involves fitting a second-order polynomial to the reciprocal power plotted against angular frequency. We demonstrate that this new method requires less than one quarter the number of frequency points as the linear method to give comparable accuracy in Q. The new method also gives comparable accuracy for signal-to-noise ratios that are approximately 8 dB greater. It is also more accurate for determination of peak frequency. Examples are given both from measured frequency responses and from simulated data obtained by the transmission line matrix method

Clegg, J

Robinson, M P

English

White Rose Research Online

This is a repository copy of Improved determination of Q-factor and resonant frequency bya quadratic curve-fitting method.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/639/Article:Robinson, M P orcid.org/0000-0003-1767-5541 and Clegg, J (2005) Improved determination of Q-factor and resonant frequency by a quadratic curve-fitting method. IEEE Transactions on Electromagnetic Compatibility. pp. 399-402. ISSN 0018-9375 https://doi.org/10.1109/TEMC.2005.847411eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 2, MAY 2005 399[16] S. M. Matytsin, K. N. Rozanov, and N. A. Simonov, “Permittivity mea-surement using slotted coaxial resonator,” in Proc. IEEE 1996 Instru-mentation and Measurement Technology Conf., Brussels, Belgium, Jun.1996, pp. 987–990.[17] M. Y. Koledintseva, K. N. Rozanov, G. Di Fazio, and J. L. Drewniak,“Restoration of the Lorentzian and Debye curves of dielectrics and mag-netics for FDTD modeling,” in Proc. 5th Int. Symp. ElectromagneticCompatibility, Sorrento, Italy, Sep. 2002, pp. 687–692.[18] M. Y. Koledintseva. Algorithm for the Lorentzian susceptibility functionextraction. [Online]. Available: http://www.emclab.umr.edu/pdf/For-mulas_Koledintseva.pdf[19] M. Y. Koledintseva, G. Antonini, J. Zhang, A. Orlandi, K. N. Rozanov,and J. L. Drewniak, “Reconstruction of the parameters of Debye andLorentzian dispersive media using a genetic algorithm,” in Proc. IEEEInt. Symp. Electromagnetic Compatibility, vol. 2, Boston, MA, Aug.2003, pp. 898–903.[20] M. Y. Koledintseva, J. Wu, J. Zhang, J. L. Drewniak, and K. N. Rozanov,“Representation of permittivity for multiphase dielectric mixtures inFDTD modeling,” in Proc. IEEE Int. Symp. Electromagnetic Compati-bility, vol. 1, Santa Clara, CA, Aug. 2004, pp. 309–314.Improved Determination of -Factor and ResonantFrequency by a Quadratic Curve-Fitting MethodM. P. Robinson and J. CleggAbstract—The -factor and peak frequency of resonant phenomena giveuseful information about the propagation and storage of energy in an elec-tronic system and therefore its electromagnetic compatibility performance.However, the calculation of by linear interpolation of a discrete fre-quency response to obtain the half-power bandwidth can give inaccurate re-sults, particularly if the data are noisy or the frequency resolution is low. Wedescribe a more accurate method that makes use of the Lorentzian shape ofthe resonant peaks and involves fitting a second-order polynomial to the re-ciprocal power plotted against angular frequency. We demonstrate that thisnew method requires less than one quarter the number of frequency pointsas the linear method to give comparable accuracy in . The new methodalso gives comparable accuracy for signal-to-noise ratios that are approx-imately 8 dB greater. It is also more accurate for determination of peakfrequency. Examples are given both from measured frequency responsesand from simulated data obtained by the transmission line matrix method.Index Terms—Electromagnetic compatibility (EMC) measurements, in-terpolation, -factor, resonance, resonant frequency.I. INTRODUCTIONResonant phenomena are encountered in the field of electromagneticcompatibility (EMC) when the dimensions of circuit boards, cables,screened enclosures, and other structures are large compared to the fre-quencies of interest. Although the Q-factors of these resonances areoften neglected, they are actually of great significance because theydescribe the energy absorption and hence the height of the peaks inthe frequency response. These are often more important than the exactfrequencies of the resonances.Q is important in the energy-balance ap-proach that Hill et al. take to characterizing the shielding effectivenessof large enclosures [1], while Dawson et al. have extracted peak pa-rameters from frequency responses in order to validate computationalManuscript received May 13, 2004; revised September 1, 2004.The authors are with the Physical Layer Group, Department of Electronics,University of York, York YO10 5DD, U.K.Digital Object Identifier 10.1109/TEMC.2005.847411Fig. 1. Simulated frequency response of electric field strength in a screenedroom, showing how a linear interpolation leads to an overestimate of half-powerbandwidth. (Solid line) Interpolated response. (Dotted line) “True” response.electromagnetic (CEM) models [2]. The Q-factors of the individualmodes are key parameters in the design of stirred-mode chambers andother reverberant environments [3]. Measurements of the changes inQ-factors and resonant frequencies are used to characterize the con-tents of shielded enclosures bymeans of the resonant perturbation tech-nique [4]. In many cases, the data are obtained by either simulation orcomputer-controlled instrumentation, and consist of scalar values ofvoltage, electric field, etc. at discrete frequency points.A simple and well-known method of calculating the Q from a peakin a frequency response is to find the maximum power, divide it bytwo, find the bandwidth at half-power, and divide this into the resonantfrequency. This “traditional” method was well suited to analogue in-strumentation that gives a continuous curve on a display as an output,and to graphical calculation techniques. However, with discrete fre-quency points and numerical calculations it can lead to errors in theresonant frequency, and more so in Q-factor, particularly if the sam-pled frequency points are sparse. This is because it is unlikely that afrequency point will lie exactly on the peak, so the peak power is un-derestimated, the bandwidth overestimated, and the Q is too low. Fur-ther errors come from linear interpolation between points—the methodused in many automated network analyzers (ANAs). This is illustratedin Fig. 1, which shows how the bandwidth is overestimated owing tothe poor frequency resolution. In this example, it is about 85% too high,and the peak frequency is also in error by 21 kHz.A better approach is to use more of the points near the peak to im-prove accuracy. A technique that applies this idea to transmission (S21)measurements of theQ of a cavity is described admirably by Leong andMazierska [5]. Their method involves fitting a circle to complex S21values plotted on a Smith Chart, and removes the effects of cables, con-nectors, and mismatches to give an accurate determination ofQ-factorsin the range 103–107. It is well-suited to precision metrology, in a setupwhere phase information is available. In the field of EMC, however, weoften have to use scalar instruments or deal with data which could havebeen recorded alongside phase information but was not. There are oftenpractical limits to the smallness of the frequency step. In computationalelectromagnetics, results from time-domain simulations are convertedto the frequency domain by Fourier transforms giving discrete points.To improve the resolution means running the model for longer, which0018-9375/$20.00 © 2005 IEEE400 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 2, MAY 2005can often take several hours. So we need a method of improving on thelinear-interpolation method without needing more points.In this correspondence, we describe a quadratic curve-fitting methodof obtaining both Q and resonant frequency, and we compare it withlinear interpolation for measurements and numerical simulation. Weconsider the effects of sparse data and poor signal-to-noise ratio (SNR)on each method.II. CALCULATION OF PEAK PARAMETERSA. Frequency Response of an OscillatorThe standard theory of an oscillator shows that the power P devel-oped in a resonant system such as a tuned circuit or shielded enclosureis given byP /!0Q1!! !!2+1Q(1)where ! is the angular frequency and !0 the peak angular frequency.This is the Lorentzian line shape familiar to spectroscopists. Hence1P= CQ!0!0! !!02+1Q2(2)where C is a constant. At frequencies near resonance!0! !!0 2 1 !!0so1P=CQ!04 1 !!02+1Q2=CQ!04 + 4!2!20  8!!0+1Q2= a!2 + b! + c (3)which is a quadratic in ! with coefficients given bya =4QC!30b =  8QC!20c =4QC!0+CQ!0: (4)Algebraic manipulation of these expressions gives the required reso-nant angular frequency and Q, and also the peak power P0!0 =  b2aQ =124acb2  1 P0 = c b24a 1: (5)A plot of 1=P against ! should therefore be a parabola, and by fittinga second-order polynomial to values of 1=P and ! we can determinetheQ-factor and the resonant frequency fres = !0=2 from (5). For afrequency response describing voltage or electric field, we should plotthe reciprocal of the square against !.Suitable routines for the curve-fitting are provided by many dataanalysis programs such as Matlab [6]. Algorithms are also availablefor those who prefer a do-it-yourself approach, such as the linear leastsquares method described by Press et al. [7]. This method uses singularvalue decomposition of the matrix before solving the linear set of equa-tions for the coefficients of the fitting curve. This is because quite oftenthe matrix can be close to singular and by using the singular value de-composition this problem can be overcome.The question arises of howmany points to include in the curve fittingstage. Empirically we have found that the most effective algorithm is toFig. 2. Data created by selecting every fourth point from the full dataset.Fig. 3. Frequency response modified by the addition of Gaussian noise at30 dB relative to the peak power.start at the maximum power Pm, then in the positive direction continueto include points until a point is reachedwhereP < 0:5Pm, then repeatfor the negative direction. This guarantees that there will be at leastthree data points, which is necessary for fitting a quadratic.The new method takes about 12 times longer to compute the param-eters than does the linear-interpolation method, with the exact differ-ence depending on the number of points between the half-power limits.However in most situations the run-time of either method is likely tobe insignificant compared to the time taken acquiring the data or per-forming the numerical simulations.B. Measured DataTo evaluate the quadratic-fit technique we used measured data fromthe frequency response of a screened room, loaded with “contents” (ac-tually a human subject) in order to reduce the Q to a value typical ofmany EMC situations. The data were obtained with an ANA coupledto small monopole antennas in the roof of the chamber, giving an S21measurement. We extracted 205 points close to the fundamental res-onance of the chamber at approximately 59 MHz. The bandwidth isIEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 2, MAY 2005 401TABLE IMEAN AND RANGE OF -FACTOR AND RESONANT FREQUENCY AS CALCULATED BY THE LINEAR-INTERPOLATION AND QUADRATIC CURVE-FIT METHODS,AS A FUNCTION OF FREQUENCY STEP SIZETABLE IIMEAN AND RANGE OF -FACTOR AND RESONANT FREQUENCY AS CALCULATED BY THE LINEAR-INTERPOLATION AND QUADRATIC CURVE-FITMETHODS, AS A FUNCTION OF SIGNAL-TO-NOISE RATIO (REFERENCED TO PEAK POWER)approximately 84 kHz, and the Q is therefore 700. The linear-inter-polation and quadratic-fit methods were implemented in Matlab [6],using the function “polyfit.” This gives the coefficients of the quadraticby means of a least mean squares algorithm. With a small frequencystep of 1.25 kHz and high SNR, this initial data yields similar valuesof peak frequency and Q for the two methods.To investigate the effect of increasing the frequency step, we “de-populated” the initial data by picking every nth value. An example isshown in Fig. 2, in which every fourth point has been chosen, thus in-creasing the step to 5 kHz. Clearly there are n ways of doing this foreach value of n. We selected n = 1; 2; 4; 8; 16; 32; and 64; and createdup to 16 new datasets for each value. We then used the linear-interpo-lation and quadratic-fit methods to calculate peak frequency and Q.To explore the effect of poor SNR, we took the above datasets, andadded Gaussian noise to each data point, at levels of  60 to  20 dBreferenced to the power at the peak. An example is shown in Fig. 3where the SNR is  30 dB. We repeated the procedure 20 times foreach SNR value. As before we compared the values of peak frequencyand Q as obtained by the two methods.C. Numerical ModelingTo compare the twomethods further, we performed a numerical sim-ulation of the screened room using the transmission line matrix (TLM)method. This is a time-domain simulation which essentially gives theimpulse response of the room; the frequency response may then beobtained by a Fourier transform. For our simulations we modeled anempty room, and increased the losses by making the reflection coeffi-cients of the walls equal to  0.999 rather than  1. This gives a Q ofapproximately 1200, a bandwidth of 49 kHz, and a resonant frequencyof 59.315 MHz.With a grid size of 50 mm, it was necessary to run the model for1:05  106 time steps in order to get a frequency resolution of 11.4kHz. This took 42 h on a computer with an Athlon 2100XP processor.To produce datasets with larger frequency steps, we took the time re-sponse and truncated it, to give durations of 1/2, 1/4, 1/8, 1/16, and1/32 of the original response, and correspondingly larger frequencysteps of up to 370 kHz.We then calculated the peak frequency andQ ofeach frequency response using the linear-interpolation and quadratic-fitmethods.III. RESULTSTable I shows the values of Q-factor as calculated by the linear-in-terpolation and quadratic-fit methods, with the frequency step rangingfrom 1.25–80 kHz. For each method the spread of Q values increaseswith frequency step. However the range of values for the quadratic-fitmethod is much less than for the linear-interpolation method. At a fre-quency step of 80 kHz, which is similar to the bandwidth of the reso-nance (84 kHz), the range is 27 for the improved method but 420 forlinear interpolation. The quadratic-fit method gives comparable accu-racy for frequency step sizes that are four to five times greater.Table I also shows a similar comparison for the peak frequency. Ata frequency step of 80 kHz, the ranges of fres are 73 and 0.5 kHz forthe linear and quadratic methods, respectively. The linear method canonly locate the peak to a precision of plus or minus half the step size,while the quadratic-fit method, which uses more points, can locate thepeak much more precisely.402 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 2, MAY 2005Fig. 4. -factor and resonant frequency of TLM simulation, calculated by thequadratic-fit (o) and linear-interpolation (+) methods, as a function of frequencystep size. (a) -factor. (b) Resonant frequency.Table II shows the values of Q-factor as calculated by the linear-in-terpolation and quadratic-fit methods, as the SNR is increased from 60 to  20 dB relative to the peak power. As might be expected,the spread of Q values increases with SNR for each method. How-ever, the quadratic-fit method is superior in both precision and accu-racy. At an SNR of  30 dB the range of Q values is 220 for thequadratic method and 360 for linear interpolation. We estimate thatthe quadratic-fit method gives comparable ranges in Q-factor for SNRvalues that are 8 dB higher. Furthermore the linear method underesti-mates the bandwidth of the resonance, givingQ values that are too high(mean 746), while the mean of the Q values at an SNR of  30 dB is706, i.e., still close to the “true” value of 701.Table II also shows the effect of increasing SNR on the peak fre-quency. The quadratic-fit method again shows superior performance.The range of values of fres at an SNR of  20 dB is 15 kHz for thequadratic method, which is similar to the range of values for the linearmethod at an SNR of  40 dB.In Fig. 4(a), the Q-factors obtained from the TLM simulation usingthe two methods are plotted against frequency step. Fig. 4(b) showsthe corresponding plots for resonant frequency. It can be seen that forthe linear-interpolation method the values of Q and peak frequencybegin to deviate from their “true” values at a frequency step of approx-imately 15 kHz, while the quadratic-fit method maintains accuracy upto a step size of about 100 kHz. Note that this is approximately twicethe half-power bandwidth of 49 kHz. As the frequency step for TLMis inversely proportional to the run time, these results show that a sim-ulation followed by linear-interpolation would need to be run for six toseven times as long as one using the quadratic method.IV. CONCLUSIONThe quadratic curve-fitting method of obtaining peak parameters isbetter than the simple linear method, because it is less sensitive to poorfrequency resolution and to the effects of Gaussian noise. The needto apply a polynomial-fitting algorithm is not a disadvantage becausethis can be done very quickly on a modern computer. The quadratic-fitmethod will give more accurate values of Q-factor and resonant fre-quency from existing data, and will enable newmeasurements and sim-ulations to be performed with less stringent requirements on frequencyresolution and signal-to-noise.REFERENCES[1] D. A. Hill, M. T. Ma, A. R. Ondrejka, B. F. Riddle, M. L. Crawford, andR. T. Johnk, “Aperture excitation of electrically large, lossy cavities,”IEEE Trans. Electromagn. Compat., vol. 36, no. 2, pp. 169–177, May1994.[2] J. F. Dawson, M. P. Robinson, and T. Konefal, “Computational elec-tromagnetic (CEM) model validation against measured and calculatedresults,” presented at the IEE Seminar on ‘Validation of ComputationalElectromagnetics’, Farnborough, U.K., Mar. 29, 2004, pp. 17–24.[3] D. A. Hill, “A reflection coefficient derivation for the of a reverber-ation chamber,” IEEE Trans. Electromagn. Compat., vol. 38, no. 4, pp.591–592, Nov. 1996.[4] M. P. Robinson, J. Clegg, and D. A. Stone, “A novel method of studyingtotal body water content using a resonant cavity: Experiments and nu-merical simulation,” Phys. Med. Biol., vol. 48, pp. 113–125, 2003.[5] K. Leong and J. Mazierska, “Precise measurements of the Q factor ofdielectric resonators in the transmission mode—Accounting for noise,crosstalk, delay of uncalibrated lines, coupling loss, and coupling reac-tance,” IEEE Trans.Microw. Theory Tech., vol. 50, no. 9, pp. 2115–2127,Sep. 2002.[6] The MathWorks Inc., Matlab Version 6.5, 2002.[7] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-merical Recipes in C—The Art of Scientific Computing. Cambridge,U.K.: Cambridge Univ. Press, 1988.

Improved determination of Q-factor and resonant frequency by a quadratic curve-fitting method

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Improved determination of Q-factor and resonant frequency by a quadratic curve-fitting method

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