# Lab 6: Sinusoidal Steady-State Analysis

## 1. Objectives

By the end of this laboratory session students will be able to:

- Measure the phase shift of a voltage/current
- Use Multisim to simulate steady-state AC circuits

## 2. Background

A sinusoid can be written in the time-domain as

$$v(t)=V_m\mathrm{sin}(\omega t + \theta)$$

where \(V_m\) is the amplitude, \(\omega\) is the frequency in rad/s, and \(\theta\) is the phase angle in degrees. A frequency domain equivalent of this sinusoid can be written in phasor notation as

$$V_m\angle\theta$$

The phase angle describes how much a sine or cosine wave is shifted along the x-axis with respect to another wave, as shown in below. A positive phase angle means the sine wave is leading and a negative phase angle means the sine wave is lagging.

The phase angle can be calculated by measuring the change in period between the two waves, \(\Delta T\), and dividing by the period of the sine wave:

$$\theta = \frac{\Delta T}{T}360^\circ$$

In the frequency domain, resistors have no imaginary component (\(Z_R=R\)) while capacitors (\(Z_C=1/\mathrm{j}\omega C\)) and inductors (\(Z_L=\mathrm{j}\omega L\)) do. Thus, capacitors and inductors shift (i.e., change the phase angle of) sinusoidal signals while resistors do not.

## 3. Laboratory Equipment

- Resistors
- Capacitors
- Oscilloscope
- Function generator

## 4. Procedure

- Assemble the circuit shown in Figure 1.
- Measure and record the component values then build the circuit.
Attach the function generator to \(\mathrm{v_s}\). Press

`Utility`

→`Output Setup`

→`High Z`

→`Done`

to make sure the proper signal is sent from the function generator. This makes the function generator assume it is delivering a signal into a load with infinite resistance. Adjust the function generator so that it outputs a 1 V peak-to-peak (\(\mathrm{V_{pp}}\)) sine wave at 10 kHz.Attach oscilloscope probe 1 across \(\mathrm{v_s}\). Make sure the probe coupling is set to

`AC`

. To do this press the`1`

button and press the top grey button under`Coupling`

until it says`AC`

.Attach oscilloscope probe 2 across \(\mathrm{C_1}\). Display both outputs by pressing

`Auto-Scale`

. Make sure the probe coupling is set to`AC`

.Adjust the offset of both waveforms to 0 V. Do this by turning the knobs directly below the

`1`

and`2`

buttons. A box in the lower-left part of the screen will appear that gives the offset.Record the voltage amplitude across \(\mathrm{C_1}\).

Adjust the horizontal and vertical scaling of each channel such that you see something similar to the figure below (Figure 2). Use the knob in the

`Horizontal`

adjustment section (with right/left arrows underneath) to center the waveform segments. Press`Run/Stop`

. This will make it easier to measure the phase shift.Measure the phase shift by displaying the cursors. Press the

`Cursors`

button. Make sure the`Type`

of measurement is set to`Time`

.Press

`CurA`

and move the A cursor until it is positioned where the signal on`CH1`

crosses the x-axis.Press

`CurB`

and move the B cursor until it is positioned where the signal on`CH2`

crosses the x-axis, just to the right of your first cursor position.Record the \(\Delta\)X value.

Move the probe to the capacitor \(\mathrm{C_2}\). Repeat steps 7–13.

Record the amplitude of \(\mathrm{v_s}\). It will be less than 1 \(\mathrm{V_{pp}}\), so be sure to take this into account during your analysis.

## 5. Results

- Analyze the circuit by hand, using the measured component and voltage source values, and calculate the voltage across each capacitor. Write your answer in phasor notation and in the time domain. Hint: assume \(\mathrm{v_s}\) is a sine wave, so you will need to convert it to cosine.
- Determine the phase angles for \(v_\mathrm{C1}\) and \(v_\mathrm{C2}\). Remember that your measurements are relative to \(\mathrm{v_s}\), which is a sine wave, so you will need to convert them to cosine.
- Compare your measured phase angles for \(v_\mathrm{C1}\) and \(v_\mathrm{C2}\) with the calculated angles. Calculate the percent error for each. Use a table to organize your data. When calculating measured phase shift, do not convert to radians, use Hz.
- Compare your measured amplitudes for \(v_\mathrm{C1}\) and \(v_\mathrm{C2}\) with the calculated amplitudes. Calculate the percent error for each. Use a table to organize your data.
- Model the circuit in Multisim and plot \(\mathrm{v_s}\), \(v_\mathrm{C1}\), and \(v_\mathrm{C2}\) in one graph and include a large, easy to read, screenshot in your report. Compare the measured phase shifts on \(\mathrm{C1}\) and \(\mathrm{C2}\) with the modeled phase shifts. See the MultiSim handout for details on performing AC analysis and measuring phase shift.
- Is the voltage source sinusoid leading or lagging the capacitor voltage?
- Is the voltage across the resistors phase-shifted too? Why or why not?