Day 23 5/10/16: Sinusoids, Phasors and Complex Numbers

All we did in class was lecture on sinusoids, phasors and complex algebra. No labs were done in class this day. We reviewed what sinusoid signals were, which are signals that have the form of a sine of a cosine function. Graphing the sinusoidal signal when given the function, and determining the function when given the graph were reviewed and practiced in a problem. We then reviewed how to switch from sine to cosine functions and vice versa using a phase angle. Polar coordinates were then reviewed and derived, and determining the resultant of two signals using polar coordinates was reviewed and practiced in a problem. Next, phasors were reviewed, which could be written in complex numbers and Euler's identity. Converting from complex numbers to phasor notation and back was also reviewed, and adding/ subtracting/ multiplying/ dividing phasors was reviewed. It was found that adding and subtracting phasors is more efficient using complex numbers, and multiplying / dividing is easier in phasor notation. Two problems applying these concepts were then performed. Another problem adding signals in phasors was also solved. Then, phasor relationships were applied to circuit elements, such as resistors, capacitors and inductors. For resistors, Ohm's law is used to convert voltage signals to time-varying current across the resistor. The voltage and current of a resistor are in phase. For a capacitor, the voltage lags the current by 90 degrees, and the current can be determined using i = c(dv/dt). Using this, the impedance of a capacitor was found to be -j/wC, and the voltage of a capacitor is the impedance of a capacitor times the current. The impedance of a capacitor is imaginary, which means realistically it does not have any impedance. Lastly, for an inductor, the voltage is L(di/dt), and using this, the impedance was found to be jwL. Again, the impedance of a capacitor is imaginary.

LECTURE:

In the above problem, phase shifts and vertical shifts for trig functions were explored, along with the base knowledge of determining an equation for a trig function by analyzing its graph. The first voltage time-varying signal was a sinusoidal wave that has an amplitude of 0.17 V and a vertical shift of 0.03 V. The function also had a period of 0.012 s.

The second voltage function was the same voltage function as the first, with the exception that the sine function is shifted a tenth of a period to the right. Therefore, for a phase shift to the right, the magnitude of the shift must be subtracted from time.

In this example, two voltage signals were added together using phasors. The angle between them was found to be 30 degrees, and the amplitude was determined by using the law of cosines.

The derivation for polar coordinated was solved for in the above picture. The relationships between x, y, r and phi were determined in these derivations.

This example involved adding sinusoidal signals together in phasor notation. First, in order to add the two signals, they were converted to complex numbers from phasor notation. Once the resultant in complex notation was found, the amplitude and the angle were determined using polar coordinates. These values were then used to determine the resultant signal in phasor notation. Then, the inverse of the phasor was determined by using Euler's identity. The inverse could have also been determined if the identity for the inverse of phasor notation was known. The inverse of a phasor is one over the amplitude and negative of the original angle.

In the above picture, a couple of different sinusoidal time-varying signals were re-written in phasor notation. In order to write a sinusoidal signal in phasor notation, it must be first converted into a positive cosine function by including a phase shift of + or - 90 degrees.

In this example, Euler's identity and a complex number were converted to a sinusoidal form. First, the Euler's identity was converted to complex number form, and then the radius and angle were determined from the complex form. These values were then used to determine the cosine function of the phasor. For the complex number, the radius and angle were first determined, and these values were used to find the cosine function of the phasor.

In this problem, the above two voltage signals were added by converting to complex number notation. Then, the radius and angle were determined of the resultant, which were used to find the resultant voltage signal in the trig form.