Relationships between Frequency, Capacitance, Inductance and ...

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LPC Physics Relationships between f, C, L and X. I o q( t) = ∫ I o sin( 2 π ft) dt = cos( πft) 2πf 2 thus Vc can be written as: I o VC = cos( 2πft) = VC cos( 2πft) . max 2πfC We define the reactance of the capacitor as: Thus, this implies that we can calculate Xc by Eq. 3. X C VCmax X C = Eq. 3 I max ⎛ I o ⎞ 1 = ⎜ ⎟ I o = Eq. 4 ⎝ 2πfC ⎠ 2πfC If an inductor is placed in a series circuit with a sinusoidal voltage supply, then the current is given by: ε o I o = where X L = 2πfL X L In the same way, an inductor can be considered an AC resistor, with effective resistance . X L Proof: The voltage drop across an inductor is given by di V L = L . dt If i( t) = I o sin( 2πft) , then VL = 2πfLI o cos 2πft and we can define the inductive resistance X L by: thus, X L max ( ) VL max X L = Eq. 5 I 2πfLI C = = 2πfL . Eq. 6 I This implies that we can calculate Xc using Eq. 5 by measuring the voltage drop across the inductor and the current through the circuit. We can also calculate X L from Eq. 6 with knowledge of f and L. o 2 of 8
LPC Physics Relationships between f, C, L and X. Experiment and Analysis: Part A 1. a. Construct the circuit shown in Figure 1 below using a 1-µF capacitor for C 1 . Use the LCR meter to determine a "good" value for the capacitance. Next, set the generator frequency to approximately 100 Hz . Adjust the output control for approximately 1- 3 V across the capacitor. Record the current flowing through the capacitor. Note: you may have difficulty using the DMM meter to record the current! It is likely to be rather small. An alternative strategy is to set up the circuit as shown in Figure 2. The resistance should be on the order of 1 / 2πfC . You can then measure the voltage across the resistance box and divide by the resistance (I = V/R) to determine the current. Keep in mind that you must use figure (1) to measure the voltage across the capacitance, and figure (1b) to measure the voltage across the resistor. Also remember not to change the resistance when you measure voltage across the capacitor. Note that the DMM measures RMS values for current: I rms = (Essentially, this means that that I o = the DMM reading multiplied by 2 ). V DMM AC Voltage readings are also rms values, so that V = o . rms I o 2 2 R Resistance Box Function generator + _ ~ C Vc DMM set on AC volts. Figure 1 Measuring the Voltage with the DMM 3 of 8
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