Fundamental Concepts in Electrical Engineering
Lab3 - AC versus DC Signals
Objectives

- Learn the fundamental physical mechanisms of AC current flow in electric circuits
- Learn the significance of root-mean-square (RMS) and average of a sinusoidal signal
- Construct an experiment and measure the significant characteristics of a sinusoidal signal, rectangular signal, and triangular signal.
- Construct a circuit stimulated by a sinusoidal signal and learn how to apply Ohm's law to the AC voltages and currents in the circuit
- Construct a circuit made of three resistors in series stimulated by a sinusoidal signal and learn how to apply Kirchhoff's Voltage law to the AC voltage drops in this circuit.

Components

- Resistors: 1Ohm (1%, 0.25W) 3 pieces,100Ohms (10%, 0.5W), 560Ohms(10%,0.5W), 820Ohms(10%, 0.5W)
- Circuit prototyping breadboard
- Electric wires

Equipment

- Tektronix TBS 1202B-EDU Oscilloscope
- Function Generator (1Hz - 2MHz)

Theory
So far we have learned about electric current and we have seen how it flows from one terminal of the battery through the external circuit and back into the battery through the other terminal.
This was a continuous current also named DC current. Let's think now at alternating current, named AC current. How do you think the AC current flows in a circuit? How do you think electrons flow in an AC electric current?
An easy way to imagine an alternating current is to look at the battery model shown in the previous figures and consider that the “pump” inside alternates the direction towards it pumps electrons. Like for example it can pump electrons towards the terminal at the bottom for one millisecond and then switch direction and pump electrons towards the terminal at the top for another millisecond, and then continuing like this back and forth one millisecond in each direction. As a result, the electrons through the entire circuit (including the load too) will move back and forth one millisecond each way. This is called alternating current and it is abbreviated AC. The voltage at the terminals of the source changes polarities as the internal “pump” changes the direction in which it pushes electrons. The term “alternating current” is typically used for currents (and voltages) that alternate their intensity following a sinusoidal function. This not always the case; sometimes the variation follows different functions like square wave, triangle wave, or just any arbitrary periodic variation. These time-varying voltages can be visually measured with an oscilloscope.
What is an oscilloscope and how can we measure time varying signals with it?
In a very simplistic way, an oscilloscope is an instrument that can be used to visualize how signals vary in time. The following figure shows the graphical representation of a square wave signal and the image of this signal as shown on the display of an oscilloscope:
We can view the oscilloscope as an instrument that captures successive snapshots of the signal waveform and display them one after another on the screen. The trigger function of the oscilloscope ensures that each snapshot starts at the same location within the signal period so the image on the display stays stable. The time base function of the oscilloscope ensures that each snapshot has the same length in time, thus when sequentially displaying these snapshots the image is clear and stable.
Here is an example of a sinusoidal signal displayed on the TBS1202B-EDU oscilloscope with my annotations in color pink:
The measurements at the bottom of the screen show the minimum and maximum voltage levels, the peak-to-peak magnitude of about 2V, the mean value of a few millivolts, the frequency of about 2MHz, and the RMS voltage of 680mV. The RMS (root mean square) voltage is what we measure with multimeters when we probe time varying signals. Not all oscilloscopes have built-in measurement for RMS voltage, so in many cases we need to use a digital multimeter (DMM) to measure the RMS value; however, DMMs are usually limited in bandwidth so they do not measure accurately the RMS value of high frequency signals. I like this TBS1020B-EDU oscilloscope since all the built-in measurements work up to the full bandwidth of 200MHz.
What is the physical meaning of the RMS value of a time varying signal?
A DMM measures the rms value of a sinusoidal signal, which is equal to the peak voltage multiplied by 0.707.
Here is a clarification for the "rms" (Vpeak*0.707) and also the "average" (Vpeak * 0.636) values of a sinusoidal waveform:
The factors 0.707 and 0.636 result from the following analysis:
0.707 comes from the rms voltage definition. V_rms, is a constant (DC) voltage that produces the same average power dissipation on a resistor as the sinusoidal voltage V_max*sin(ω*t). The power dissipation P = V*I = V^2 / R . So for V_rms, which remember is a DC voltage, it’s easy: P_avg = V_rms^2 / R; however, for a sinusoidal voltage P_avg = ( V_max*sin(ω*t) )^2 / R = [V_max^2 * (sin(ω*t))^2] / R . Since these two average powers have to be equal, it results that: V_rms^2 = V_max^2 * (sin(ω*t))^2. The sinusoidal term is an average over many periods of a sin square function, which from trigonometry or from an intuitive graphic (similar to a sinusoid but shifted up and varying between 0 and 1) has the value of 0.5. Inserting 0.5 in the above equation V_rms^2 = V_max^2 * 0.5, and taking square roots on both sides V_rms = V_max * squareroot(0.5) = V_max * 0.707
0.636 comes form calculating the time average of voltage (not power like in the rms case above) for one half cycle. By integrating the sinusoidal voltage over half of period we obtain V_max * (2/pi) = V_max * (2/3.14) =V_max * 0.636