1. Study Guide Final

2. Closed book/Closed notes Bring a calculator or use a mathematical program on your computer The length of the exam is the standard 2 hours for a final. – Hopefully, you will not take that long.

3. Conversions Sinusoids – Sine and Cosine Know the transformation between angular frequency (  ) and frequency (f) Know how to calculate the period Phasor Notation Exponential Form Rectangular Coordinates

4. Conversions for Sinusoids A sin(  t +  )A cos(  t +  - 90 o ) - A sin(  t +  )A sin(  t +  + 180 o ) Or A sin(  t +  - 180 o ) - A cos(  t +  )A cos(  t +  + 180 o ) Or A cos(  t +  - 180 o ) A sin(  t +  )A sin (  t +  - 360 o ) Or A sin (  t +  + 360 o ) A cos(  t +  )A cos (  t +  - 360 o ) Or A cos (  t +  + 360 o )

5. Sinusoid to Phasor The sinusoid should be written as a cosine Amplitude or magnitude of the cosine should be positive – This becomes the magnitude of the phasor Angle should be between +180 o and -180 o. – This becomes the phase angle of the phasor.

6. Phasor to Exponential Form The magnitude of the phasor is the magnitude of the coefficient. The phase angle multiplied by j is the exponent of e. B  = B e j 

7. Rectangular to Phasor Amplitude of phasor is equal to the square root of the sum of the square of the real component and the square of the imaginary component. Angle of the phasor is equal to the inverse tangent of the imaginary component divided by the real component

8. Phasor to Rectangular Real component is the product of – the amplitude of the phasor and the cosine of the phase angle. Imaginary component is the product of – the amplitude of the phasor and the sine of the phase angle.

9. Steps to Perform Before Comparing Angles between Signals The comparison can only be done if the angular frequency of both signals are equal. Express the sinusoidal signals as the same trig function (either all sines or cosines). If the magnitude is negative, modify the angle in the trig function so that the magnitude becomes positive. If there is more than 180 o difference between the two signals that you are comparing, rewrite one of the trig functions Subtract the two angles to determine the phase angle.

10. Lagging or Leading v 1 (t) = V m1 sin(  t +   ) v 2 (t) = V m2 sin(  t +   ) If       v 1 (t) leads v 2 (t) and v 2 (t) lags v 1 (t) If       v 2 (t) leads v 1 (t) and v 1 (t) lags v 2 (t)

11. Impedances Phasor Notation Admittances Phasor Notation Z R = R = 1/G Z R = R  0 o Y R = 1/R = G Y R = G  0 o Z L = j  LZ L =  L  90 o Y L =-j/(  L)Y L = 1/(  L)  -90 o Z C = -j/(  C)Z C = 1/(  C)  -90 o Y C = j  CY C =  C  90 o

12. Ohm’s Law in Phasor Notation V = I ZV = I/Y I = V/ZI = V Y

13. Equivalent ImpedancesEquivalent Admittances In Series: Z eq = Z 1 + Z 2 + Z 3 ….+ Z n Y eq = [1/Y 1 +1/Y 2 +1/Y 3 ….+ 1/Y n ] -1 In Parallel: Z eq = [1/Z 1 +1/Z 2 +1/Z 3 ….+ 1/Z n ] -1 Y eq = Y 1 + Y 2 + Y 3 ….+ Y n

14. Voltage Division: Impedances and Admittances The voltage associated with one component in a chain of multiple components in series is V z = [Z/Z eq ] V total V z = [Y eq /Y] V total where V total is the total of the voltages applied across the resistors.

15. Current Division: Impedances and Admittances The current associated with one component in parallel with one or more components is I Z = [Z eq /Z] I total I Z = [Y/Y eq ] I total where I total is the total of the currents entering the node shared by the resistors in parallel.

16. Transformations to Simplify Circuits The relationship between the sources used in Thévenin and Norton transformations is: – V th = Z th I N If Z th has an non-zero phase angle, then the phase angle of V th is equal to the sum of the phase angle of the impedance plus the phase angle of the Norton current source.

17. Circuit Analysis The other circuit analysis techniques that were discussed in the first half of the course can be applied when a circuit contains resistors, capacitors, and/or inductors. – These do not have to be applied when calculating the currents and voltages in the circuits on the final. You should be able to find thes values of these parameters using equivalent impedance, voltage and/or current division, and Thévenin and/Norton transformations.