# Modeling Dynamic Hysteresis Curves in Amorphous Magnetic Ribbons
## 2 Measurements  
Measurements of hysteresis curves and related quantities (e.g., coercive field strength and power losses) for a cylinder-shaped commercial Metglas sample (dimensions: inner diameter, d = 40 mm; outer diameter, D = 48 mm; height, h = 10 mm; mass, m = 24 g; density, \(\rho=7180\) kg/m\({}^{3}\); and number of windings, 100/100 for the excitation/measurement circuit) were carried out in accordance with the international standard IEC 60404-6 [16]. In particular, the form factor of the secondary voltage was kept within the prescribed value \(1.111\pm 1\%\). Figure 1 depicts a general overview of the measurement setup used. The measured signals are the voltage drop \(u(t)\) on the resistor \(R\) in the magnetizing circuit of the sample and the electromotive force \(e(t)\) induced in the measurement winding \(z_{\text{p}}\) of the sample. The relationship between instant values of voltage \(u(t)\) and magnetic field strength \(H(t)\) is determined from [PERSON]'s law. \[u(t)=\frac{l\ H(t)\ R}{z_{\text{m}}} \tag{1}\]  
where \(l\) is the magnetic path length in the sample and \(z_{\text{m}}\) is the number of windings in the magnetizing coil, whereas \(R\) is the value of resistance. The relationship between instant values of induced electromotive force \(e(t)\) and induction \(B(t)\) follows from the law of electromagnetic induction and is given as  
Figure 1: A general view of the measurement setup. \[e(t)=z_{\rm p}\ \rm S\ \frac{dB}{d\rm f} \tag{2}\]  
where \(z_{\rm p}\) is the number of windings in the measurement coil, and \(S\) is the cross-section area of the measured sample. The measured signals allow one to determine the quantities and the parameters that characterize magnetic properties of the examined sample. During processing, they are forwarded to input amplifiers, which adjust the levels of analogue input signals to levels required by digital circuits. After adjustment, the signals are subject to sampling in Sample and Hold (S/H) converters; subsequently, they are quantized in the Analogue/Digital (A/D) converter switched to successive measurement channels by the multiplexer. The time dependencies of the measured signals in the digital form are subject to post-processing and, after some computations, the parameters of individual quantities are determined. This is the so-called method of signal representation. The system operates in the on-line mode, which is supervised using a digital computer, which additionally carries out necessary computations and controls the setting of a prescribed magnetization level, the presentation and archiving of the results, etc. In the system, all signal processing operations are carried out in the digital form. Pre-magnetization to a prescribed level is carried out using the block of signal generation with the digital synthesis method. More details (capabilities, uncertainties, etc.) on the measurement setup used in measurements may be found from the producer's website [17]. The main features are a peak magnetic flux density of more than 2 T; peak magnetic field strength of several kA/m (depending on the core dimensions); maximal frequency of 2000 Hz; preservation of the sinusoidal shape of \(B(t)\); and high stability, resolution, and accuracy of measured signals for a wide range of input excitations. Figures 2 and 3 depict hysteresis curves for two levels of an induction amplitude (\(B_{\rm m}=1.2\) T corresponds, approximately, to technical saturation) and for three excitation frequencies. A frequency of \(f=10\) Hz is chosen to mimic quasi-static excitation conditions, and \(f=50\) Hz is the working frequency in most countries worldwide, excluding the Americas; thus, it is of practical importance, whereas \(f=400\) Hz is the highest excitation frequency in the measurement setup for which the form factor could be preserved within the prescribed limits for the examined sample and the sample could be driven up to conditions approaching technical saturation.