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.
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!!
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
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 !,