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Introduction:
Phasor analysis is a useful tool to analyze the sinusoidal steady state behavior of circuits containing capacitors and inductors. In phasor analysis the information about the frequency of the applied signal is suppressed and only magnitude and phase are analyzed. This makes sense since linear circuits (i.e. circuits containing resistors®, capacitors© and inductors (L)) can only affect the magnitude and phase of the input signal and the frequency of the sinusoid remains the same throughout the circuit. Using phasor technique we transform the time domain V-I relations into phasor domain relations in the complex domain. This converts all the differential equations that describe the circuit in time domain to phasor equivalent linear equations in complex frequency domain. In phasor domain all passive elements (R, L and C) are converted into respective impedances Z. All impedances in phasor domain follow the simple relation
V ̃=I ̃Z
Where V ̃ and I ̃ are complex values of phasor voltage and current. The impedance Z is a complex number defined as the ratio between the voltage and current in phasor domain. Since all elements obey this form of Ohm `s law in phasor domain hence all the techniques of resistive circuit analysis are valid for R,L and C circuits in phasor domain. The impedance of resistor is purely real whereas that of capacitors and inductors is purely imaginary. The equivalent impedance of a network of RLC components is a complex number, the real part of such impedance is called “Resistance” and imaginary part is called “Reactance”.
Pre-Lab Task 1:
A phasor quantity (Voltage or Current) is a complex value which consists of two parts; Magnitude and Phase. The magnitude is equal to the amplitude of the sinusoidal wave and phase is equal to the phase shift of the sinusoidal wave from the origin. In the above section it is discussed how to calculate the phase difference between two waveforms. Since we had set the phase shift of the input voltage source equal to zero in the previous section (we left the value blank which by default means a zero value) hence the phase difference is in fact equal to phase shift of the capacitor voltage waveform. Hence we can calculate the phasor magnitude and phase from the waveform generated.
EQUIPMENT AND MATERIALS
1. Digital Oscilloscopes with Probes
2. Digital Multi-meter.
3. Digital Function Generator with Probes.
4. 2.2 uF Capacitor, 2.2mH inductor, Resistors
5. DC power Supply.
SECTION I (LAB TASKS)
Task 1: (Response of C and L for DC and High Frequency)
1. Create the simple RC circuit as shown in Fig No.3 on a breadboard. (Your actual values of R and C might be slightly different. Make sure to find the actual values using RLC meter).
2. Apply 5V dc as V1 using a DC power Supply.
3. Determine the voltage and current through the capacitor using either oscilloscope or a multi-meter.
4. Now apply a sinusoidal signal of very high frequency e.g. 10Khz or more as V1 using the function generator.
5. Now determine the voltage and current through the capacitor using oscilloscope. To determine current determine the voltage through the resistor. Now find the current through the resistor using ohm `s law. Since the elements are in series so the same current flows through the capacitor and the resistor.
6. Now create the simple RL circuit as shown in Fig No.4 on bread board. Repeat the above experiment for this circuit as well.
Task 2: (Phasor Impedance of a Capacitor)
1. Create the simple RC circuit as shown in Fig No.3 on a breadboard.
2. Apply a sinusoidal signal of 5V peak (10 Vpp) with a frequency of 1kHZ as V1.
Voltage Measurements (Magnitude):
3. Determine the magnitude (i.e. peak) of the output voltage across the capacitor using the oscilloscope.
SECTION II (Post Lab Discussion)
1. What is the behavior of Capacitor for DC, what is its behavior for very high frequencies?
2. What is the behavior of Inductor for DC, what is its behavior for very high frequencies?
3. What is meant by impedance? What is meant by admittance? What is meant by Reactance?
4. Why does when a circuit is transformed into phasor domain all the techniques of resistive circuit analysis become valid.
5. What are the impedance and admittance and reactance of R, L and C.?
6. What is the phase difference of a voltage and current across a resistor? Which one is leading?
7. What is the phase difference of a voltage and current across a capacitor? Which one is leading?
8. What is the phase difference of a voltage and current across an inductor? Which one is leading?
9. What is the physical interpretation of phase in time domain?
10. Let the phase difference between two cosine waves of frequency 50Hz is 30 degrees. What would be the corresponding time delay?