Hysteresis Report

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Electrical resonance and signal filtering

1 Abstract In this experiment we investigated the following properties of a resonant circuit consisting of an inductor, a capacitor and a resistor (LCR): 1. The voltage response of the LCR to the falling edge of a square wave- We analyzed transients obtained for 5 different resistances, paying particular attention to the change in transient behavior with resistance. 2. The impedance response of the LCR to sinusoidal signals of varying frequency- By identifying the resonant frequency we determined the unknown inductance to be 0.124H. This value was used to calculate the capacitance in the signal filtering circuit we constructed. 3. Using the resonance of the LCR to separate a Morse code signal from electrical noise (signal filtering). 2 Introduction The behavior of an LCR circuit in Figure 1 is one instance of a system present in numerous contexts, such as the damped motion of a mass hanging from a spring. The same form of differential equation characterizes the behavior of both systems, and hence their behavior is essentially the same, though the physical quantities involved are different. Spring-mass system: !!!!!! + !"!" + = where is the position of the mass, m is the mass, b is the damping constant, k is the spring constant and F(t) is the driving force. LCR system: !!!!!! + !"!" + !! = (), where q is the charge on the capacitor, L is the inductance, R is the resistance, C is the capacitance, and V(t) is the input voltage. Hence L, R, 1/C and V(t) are the electrical analogues of m, b, k and F(t).Therefore the LCR circuit is a useful way to study mechanical systems, which can often be difficult to construct and analyze.

2 In order to characterize the given system, we vary the voltage input using certain test inputs. First we determine the voltage response of the system to an impulse input; next we study the impedance response of the system using various sinusoidal inputs. Finally, we use the knowledge gained to build a signal filtering system, which can be used to separate messages from noise picked up during transmission.

3 3 Theoretical background A magnetic material gets magnetized when placed in an external magnetic field. This process involves energy losses, resulting in the material retaining some magnetization even when the external field is turned off. This is the phenomenon of magnetic hysteresis. Crystals of magnetic materials such as iron contain domains of particular magnetizations, separated by domain walls. When an external magnetic field is applied, these walls shift to increase the size of those domains whose magnetization is more favorable to the field. An ordinary piece of iron is polycrystalline, in which each crystal has its own set of magnetic domains. Additionally each crystal contains impurities and imperfections, which are the source of the hysteresis effect. In weak magnetic fields, the domain walls within each crystal reversibly shift very slightly. As the field is made stronger, the domain wall rearrangement is hindered by crystal impurities. Domain walls get stuck at such impurities, and will only move past if the field is raised further. Thus the motion of the domain wall is not smooth, but involves a series of jerks and breaks. As the domain walls quickly shift from one impediment to the next, the changing magnetization produces rapidly varying magnetic fields in the material. These induce eddy currents in the crystal that lose energy by heating the metal. The domain wall movements also alter the dimensions of the crystal, generating small sound waves that dissipate energy. Due to these frictional losses, when the external field goes to zero, the domains do not all return to their original configurations and the iron block gains a net magnetization. Therefore, a graph of B against an alternating H produces a loop. The area contained within the loop gives the energy lost per cycle by a unit volume of the material.

4 4 Experimental background Our primary goal is to observe the hysteretic properties of three materialsmild steel, transformer iron and a Cu/Ni alloy. Consequently, we require a way of plotting the behavior of flux density B within these materials, versus magnetizing force H. This is achieved through the circuit in Figure . The voltages ! and ! are measured by the oscilloscope probes and displayed as ! versus ! on the oscilloscope output. ! = !!! ! = !!! where ! - Number of turns in primary coil ! - Number of turns in secondary coil ! - Cross sectional area of secondary coil ! - Length of primary coil Since these quantities are constants, the graphs of ! against ! and of B against H are equivalent. Accordingly, with the use of appropriate scale factors on the ! and ! axes, the hysteretic energy loss can be calculated as the area enclosed in the loop of the ! -! graph.

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4 Methods and Results 4.1 Building and testing the circuit We built the circuit as in Figure with component values given in Table 1 The integrator is built as in Figure We set the gain at 0.4, considering the following: 1. The integrator should produce an output that is easily measurable on the oscilloscope. 2. The gain should not be large enough to saturate the integrator output given its 15V power supply. A 50Hz sine wave was directly input into the integrator from the signal generator. !" and !"# are displayed simultaneously on the oscilloscope output, and !"# is obtained as a 50 Hz cosine wave. We conclude that the integrator functions as desired at 50 Hz. The impedance ratio of ! to C for this frequency is 47, large enough to stabilize the DC conditions required for the correct functioning of the integrator.

6 4.2 Relationship of and for air cored secondary coil The flux density B and magnetizing force H are related by = !! , where ! generally varies with H. For air !=1, and therefore for an air cored secondary coil ! = !!!!!!! ! = !

4.2.1 Theoretical calculation of As = !!!!!!!!!!!!!!! , we calculate using the values of its constituent quantities. Our measurements of these quantities are tabulated in Table 1

Table 1 Values of quantities required to calculate Quantity Value ! 241 2 10!!! ! 4.314 0.002 10!! ! 400 ! 500 ! 2.2 ! 9.852 0.001 959.7 0.1 Accordingly, we obtain = (6.75 0.06)10!!

4.2.2 Measurement of from oscilloscope output We calculate as the gradient of the ! -! graph displayed on the oscilloscope screen. We obtain = (8.26 0.26)10!!

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4.3 Hysteresis Loops and energy calculations We observe hysteresis in three materials of dimensions tabulated in Table 2. Table 2 Dimension and Area of Samples

Sample Radius/m Cross sectional Area/ Mild steel (0.164 0.001)10!! (84.5 0.1)10!! Transformer iron (30.2 0.4)10!! Cu/Ni alloy (0.255 0.001)10!! (204 16)10!!

Each sample is inserted into the secondary coil, and the hysteresis loop seen on the oscilloscope display is plotted on graph paper with the axes scaled as follows = !! = !! Where ! = !!! = 4160 and ! = !! A is the cross sectional area of the sample inserted into the secondary coil. The values of ! are tabulated below in Table 3. Table 3 values for the three samples

Sample Mild steel 2.24 Transformer iron 6.26 Cu/Ni alloy 0.926

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9 5 Discussion 5.1 Determination of In Section 4.2 we found that our two measurements of did not agree within the bounds of experimental error. We identified the following sources of error-

1. Uncertainty in the Cross sectional area of the secondary coil To calculate the Area of the secondary coil we required measuring its radius. Accordingly we measured the radius of the coil cavity using a pair of Vernier calipers. However, we note that the cross section of the coil probably looks like Figure , and so in measuring the cavity radius we have not actually measured the effective radius of the coil. An improved procedure would have been to measure both the cavity radius and the radius of the entire coil, and include the effect of this range in the error in !. 2. Uncertainty in the value of In our calculation of we used ! = 2.2 , as this was its recorded value. We attempted to verify this in the following ways- First we measured ! on the bridge while the resistor was still hot and found ! = 2.1992 . Second we measured ! in the active circuit with a pair of multimeters to measure the current through it and the voltage across it. This process yielded ! = 2.23 . Thus we found a range in the value of ! which was not taken into account in our calculations. 3. Uncertainty in We took the values of ! and ! to be 400 and 500 respectively. We note, however, that these are nominal values, and expect the true values to be slightly different. Due to the casing around the coils we are unable to directly measure these quantities, and so are unable to calculate a plausible error in them. Of these three errors we believe our uncertainty in determining the radius of the

10 secondary coil to be the most significant error in our calculation of . 5.2 Hysteresis loops and energy calculation In working with our samples of Section 4.3, we noted that , the cross sectional area of the sample (which is used in the calculations that followed), was not the same as !, the cross sectional area of the coil. We justify this by the fact that !(!"#$%&) !(!"#), allowing us to neglect the area of the secondary coil not filled by the sample. The are contained in the hysteresis loops of Figures is estimated by counting squares, and scaling by the energy content of a unit square on the graph. We find mild steel shows the greatest rate of energy loss (36.3 ), followed by transformer iron (15.1 ). For the Cu/Ni alloy shows the least degree of hysteresis, with no hysteresis loop for a sample temperature ~40C, as displayed by the linear graph of Figure . However, once the Cu/Ni alloy was cooled to ~10C by immersion in a beaker of ice water, a relatively small hysteresis loop was seen (Figure ) , whose rate of energy loss is found to be only 241J per cycle, much lower than the values for mild steel or iron.

Using water bath to observe hysteresis change over a few degrees of temperature We do not feel this method is adequate to observe hysteresis variations over the range of a few degrees of temperature for the following reasons

We are unable to measure the sample temperature accurately by measuring the temperature of the water. It would be advisable to use a thermocouple to track the changes in sample temperature

Secondly, we are unable to precisely control the temperature of the water bath well enough to observe hysteresis variations over the small range of a few degrees.

11 5.3 Signal filtering When filtering a signal using the LCR system in Figure 7, we want the noise voltage to be dropped across the 100k resistor, and the message voltage dropped across the LCR. As explained in the Theoretical Background, matching ! to the signal frequency partly achieves this. To produce a LCR network with != 5.0kHz, we kept the original L=0.124H, and used a new 8.2nF capacitance, which was calculated using the equation ! = !!! !" Matching the resonant and message frequencies doesnt cleanly filter the signal, as the noise with frequencies close to 5.0kHz would also come through the LCR. To eliminate this noise we required a LCR with a tall, narrow resonance peak so that the noise frequencies would produce much lower impedance than the message frequency, and hence a much less significant voltage output. By considering the graphs in Figure 6 we decided R=0.5k was the most suitable resistor available.

12 6 Conclusions We set up the LCR circuit in Figure 1 and observed its transient response for 5 resistances. By increasing the resistance we changed the transient from underdamped to overdamped, demonstrating that resistance is indeed the damping effect in this system. The steady state impedance response of this LCR network was investigated for a range of sinusoidal input frequencies. We repeated this using R=1.5k and R=4.0k, and determine that damping reduces the strength of resonance. From the results of the impedance response experiment, it was concluded that the 0.5k resistor produced the sharpest resonance peak. This was deemed a desirable property for the signal filtering system that was designed Section 4.4 We successfully built a signal filtering circuit, matching the resonant frequency to the message frequency at 5.0 kHz and selecting the resistance corresponding to the sharpest resonance peak. This setup separated the Morse code from the noise adequately, such that the message could be detected and decoded. 7 References 1. NST Part IA Physics Practicals Class Manual, Lent term 2011, pp30-35.

13 . Although the magnetization vector alignment is slight for a general weak field, much stronger fields induce a magnetization parallel to the field. When a small magnetic field is applied to a piece of polycrystalline material, the domain walls shift slightly and so domains with more favorable magnetizations grow larger. This growth is reversible as removing the magnetic field would restore the initial magnetization state. However, for stronger fields, the shifting of the domain walls involves interaction energy with the crystals impurities. For particular field strength, the domain wall may get stuck at such an impurity, and can only move past if the field is raised further. Thus the motion of the domain wall is not smooth as for a pure crystal, being rather jerky instead. When a domain wall moves past one impediment, it moves quickly into the next, and so produces rapidly changing magnetic fields inside the material. These changing fields produce eddy currents in the crystal that lose energy by heating the metal. Secondly a domain change alters the dimensions of a crystal, setting up a small sound wave that carries further energy away. As a consequence of these energy losses, when the external field is made zero, all the domains do not return to their initial states, and so the iron block retains some magnetization. This magnetization can be reduced by increasing the external field in the opposite direction. The magnetization follows the curve in Figure . It can be seen that after the material is initially magnetized, it retains magnetization whenever the external field goes to zero. The area under the resulting loop is proportional to the energy loss per unit volume per cycle. Energy losses during the process of magnetization result in retention of magnetization even in when the external field is zero. This phenomenon is magnetic hysteresis. Crystals of magnetic materials such as iron contain domains of particular magnetizations, separated by domain walls. The precise arrangement of domains within a crystal depends on a balance of a number of energy factors involving stresses due to magnetostriction, domain wall energy, and the energy in the materials magnetic field itself. The crystals goal is to reach an energy minimum, and consequently a stable condition. (is explaining the nature of the domain walls really relevant to the main text?) On application of an external field to such a crystal, its domain walls shift to increase the size of domains whose magnetization is more favorable to the field. This alignment is slight for a general weak field, but for much stronger fields the magnetization becomes parallel to the field.

14 Our secondary goal is to evaluate and assess the errors present in a physical measurement. This is done in Section by testing the following relationship between ! and ! . ! = !!!!!!! ! Explanations and derivations of these quantities are given in Appendix We want to look at hysteresis loops, and so we need a way of measuring B and H. The circuit in Fig is used throughout the experiment. Part A and B of the circuit are connected only through the magnetic flux between the primary and secondary coils. The alternating voltage supply creates a time varying magnetic field through the primary coil, which induces an emf in the secondary coil. This voltage passes through the integrator to produce ! , such that ! = !!!!!!!! Similarly the voltage in part A, ! = !!!!!!!!!!! ,

15 And so ! and ! are related through the equation ! = !!!!!!!!!!!!!!! ! The oscilloscope displays ! against ! , which is essentially a graph of B against H, where H is the magnetizing field and B is the magnetic field inside the core of the primary and secondary coils, as ! and ! are proportional to B and H respectively. ! = !!!!! ! = !!! The graphs in Figures 3 and 4 show the systems transition from underdamped to overdamped with increasing resistance.

Underdamped motion

For resistances of 65.1 and 502.1, the system clearly oscillates more than once, allowing us to determine their oscillation periods to be 532.0s and 537.0s. This is in agreement with the theoretical prediction that the period of oscillation should increase with resistance. We later calculated the time periods using the formula = !!"!!! !!! to be 568.6 and 578.2 respectively. The time period was calculated using the picoscope program by taking the difference between two markers placed on successive crests. The placing of the markers could be the source of error which resulted in the ~40s difference between theoretical predictions and experimental readings. A better method would possibly have been to place the markers at points where the oscillation was zero. For R=1.5k, only a single oscillation occurred. The period of this oscillation (467.9s) was much lower than the theoretical prediction of 679.1s. However it is possible that the second zero on the graph was produced by the exponential term going to zero rather than the cosine term. This would result in one underestimating the period as we have done in the experiment. The half-lives decreased with increasing resistance, as expected from the theoretical prediction ! ! = !"#!! . However it is hard to confirm this trend as only three values of ! !were recorded.

Overdamped motion

16 The system is clearly overdamped for resistances of 10k and 100k, with half-life increasing with resistance. As the system was underdamped at R=1.5k and overdamped at R=10k, we conclude that the resistance producing critical damping is in the region 1.5k < R < 10k. We expect LCR systems with low resistances to produce a strong and narrow resonance peak. As the resistance is increased the damping in the LCR becomes more significant, and subsequently its resonance peak becomes shorter and flatter. This effect is clearly illustrated by the three graphs in Figure 6. The LCR network is only lightly damped when R=0.5k, and so it produces a tall and narrow resonance peak. However when the resistance is raised to 1.5k, only a small, flat resonance peak can be observed, shifted to the left of the previous peak. For R=4k, the circuit is so heavily damped that there is no observable peak; in fact the circuit barely resonates and Z /! decreases to zero with increasing frequency. These observations played an important role when we built our signal filtering LCR system, as explained Section 5.3

4.4 Signal filtering A coaxial cable carried a Morse code signal in a modulated carrier wave of frequency 5.0kHz. The signal is buried in electrical noise with a wide frequency spectrum. Therefore to separate the signal from the noise using the LCR resonance, we need ! to be 5.0kHz. To construct this LCR network we used the same inductance L=0.124H, with a new 8.2nF capacitor and a 0.5 k resistor. The LCR network was connected in series with a 100k resistor. The diagram of this setup is shown in Figure 7.

17 Figure 1: Signal filtering LCR system We connected the filter to the coaxial cable and set up the picoscope as instructed on pg34 of the lab manual [1]. We decoded the Morse code using the key on pg35 of the lab manual [1]. The message was MISSION OVER.

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