Lab Report 4

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P a g e | 1 Iain Copland David Kim Bench 10 2019-03-07 Physics 115 – L1 Instructor: Derek Howell Determination of the internal resistance/inductance of an inductor and their applications as a passive filter Abstract Using an LR circuit, values of time (t) and potential (V) were taken using a set of resistors and LoggerPro. The time constants, τ were calculated using τ = t 2 − t 1 ln ( V 1 V 2 ) A graph of 1/τ and the related resistance from the circuit was created. The reciprocal slope, equal to the inductance of the inductor, 0.841 ± 0.003 H, and the y-intercept, 130 ± 4, were used to calculate the internal resistance of the inductor to be 109 ± 3 Ω. Using these values in f c = R 2 πL the cut-off frequency, f c , for the circuit was determined to be 210 ± 1 Hz. Using simple analysis and reasoning, the LR circuit was determined to act as a low-pass filter.

P a g e | 2 Iain Copland David Kim 2019-03-07 Determination of the internal resistance/inductance of an inductor and their applications as a passive filter Objective 1) To graphically determine the internal resistance and inductance of an inductor from a 1 τ versus R graph. 2) To discuss the presence and significance of the y-intercept from the 1 τ versus R graph. 3) To determine the theoretical cut-off frequency of an LR filter. Introduction An inductor, L, is a fundamental electronic component, alongside the resistor, R . An inductor acts similarly to a capacitor in the sense that a capacitor stores an electric charge, whereas an inductor is able to essentially store current, I . The way an inductor stores a current is due to its tendency to oppose changes in current. Without a change in current, an inductor does not play much role in a typical circuit. When the

P a g e | 3 current in an inductor (in many cases a solenoid) is changed, a magnetic field is created by the coils to induce a current to oppose this change. This induced current establishes a potential difference across the solenoid. [1] As the inductor resists changes in current, this produces an effect in which the inductor gains and releases current over a period, rather than instantaneously. Due to this gradual process, a derivative is involved in determining the potential difference produced across the inductor, V L, given by the equation V L = L dI dt (1) where I is the current, t is the time, and L is the inductance of the inductor. [1] With a resistor and inductor in parallel, as seen above in Figure 6.1, Kirchoff’s loop rule gives the equation V R + V L = 0 (2) where V R is the voltage drop across the resistor (given by V R = IR). Combining and manipulating equations (1) and (2) gives dt L / R = dI I (3)

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P a g e | 4 Considering the circuit from figure 6.1, with the switch left closed, current has an adequate amount of time to build up. When the switch is opened, due to the properties of an inductor, the current will begin to gradually decay. If the time when the switch is opened is considered as time t = 0, and by Ohm’s law, the current at this time, I 0 follows I 0 = V/R. Using this information, equation (3) can be integrated to give I ( t ) = I 0 e ( − t L / R ) = I 0 e ( − t τ ) (4) where τ = L/R is the time constant. The exponent must be unitless, therefore L/R must have units of time, hence it being defined as the time constant, τ. Equation (4) can also be expressed similarly for potential by simply multiplying both sides by a specific constant resistance, giving V ( t ) = V 0 e ( − t τ ) (5) The time constant can be isolated by taking the natural logarithm of both sides and solving algebraically, which gives a simple equation to solve for τ as τ = t 2 − t 1 ln ( V 1 V 2 ) (6) When a resistor and an inductor are placed in series, the created circuit is commonly known as an LR circuit (or an RL circuit). This circuit has many applications, primarily involving the magnetism created in the inductor, however for the purpose of this experiment, only the application as a passive filter is relevant.

P a g e | 5 There are two primary types of passive filters, the low-pass filter (LPF) and the high-pass filter (HPF). Passive filters reduce the amplitude of signals outside a specific range of frequencies to zero (ideally). The primary parameter involved in passive filters is the voltage transfer function in the range of frequencies, H v ( jω ) = V 0 V i (7) where V 0 and V i are potential values measured across either the resistor or the inductor. The subscript v of H v is commonly dropped. [2] In a low-pass filter, the maximum frequency allowed to pass through is called the cut-off frequency (denoted ω c in rads/s). In a practical filter, as the amplitude isn’t immediately zeroed (versus an ideal filter), the cut-off frequency is defined as the frequency at which H(j ω ) is reduced to 1 √ 2 . [2] A high-pass filter acts similarly to a low-pass filter, with the difference that the cut-off frequency is the minimum frequency allowed to pass through, and any above that level will be allowed. Similar to a low-pass filter, the amplitude isn’t immediately zeroed and the cut-off frequency is defined in the same method. [2]

P a g e | 6 The LR Circuit can act as either a low-pass filter or a high-pass filter, dependant on the set up. To act as a low-pass filter, H(j ω ) is measured across the resistor. In a high-pass filter, H(j ω ) is measured across the inductor. The equation in LR circuits used to determine the values for H(j ω ) is the voltage divider formula V 0 = R R + jωL V i (8) (where j is the imaginary unit) which when combined with equation (2) gives H ( j ω ) = V 0 V i = R R + j ωL = 1 1 + j ( ωL R ) (9) Noting that to derive the cut-off frequency gives ωL R ¿ 2 ¿ 1 + ¿ √ ¿ | H ( jω ) | = 1 ¿ (10) Noting from above that the cut-off frequency is given when H(j ω ) = 1 √ 2 gives ω c L R ¿ 2 ¿ 1 + ¿ √ ¿ 1 ¿  ω c L R ¿ 2 = 2 1 + ¿  ω c L R = 1  ω c = R L (11) Since the above is in rads/s, in terms of Hz this gives us the cut-off frequency f c as

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P a g e | 7 f c = R 2 πL (12) [2] For this experiment, the potential is measured across the resistor, therefore the LR circuit acts as a low-pass filter. The theory outlined can be used to determine the inductance of the inductor, which can then be used to determine the cut-off frequency of the low-pass filter created with the circuit. Procedure The first step of our experiment was to collect several different resistors and record accurate values for their resistance using a digital multi-meter. In our case, we collected 9 resistors ranging from 100 – 2000 ohms. A function generator was set up according to the circuit in figure 6.2, however in place of an oscilloscope, we carried out the experiment using LabQuest Mini (alongside LoggerPro v3.15) to be able to record precise values for potential and time.

P a g e | 8 LoggerPro was setup by using the min() function (from the statistics subsection) in a calculated column to treat the value that the graph leveled off at the bottom as our “zero” reference point. This calculated column was then graphed versus the elapsed time. Using this graph, we took 2 values for potential at different time values and entered them directly into Microsoft Excel. We then used excel to automatically calculate the value for the time constant (τ) for each separate resistance (R). Using these values, we created graphs of τ versus R, as well as a graph of 1/τ versus R, which updated constantly as we entered new values. We noticed the trend of our τ versus R graph seemed to follow an inverse relationship, which our other graph proved by showing a linear relationship. Using the slope of the (linear) inverse graph (with an R 2 value of 0.9999), we determined the inductance (L) of our coil to be (1.189±0.004) H. Using equations outlined in the theory, the internal resistance was determined to be (109±3) Ω and the cut-off frequency to be (210±1) Hz. Data Table 1: Resistors, measured using a Digital Multi Meter and uncertainty for that DMM Resistor (Assigned Number) Resistance, R (Ohms) Uncertainty 1 99.790 0.389 2 217.80 0.743 3 329.69 1.079 4 466.20 1.799

P a g e | 9 5 1011.1 3.433 6 1193.7 3.981 7 1481.0 4.843 8 1782.8 5.748 9 681.50 2.445 Table 2: Recorded Data from Logger Pro as outlined Resistance, R Voltage 1, V 1 (V) Voltage 2, V 2 (V) Time 1, t 1 (s) Time 2, t 2 (s) 99.790 7.968 0.515 0.41003 0.42081 217.80 11.135 0.491 0.41594 0.42398 329.69 13.137 0.350 0.40831 0.41531 466.20 14.148 0.408 0.42030 0.42553 1011.1 15.605 0.957 0.41157 0.41366 1193.7 15.226 0.889 0.41340 0.41523 1481.0 15.236 0.860 0.41253 0.41401 1782.8 15.153 0.986 0.41134 0.41256 681.50 14.594 0.909 0.41152 0.41447 Note: All voltage uncertainties = 0.01; all time uncertainties = 0.00001 Results Using the values for the resistance, R, of 99.790 from table 2, an example for equation (6) to calculate the time constant, τ gives τ = t 2 − t 1 ln ( V 1 V 2 ) = 0.42081 − 0.41003 ln ( 7.968 0.515 ) = 0.003936 s ( V 1 V 2 )∗ √ ( δ V 1 V 1 ) 2 + ( δ V 2 V 2 ) 2 V 1 V 2 ln ( V 1 V 2 ) ¿ 2 ( √ ( δ t 1 ) 2 + ( δ t 2 ) 2 t 2 − t 1 ) 2 + ¿ δτ = τ ∗ √ ¿

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P a g e | 10 ¿ ( 0.003936 ) ∗ √ ( √ ( 0.00001 ) 2 + ( 0.00001 ) 2 ( 0.42081 − 0.41003 ) ) 2 +( ( 7.968 0.515 ) √ ( 0.01 7.968 ) 2 + ( 0.01 0.515 ) 2 7.968 0.515 ln ( 7.968 0.515 ) ) 2 = 0.000028 Therefore, the calculated time constant value was determined to be 0.00394 ± 0.00003 s. Repeating this process for all recorded values of resistance gives the following values: Table 3: Calculated values of time constant with uncertainties. Resistance, R (Ohms) Uncertainty, δR Time constant, τ (s) Uncertainty, δ τ 99.8 0.4 0.00394 0.00003 217.8 0.7 0.00258 0.00002 330 1 0.00193 0.00002 466 2 0.00418 0.00001 1011 3 0.00075 0.00001 1194 4 0.00064 0.00001 1481 5 0.00053 0.00001 1783 6 0.00045 0.00001 682 2 0.00106 0.00002 Plotting these data values in a graph of 1 τ versus R gives Figure 1.

P a g e | 11 The slope determined from Figure 1 was 1.189, and the uncertainty determined using the linest function in excel was 0.004. Using the previous definition that τ = L/R the inductance can be determined from the listed slope slope = 1 L  L = 1 slope = 1 1.189 = 0.841 H The y-intercept (with uncertainty from linest) was determined to be 130 ± 4. The y- intercept allows us to determine the internal resistance, R int , of the inductor. ∫ ¿ = 109.33 Ω 1 τ ∗ L = 130 ∗ 0.841 = R ¿ 4 130 ¿ 2 ¿ ( 0.003 0.841 ) 2 + ¿ δR = ( 109.33 ) √ ¿ Therefore, the internal resistance of the inductor is determined to be 109±3 Ω To determine the cut-off frequency using a (theoretical) 1000 ohm resistor, treat the resistor as if it were read with a digital multimeter, which can read 5 digits, therefore R res = 1000.0 ± 3 Ω The total R for the cut-off frequency is given by adding the internal resistance and the theoretical resistor, giving R = R res + R int = 1000.0 + 109 = 1109 Ω δR = √ 3 2 + 3 2 = 4

P a g e | 12 Therefore R = 1109 ± 4 Ω . Substituting our values into equation (11) we get f c = R 2 πL = 1109 2 π ( 0.841 ) = 209.87 Hz 0.003 0.841 ¿ 2 ¿ 4 1109 ¿ 2 + ¿ ¿ δ f c = 209.87 ∗ √ ¿ Therefore, the cut-off frequency for our LR circuit, with a theoretical 1000 ohm resistor is 210 ± 1 Hz.

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P a g e | 13 Figure 1 : Graph of the inverse of the calculated time constant versus the added resistance in an LR Circuit. A function generator was used as an AC voltage source. The graph demonstrates a linear relationship between 1/τ and R (or an inverse relationship between τ and R). It’s slope and y-intercept were determined to be 1.189 and 130 respectively.

P a g e | 14 Discussion The results determined throughout this lab fell in line with all the theory. The inductance of the inductor was determined to be 0.841 ± 0.003 H, which falls in line relatively well with the instructor’s statement that the value would be in the range of one Henry. Other than this, it is difficult to compare the accuracy of this value in any other way. A large majority of other inductors produce much smaller ranges of inductance, erring into the micro or nanohenries. All uncertainties throughout this experiment were relatively small. This precision is largely due to the use of Logger Pro for determining voltage and time values. With the settings used for this lab, an uncertainty for the time values was able to be as little as 0.00001 s, a miniscule amount in comparison to the actual time values. The voltage was also relatively precise, with an uncertainty of 0.01 V. These uncertainties for the raw data led to the 0.36% uncertainty for the inductance value. As the time constant, τ, was defined previously as being equal to L/R, the graph produced of 1/τ versus R should have demonstrated linear qualities, which was clearly visible, notably giving an R 2 value of 0.9999. The y-intercept gave information pertaining to the internal resistance of the inductor. This internal resistance was relevant in calculating the cut-off frequency, as well as further understanding the values of the inductor. The calculated internal resistance of 109 ohms seems reasonable. According to other studies, inductors of around 0.5H possessed an internal

P a g e | 15 resistance of around 30 ohms. It is likely that the solenoid used in this experiment was far less ideal than those used in other professional settings. The calculated cut-off frequency of 210 Hz confirms the hypothesis that the setup of the LR circuit would create a low-pass filter. As the frequency generator was able to create frequencies up to the hundreds of thousands, 210 Hz is (relatively) a very small frequency, and thus is much more suitable as a maximum cut-off frequency than a minimum. It’s notable that this experiment was carried out using frequencies ranging from 20 – 100 Hz, further confirming the calculated cut-off frequency being a maximum value. When the frequency was raised to much higher thresholds, the image produced on logger pro was impossible to read, which also falls in line with the theory behind low-pass filters. Conclusion The experiment carried out was successful in meeting the objectives. The methods were able to accurately develop a graphical representation of the relationship between τ and R. This graph was then useful in determining the inductance of the inductor from the slope to be 0 .841 ± 0.003 H, as well as the internal resistance of the inductor from the y-intercept to be 109 ± 3. These values were then used to determine the theoretical cut-off frequency to be 210 ± 1 Hz. These determined values, alongside the performed experiment, provide an accurate representation of the fundamentals behind LR filters, specifically in regard to the use of an LR circuit as a low-pass filter. With accurate values, it is easy to see the correlation between the inductance, the internal resistance, and the cut-off frequency of this specific circuit. This is obvious as the cut-off frequency is determined from the inductance and internal resistance

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P a g e | 16 themselves. All these values in conjunction facilitate answering whether the specific circuit acted as a high-pass or low-pass filter as well. As the experiment was so successful, with relatively low uncertainties for the final values, no large changes would be required were this experiment to be carried out again. References [1] Knight, Randall D., Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. 4 th Edition., (Pearson, 2016) [2] Najmabadi, Farrokh., ECE60L Lecture Notes: Passive Filters., (University of California San Diego, 2002)

Lab Report 4

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