A. Study of phase difference between voltage and current in series RC and LR circuit B. Study of resonant frequency in parallel LC circuit

1100-turn coil with resistance 2.2kΩ:

             

→ The phase difference between the voltage and the current was about 0 rad. (in phase)

1100-turn coil with resistance 100Ω:

             

→ The phase difference between the voltage and the current was aboutπ/2 rad.

        The voltage leads the current by π/2 rad.

1100-turn coil with resistance100kΩ:

             

→ The phase difference between the voltage and the current was about 0 rad. (in phase)

Disscusion:

1. The channel 1 of the CRO should be connected across both the capacitor and the resistor but not the capacitor only. Because if the cannel 1 of the CRO is connected across the capacitor only, it just shows the trace of VC but not applied voltage, and the trace will not change whether changing the resistance of the resistor.

2. The channel 2 of the CRO could give the current through the capacitor beause the current across the resistor is in phase with that of voltage. By measuring the voltage across the resistor, the current across the capacitor can be determined.

3. The phase differences between the voltage and the current for both RC and LR circuits in steps 5 and 11 were not equal to 90°. It was because the applied voltage V was the sum of VR and VC, and there is always a phase differenceδbetween the applied voltage V and the current I. But since tan δ= VC/VR, It is impossible to have the phase difference 90°.

4. From the results in step 6, it was found that the greater the resistance of resistor is, the smaller the phase difference between the voltage and the current in the capacitor is. The phenomenon is due to the fact that when the resistance of the resistor is increased, the voltage across resistor will be increased while across capacitor decreased. Refer to the diagram before and tan δ= VC/VR, δwill become smaller which mean the phase difference will become smaller. On the other hand, if resistor with smaller resistance is used, the phase difference will become larger.

5. From the results in step 12, it was found that the greater the resistance of resistor is, the smaller the phase difference between the voltage and the current in the inductor is. The phenomenon is because when the resistance of the resistor is increased, the voltage across resistor will be increased while across inductor decreased. Refer to the diagram and tan δ= VL/VR, δwill become smaller which mean the phase difference will become smaller. On the other hand, if resistor with smaller resistance is used, the phase difference will become larger.

Sources of error and improvement:

1. The instruments are not accurate and precise enough. For example, the actual voltage supply is not the same as that stated on the power supply, so the result is not accurate. To improve this, the actual voltage supply should be measured by CRO first.

2. The phase differences were not easily observed by human naked eyes on the CRO. To have a more accurate result, fine adjustments of CRO must be taken to freeze the traces or use camera to capture the display of the CRO.

Conclusion:

        

Through the experiment, the waveforms of voltage and currents in series RC and LR circuits by CRO were observed. And it was found that in RC circuit with small resistance, the current leads the voltage byπ/2 rad, and in LR circuit with small resistance, the voltage leads the current byπ/2 rad. Also, if a resistor with larger resistance is used, the voltage and current would tend to be in phase in both RC and LR circuits.

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Part B:

Objective:

1.         To determine the resonant frequency of a parallel LC circuit by a CRO.

2.         To determine the factors affecting the resonant frequency of the parallel LC circuit.

Procedure and Results:

A. Parallel LC circuit

        

The parallel LC circuit was set up firstly. And the setting of the CRO was adjusted to display some whole waves on its screen.

B. Capacitance dependence of resonant frequency

        The waves shown on the CRO:

           

        The steps were repeated with the other capacitors of different capacitance (C), and the results were tabulated.

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3. Inductance dependence of resonant frequency

After removing one C-core,

             

After removing two C-cores,

           

Disscusion:

1. From the result, it was found that decreasing in capacitance or inductance in a parallel LC circuit would both increase resonant frequency of the circuit. However, the amplitude of the voltage across the circuit would remain unchanged. It was because the voltage is independent to the resonant frequency.

2.    Refer to equation (1) ,  when the inductance remains constant, which mean the smaller the capacitance is, the higher the resonant frequency is . From the experimental results in step 6, the data was found to follow the equation and showed the expected results.

Similarly, refer to equation (1) ,  when the capacitance remains constant, which mean the smaller the inductance is, the higher the resonant frequency is. From the experimental results in step 10, since inductance will decease when core is removed, the data was found to follow the equation and showed the expected results.

Sources of error and improvement:

1. The instruments are not accurate and precise enough. For example, the capacitors may not have the same capacitance as it showed. So it made the trend not accurate. So the capacitance of capacitors should be tested first if it is available. Also , if graphs could be plotted to observe the relationships, the equation could be proved clearer.

2. Since the capacitance substitution box and inductor were not pure and may have some resistance, so some of the voltage may be dissipated by them. And it made the voltages vary in different times. So purer capacitors and inductors should be used.

Conclusion:

The resonant frequency was found from the results with different capacitance and inductance and it was determined that the inductance and capacitance are the factors affecting the frequency of oscillations in the parallel LC circuit.