A Short Note on Hysteresis and Odd Harmonics

Transcription

1 1 A Short Note on yterei and Odd armonic Peter R aoput Senior Reearch Scientit ubocope Pipeline Engineering outon, X 7751 USA pmaoput@varco.com June 19, Abtract hi hort note deal with the exitence of odd harmonic in a nonlinear ytem driven by a hyterei curve. Keyword and phrae: hyterei, harmonic, Fourier erie, half-wave ymmetry. 1 Introduction he knowledge of the frequency pectrum of the output of a nonlinear ytem exhibiting hyterei given a inuoidal input i of importance in applied cience. An example of uch a ituation i found in ferromagnetim, where the magnetization proce exhibit hyterei when a varying magnetic field i applied. If thi varying magnetic field i inuoidal in time, then the quetion arie a to whether the output ignal conit of all harmonic or only odd/even harmonic. In thi hort note, we how that under the aumption that the nonlinear ytem = hyterei curve) poee half-wave ymmetry, only odd harmonic are preent in the frequency pectrum of the output. armonic Repreentation Any piecewie continuou and periodic function or ignal may be decompoed into it harmonic component via expanion into a Fourier erie. o thi end, let f be uch a ignal with period, i.e., ft + ) = ft), for all t. Let ω = π/ the frequency of f). he Fourier erie of f i defined by ft) = a + a ν coνωt) + b ν inνωt).1)

2 Peter R aoput where the Fourier coefficient a ν and b ν are given by a ν = b ν = ft) coνωt)dt, ν =, 1,....) ft) inνωt)dt, ν = 1,,....3) he number νω, ν = 1,,..., are called the harmonic frequencie or harmonic of f. Definition.1. A piecewie continuou and periodic function of period i aid to have half-wave ymmetry if f t + ) = ft) t ).4) he following reult characterize ignal with half-wave ymetry. heorem.1. he ignal f ha half-wie ymmetry if and only if it Fourier erie conit only of odd harmonic. Proof. he neceity follow from the definition of the Fourier coefficient. a ν = ft) coνωt)dt = / ft) coνωt)dt + ft) coνωt)dt / = / / ft) coνωt)dt + ft + /) coνω[t + /])dt = / / ft) coνωt)dt ft) coνωt + νπ)dt {, if ν i even; = / ft) coνωt)dt, if ν i odd. he argument for the Fourier coefficient b ν i proven analogouly and therefore omitted. o how ufficiency, let f be given by it Fourier erie coniting only of odd harmonic. hen ft + /) = = a ν co[ν + 1)ωt + /)] + b ν in[ν + 1)ωt + /)] a ν co[ν + 1)ωt + ν + 1)π] + b ν in[ν + 1)ωt + ν + 1)π] Since co[ν + 1)ωt + ν + 1)π] = coν + 1)ωt and in[ν + 1)ωt + ν + 1)π] = inν + 1)ωt the reult follow. 3 yterei and Odd armonic hi ection briefly review the concept of hyterei and preent ome of the propertie relating to the exitence of odd harmonic in the output ignal.

3 yterei and Odd armonic 3 Definition 3.1. he effect that the repone of a ytem not only depend on it preent tate and configuration but alo on it pat hitory i called hyterei. In particular, the magnetization tate of ferromagnetic material due to a varying magnetic field depend on it pat hitory and lag behind the field. he figure below depict a typical hyterei curve for a ferromagnetic material. he quantity denote the varying magnetic field and the magnetization of the ferromagnetic field due to thi applied field. Denote the value of at which the magnetization attain it minimum and maximum value by and, repectively. Note that the hyterei curve i ymmetric about the origin.) For a more detailed decription about the phenomenon of hyterei we refer the intereted reader to [1] or, if a more mathematical decription i wanted, to [3]. Figure 3.1: A hyterei curve. Clearly, the relationhip between and i decribable by a multi-valued function = ). hi multi-valuedne i reolved, if we parameterize the hyterei curve. One may think of the parameter a decribing the magnetization proce a a function of time. Figure 3. how the hyterei curve a a three-dimenional curve parameterized by time. he projection of the parameterized hyterei curve onto the plane yield a curve a in Figure 3.1. Figure 3.: he hyterei curve a a parameterized three-dimenional curve left) and it projection onto the plane right). Employing the parameterization, the hyterei curve = ) can be decribed a a periodic function in the parameter of period 4 and a uch it i eay to ee that it ha half-wave ymmetry ee Figure 3.3). In light of thi periodicity, reference to value of for negative -value have to be interpreted a follow. ) = + 4 ) = ), ). 3.1) 3

4 4 Peter R aoput Figure 3.3: he half-wave ymmetry of the hyterei curve. Note that equation of the above type make only ene if ) i ingle-valued. Now uppoe that the applied field i of the form t) = in ωt. 3.) In other word, the magnetization i driven by a time-varying inuoidal with frequency ω. herefore, = ) = in ωt) become a function of time. Denote by = π/ω the period of the applied field. hen, ence, we etablihed the following reult. t + /) = in ωt + /)) = inωt + π)) = in ωt) = ) = + ) = ) = t). heorem 3.1. If the input ignal to a hyterei curve = ) i inuoidal with period frequency ω = π/ ) then the output ignal conit entirely of odd harmonic. In Figure 3.4 the output t) of a inuoidal input t) = in ωt to a hyterei curve i hown Figure 3.4: he output of a inuoidal input to a hyterei curve. Regarding the above theorem, a few remark are in order. Remark he above reult alo hold for the magnetic flux denity B. Recall that B = + 4π and ince i inuoidal it ha half-wave ymmetry.. For a inuoidal input whoe amplitude atifie < <, the theorem alo hold. he minor loop of the hyterei originating on the initial magnetization curve have half-wave ymmetry.) 3. heorem 3.1 i true for all other minor loop, provided they exhibit half-wave ymmetry. 4

5 yterei and Odd armonic 5 Reference [1] A. Iványi, yterei odel in Electromagnetic Computation, Akadémiai Kiadó, Budapet, ungary, [] årten Sjötröm, Frequency Analyi of Claical Preiach odel, IEEE ran. ag., Vol. 35, No ), [3] A. Viintin, Differential odel of yterei, Applied athematical Science, No. 111, Springer Verlag, Berlin eidelberg, Germany,