PURPOSE:

To study the voltage current relationships of a first order system; and to study phase angle measurement techniques.

EQUIPMENT:

• Oscilloscope
• Function Generator
• Multimeter

GENERAL THEORY:

1. Introduction

The voltage and current of a first order system excited by a sinusoidal signal can be represented by two phasors which are not in phase. In this experiment we will study the steady-state response and the phase angle between voltage and current of a first order system.

2. Phasor

The complex voltage or current at a given frequency is characterized by only two parameters, amplitude and phase angle. For example the complex voltage V_s(t) = V_m e^{j(wt + q)} can be exactly defined if V_m and q are known. Therefore, for any linear circuit operating in the sinusoidal steady-state at a single frequency w, every current or voltage may be characterized completely by its amplitude and phase angle; and their complex representation will contain the same factor e^jwt. Thus, it is possible to simplify the complex representation by omitting the factor e^jwt. Also, additional time and effort may be saved if these complex quantities are written in polar form.

Let us define impedance Z as the ratio of the phasor voltage to phasor current. Impedance is a complex quantity having dimensions of ohms. Think of an inductor L as being represented in the time-domain by its inductance L and in the frequency domain by its impedance jwL. The same can be said about C and 1/jwC. Also, note that the impedance of an inductor and capacitor are functions of frequency.

Component	Voltage	Phase	Impedance
R		in phase
C		lead 90
L		lag 90

3. Phase Measurement

Phase comparison of two signals of the same frequency can be made using the dual trace feature of the scopes in the lab. To make the comparison, proceed as follows:

1. Connect the Oscilloscope Probes so that channel 1 is across the input voltage source and that channel 2 is across the resistor. Note: One end of the resistor must be grounded.
2. Set the vertical Ch1 and Ch2 scales so that each waveform is aproximately equal size on the display. Note, this does NOT mean that each channel has the same scale!
3. Set the horizontal scale to obtain a display of slightly more than one period of the waveform.
4. Use the vertical position controls to ground and center each waveforms vertically.
5. Turn on the Cursor and select vertical bars.
6. To Calibrate the Phase, find the period of the reference signal.
7. Use the Cursor to measure the difference between the zero crossings of the two signals. Find the phase by converting the time measurement into a degrees measurement.
8. Note whether the voltage across the resistor Leads or Lags the reference signal.

Pre-Lab:

1. Find the general equation for impedeance at a given phase angle and frequency in a RC ciruit in terms of R and C.
2. Calculate resistor values that will result in phase angles of approximately 30^o, 45^o and 60^o between i and V_S when the source in figure 1 is replaced with a 1KHz sinusoidal source.

PROCEDURE

1. Connect the circuit of figure 1 using a sinusoidal voltage source with an amplitude of 5 Volts.


2. Find the RMS voltage across the source, resistor and capacitor for each of the following frequencies(Hz): 100, 200, 500, 1K, 2K, 5K, 10K.
3. Measure the phase between the source voltage and the circuit current for each of the frequencies listed in step 2. Hint: Make the measurement concurrent with the previous step.
4. Design a simple RC circuit for a specific phase angle at 1 kHz
   1. Use the RLC meter to find the value of your 100 nF capacitor
   2. Find a R value by using the Pre-lab equations for a 30^o phase angle
   3. Find the closest resistors to build the calculated R value
   4. Connect the circuit and measure the actual phase angle
   5. Repeat the above steps for both 45^o and 60^o

Analysis:

1. Make a plot of |V_C/V_IN| and |V_R/V_IN| versus frequency for the collected data.


2. Plot the phase between the source voltage and the circuit current as a function of frequency.


3. Compare the measured phase difference for the calculated RC phase angles and the actual value.


4. Compare the results with those optain in the Advanced PSpice experiment.

**Note: Use a semi-log scale for the frequency axis of all of the plots.