ANALELE STIINTIFICE ALE UNIVERSITATII "AL.I.CUZA" DIN IASI Tomul XLV-XLVI, s. Fizica Stării Condensate, 1999 – 2000, p. 61 – 66.
MODELING RL CIRCUITS. FREQUENCY AND WAVEFORMS-DEPENDENCE OVIDIU CALTUN♣, ALIN APETREI♣ KEYWORDS: RL circuits, magnetic measurements The problem of analysis and modelling the RL circuit becomes a difficulty if the excitation voltage is a non-sinusoidal one. RL circuits with ferrite cores driven with sinusoidal, square and triangular waveform was studied. The Jiles-Atherthon model is applied to the problem of describing the dynamics of circuit driven by a square wave voltage source. The drawbacks of J-A model are discussed.
INTRODUCTION In this paper we shall address the problem of analysis and modelling the apparently simple RL circuit which comprise a linear resistor in series with a nonsaturating and hysteretic inductor driven by a sinusoidal, square or triangular wave voltage source. Many different models of hysteresis [1, 2, 3, and 4] have been proposed to describe the RL or RLC circuits. The model of Jiles-Atherthon with the correct choice of the parameters gives a good agreement with experimental results when the circuit is driven with a sinusoidal voltage source [5]. A modification of the J-A model for high frequency of sinusoidal excitation voltage is discussed in [6]. In order to prove the intricacy of the experimental and theoretical study of nonlinear RL circuits driven by different voltage waveforms at high frequency the experimental and simulation results were compared for four categories of Bi substituted ferrite cores [7]. EXPERIMENTAL For simplicity a RL circuit is chosen. The resistor is linear and includes the ohmic looses in the inductor winding and serves to measure the current across the circuit. The cores are toroidal in shape and are made of Mn-Zn ferrite with different grain size [5]. The outer diameter D, the inner diameter d, the height h, the number of winding in the primary coil and the secondary coils (serves for inductance measurements only) nP and
♣
”Al.I.Cuza” University, Faculty of Physics, Blvd. Carol I, 11, Iasi, 6600, Romania
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nS, are summarized in Table 1. The last column represents the average particle size and the last but one column represents the total resistance of the RL circuit in each case.
I0 (2) Y1
(1)
(2)
Y2
L
RC Power Amplifier
Sine wave Source
Fig. 1 RL circuit, excitation voltage supply and leads to the oscilloscope Table 1 D (mm) 20.8 20.6 20.7 20.8
35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -35 0.0
h (mm) 8.6 8.5 8.5 8.5
nP=nS (turns) 24 24 24 24
R (Ω) 100.5 100.5 100.5 100.5
40 30
Magnetic Field (A/m)
Magnetic Field Strength (A/m)
SX S1 S2 S3
d (mm) 11.0 10.8 10.9 10.9
alt35kHz alt30kHz alt25kHz alt20kHz alt15kHz alt10kHz
-6
2.0x10
-6
4.0x10
-6
6.0x10
-6
8.0x10
-5
1.0x10
Magnetic Field Strength (A/m)
20 10 0
dr40k dr35k dr30k dr35k
-10 -20 -30 -40 0.0
1.0x10
-6
2.0x10
-6
Time (s)
3.0x10
-6
4.0x10
-6
Magnetic Field Strength (A/m)
Sample
Grain size (µm) 20±1.0 10±0.7 40±2.7 25±1.3
30 25 20 15 10 5 0 tri40KHz tri3oKHz tri25kHz tri20kHz tri15kHz tri10kHz
-5 -10 -15 -20 -25 -30 0.0
-6
2.0x10
-6
4.0x10
-6
6.0x10
-6
8.0x10
-5
1.0x10
Time (s)
a) a)
b) c) Fig. 2 Magnetic field strength vs. time sine; b) rectangular alternating pulse; c) sawtooth (triangular) voltages applied in the primary circuit
The voltage waveform across the sensing resistor R=100Ω, proportional to the current and the voltage waveform across the secondary coil proportional to the time derivative of the flux density are stored by oscilloscope. These signals are used to display the magnetization curves [3]. A frequency-variable voltage generated by the synthesised function generator excites the test core. Different voltage waveforms were applied to the primary winding, magnetic flux being generated in the core. The magnetic field strength vs. time for different frequencies and different excitation waveforms are featured in figure 2, while the magnetic flux density rate, magnetic flux density and hysteresis loops are presented
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MODELING RL CIRCUITS. FREQUENCY AND …
1500 1000 500 0 -500 -1000 -1500 -2000 0.0
2.0x10
-6
4.0x10
-6
6.0x10
-6
8.0x10
-6
1.0x10
-5
4000 3000 2000 1000 0 -1000
dr40kHz dr35kHz dr30kHz dr25kHz
-2000 -3000 -4000 0.00
2.50x10
Time(s)
-6
-6
5.00x10
Magnetic Flux Density Rate(T/s)
2000
Magnetic Flux Density Rate(T/s)
Magnetic Flux Density Rate (T/s)
in figs 3, 4 and 5 respectively.
1500 1000 500 0
tri40kHz tri30kHz tri25kHz tri20kHz tri15kHz
-500 -1000 -1500 0.00
-6
-6
2.50x10
Time (s)
-6
5.00x10
7.50x10
Time (s)
a)
b) c) Fig. 3 Magnetic flux density rate vs. time b) sine; b) rectangular alternating pulse; c) sawtooth (triangular) voltages applied in the primary circuit
alt.40kHz alt.35kHz alt.30kHz alt.25kHz alt.20kHz alt.15kHz alt.10kHz
0.0000
-0.0005
-0.0010 0.00
-6
2.50x10
5.00x10
-6
7.50x10
-6
0.0010
0.0005
0.0000
dr40kHz drt35kHz dr30kHz dr25kHz
-0.0005
-0.0010
0.0
-5
1.00x10
-6
-6
4.0x10
2.0x10
Time (s)
Magnetic Flux Density (T)
0.0005
Magnetic Flux Density (T)
Magnetic Flux Density (T)
0.0010 0.0010
0.0005
tri40kHz tri30kHz tri25kHz tri20kHz tri15kHz tri10kHz
0.0000
-0.0005
-0.0010 0.00
-6
2.50x10
5.00x10
Time (s)
-6
7.50x10
-6
1.00x10
-5
Time (s)
a) c)
b) c) Fig. 4 Magnetic flux density vs. time sine; b) rectangular alternating pulse; c) sawtooth (triangular) voltages applied in the primary circuit
0.0004
p40k p35k p30k p25k P20k p15k p10k
0.0000
-0.0004
-0.0008
-30
-20
-10
0
10
20
30
Magnetic Field Strength (A/m)
0.0010
Magnetic Flux Density (T)
0.0010
0.0008
Magnetic Flux Density (T)
Magnetic Flux Density (T)
0.0012
dr40k dr35k dr30k dr25k dr15k
0.0005
0.0000
-0.0005
-0.0010
-30
-20
-10
0
10
20
30
MAgnetic Field Strength(A/m)
0.0005
tri40k tri30k tri25k tri20k tri15k tri10k
0.0000
-0.0005
-0.0010 -30
-20
-10
0
10
20
30
Magnetic Field Strength (A/m)
a)
b) c) Fig. 4 Hysteresis loops d) sine; b) rectangular alternating pulse; c) sawtooth (triangular) voltages applied in the primary circuit When a square waveform of excitation voltage is impressed on the test circuit the shape of the waveform of the magnetic field applied changes. Consequently the waveform of the magnetic flux density rate and the waveform of the magnetic flux density changes. A special problem in recording dynamic hysteresis loops is the exact
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phase relationship of the primary current referred to the induce voltage. The secondary voltage is the derivative of the magnetic flux density, the passing of the voltage trough zero and the maximum of the current should coincide. The applicability of this criterion is facilitated by the saturation of the material. Our program operates the control of the hysteresis loop also for the non-saturation region. THEORETICAL RESULTS In the Jiles-Atherton model the magnetic susceptibility is a function of the magnetization M and the applied magnetic field H [8, 9]:
dM (H ) M an (H e ) − M (H ) dM an (H e ) (1) = (1 − c )δ M +c & dH k (1 − c )sgn H − α[M an (H e ) − M (H )] dH e
( )
where He is the effective field given by: H e = H + αM , (2) and Man is the anhysteretic magnetization, which is considered a Langevin function:
H a H M an (H e ) = M s L e = M s coth e − a a He
(3)
and δ M defined as follows:
0,if H& < 0and M an (H e ) − M (H ) ≥ 0; δ M = 0,if H& > 0and M an (H e ) − M (H ) ≤ 0; 1,otherwise.
(4)
In the above equations a is the form factor for the anhysteretic curve, c is approximately the ratio of the initial susceptibility on the first magnetization curve to the initial anhysteretic differential susceptibility, α is a parameter representing the coupling between domains, k is the pinning constant (it gives a measure of the width of the hysteresis loop) and Ms is the saturation magnetization. All these parameters can be calculated [8] from experimental values of the coercive field (Hc), remanent magnetization (Mr), saturation magnetization (Ms), initial anhysteretic susceptibility ( χ an ), initial susceptibility measured on the first magnetization curve ( χ in ), maximum differential susceptibility ( χ c ) and differential susceptibility at remanence ( χ r ). In Figs. 4 and 5 are presented the experimental and calculated hysteresis loops obtained at the same excitation frequency ν=30kHz and magnetic field strength (maximum value 2A/m) for all the waveform at room temperature (Sample 1). The identification has been performed in each case for all the samples.
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MODELING RL CIRCUITS. FREQUENCY AND …
Fig. 4 Hysteresis loop for Sample 1 at 250C (black) experimental and (gray) calculated calculated sine waveform
Fig. 5 Hysteresis loop for Sample X at 250C (black) experimental and (gray) triangular waveform
Table 1 Jiles-Atherthon model’s parameters for four frequency at room temperature (Sample 1) and sine waveform f MS k k0 kS a S α (kHz) (A/m) (A/m) (A/m) (A/m) (A/m) 40 17800 0.77 0.15 1.40 19.6 1 0.00095 30 14500 0.57 0.10 1.60 20.3 1 0.00098 20 14000 0.36 0.08 1.92 22.5 0.9 0.00112 10 13800 0.29 0.04 2.08 33.0 0.9 0.00143 Table 2 Jiles-Atherthon model’s parameters for four frequency at room temperature (Sample 1) and triangular waveform k k0 f MS kS a S α (A/m) (A/m) (A/m) (A/m) (A/m) (kHz) 40 13000 0.57 0.008 1.62 16.5 1 0.00072 30 13000 0.49 0.004 1.60 18.3 1 0.00087 20 13000 0.46 0.08 1.59 21.5 1 0.00090 10 13000 0.42 0.1 2.00 25.0 0.9 0.00110 CONCLUSIONS The experimental study has demonstrated that the shape of the hysteresis loops depends on the waveform of the excitation voltage. The program realised in Delphi 4 allows the correct plot of the hysteresis loops for particular measurement conditions. The experimental hysteresis was taken in to account in the identification of the parameters of the Jiles-Atherton model. The parameters change for the particular condition as frequency and shape of the waveforms of the excitation. The parameter α as well k as monotonous decreases by increasing the frequency while a, k0, kS increase by increasing the frequency. The hysteresis loops obtained by exciting the primary windings with rectangular alternating pulse can’t be compared with the calculated ones cause of the drawback of Jiles-Atherton model.
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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
J. H. B. Deane, IEEE Trans. on Magn., 30 (5) (1994) 2795. P. Andrei, O. Caltun, Al. Stancu, IEEE Trans. Magn., 34 (1) (1998) 231. N. Schmidt, H. Guldner, IEEE Trans. on Magn., 32 (2) (1996) 489. P. Andrei, Al. Stancu, O. Caltun, J. Appl. Phys., 83(11) (1998) 6359. P. Andrei, O.F. Caltun, C. Papusoi, Al. Stancu, M. Feder, JMMM 196-197 (1999) 362. D. C. Jiles, IEEE Trans. on Magn., 29 (6) (1993) 3990. O. Caltun, M. Feder, P. Andrei, Al. Stancu, An. St. Univ. “Al. I. Cuza”, Tom, XLIII-XLIV, s.l. b.fasc.2 Fizica Solidelor- Fizica Teoretica, 27. D.C. Jiles, J. B. Thoelke, and M.K. Devine, IEEE Trans. on Magn, 28, 1992, p. 29. J.H.B. Deane, IEEE Trans. on Magn., 30-5, 1994, p. 2795.
