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Experiment 8: LRC Circuit
Aim
The aim of this experiment is to measure and calculate the resonance frequency in different ways. Meanwhile, there is a requirement to use the apparatus proficiently. For the last part of the experiment, there is a demand to analyze the phenomenon and get a better understanding. Moreover, from this experiment, we can understand the principle of this experiment and learn some circuit knowledge.

Background/ Theory
In the experiment, in an LRC circuit, a resistor(R), an inductor (L) and a capacitor (C) are required to connect in series. Just like the figure shows. In the experiment, we want to calculate the resonance frequency. We know the current will be maximum when the circuit is driven at its resonance frequency. In addition, the amplitude I0 of the AC current in a series LRC circuit depends on the amplitude V0 of the applied voltage and the impedance Z, which is a measure of the overall opposition of a circuit to the flow of an electrical current as , So we just need to use the apparatus to measure the maximum voltage. As we know, at the resonance frequency, we have XL=XC and the impedance, Z is equal to the resistance R, where Z=R. Because the capacitative and inductive reactances vary with the frequency of the AC current, the impedance of a circuit containing capacitors and inductors also varies with AC frequency. For a circuit with AC current flowing at angular frequency ω, its impedance is given by

where XL = ωL is the inductive reactance, XC = 1/ωC is the capacitative reactance, R is the resistance, and ω = 2π f ( f is the linear frequency).

Apparatus

• PC with DataStudio installed
• Science Workshop 750 USB Interface Box

• Power Amplifier

• Voltage Sensor

• AC/DC Electronics Lab Board

• LCR meter

• Connecting patch cords

Experimental Procedure
The experimental procedure can be divided into three parts: Part I: Using a Frequency Scan to Determine the Resonance Frequency • The first step was to check all the apparatus were there and well. • The Power Amplifier was connected to Analog Channel A and Voltage sensor was connected to Analog Channel B of the Science Workshop 750 USB Interface Box. • The Signal Output of the Power Amplifier was connected to the two banana jacks on the AC/DC Electronics Lab Board and the banana plug patch cords that provided are used to connect. • Prepare a 10 Ω resistor, a 100 μF capacitor and the inductor. They were connected with the Power Amplifier in series on the AC/DC Electronics Lab Board as shown in the figure.

• Analog sensors was accurately added to Channel A and B into DataStudio which was in the computer and set up appropriately. • The following settings were used for the signal generator: • Output: Sine Wave.

• Amplitude: 3.0 V.
• Frequency: 10 Hz.
• Increment factor: 10.
• In the same Scope display, both Voltage Channel B and Output Voltage were added as measured quantities versus time. And the graph was observed. • Through the graph, the amplitude of the voltage across the resistor was easily observed and determined. • The frequency was changed from 10.0 Hz to 270 Hz to repeat the measurement in order to find the resonance frequency. • The 47 μF and 330 μF capacitor was changed to repeat the experimental procedures in order to find their own resonance frequencies. • The resonance frequencies of the capacitors were determined by the Frequency Scan as required.

Part II: Using a Phase Portrait to Determine the Resonance Frequency • The LCR Meter was set to Inductance to measure and record the inductance of the inductor. • The LCR Meter was set to Capacitance to measure and record the capacitances of the 47 μF, 100 μF, 330 μF capacitors. And different numerical values were recorded. • The apparatus was set up on the AC/DC Electronics Lab Board in series as shown in the figure.

• The Scope display was used to monitor Voltage Channel B. And the x-axis was changed to measure Output Voltage. •...
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