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Week 8

Week 8. Complex Numbers. Introduction. To analyze ac circuits in the time domain is not very practical It is more practical to: Express voltages and currents as phasors Circuit elements as impedances Represent them using complex numbers. Introduction. AC circuits

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Week 8

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  1. Week 8 Complex Numbers

  2. Introduction • To analyze ac circuits in the time domain is not very practical • It is more practical to: • Express voltages and currents as phasors • Circuit elements as impedances • Represent them using complex numbers

  3. Introduction AC circuits • Handled much like dc circuits using the same relationships and laws

  4. Complex Number Review A complex number has the form: • a + jb, where j = (mathematics uses i to represent imaginary numbers) • a is the real part • jb is the imaginary part • Called rectangular form

  5. Complex Number Review Complex number • May be represented graphically with a being the horizontal component • b being the vertical component in the complex plane

  6. Conversion between Rectangular and Polar Forms If C = a + jb in rectangular form, then C = C, where

  7. Complex Number Review • j 0 = 1 • j 1 = j • j 2 = -1 • j 3 = -j • j 4 = 1 (Pattern repeats for higher powers of j) • 1/j = -j

  8. Complex Number Review • To add complex numbers • Add real parts and imaginary parts separately • Subtraction is done similarly

  9. Review of Complex Numbers To multiply or divide complex numbers • Best to convert to polar form first • (A)•(B) = (AB)( + ) • (A)/(B) = (A/B)( - ) • (1/C) = (1/C)-

  10. Review of Complex Numbers Complex conjugate of a + jb is a - jb • If C = a + jb • Complex conjugate is usually represented as C*

  11. Voltages and Currents as Complex Numbers • AC voltages and currents can be represented as phasors • Phasors have magnitude and angle • Viewed as complex numbers

  12. Voltages and Currents as Complex Numbers • A voltage given as 100 sin (314t + 30°) • Written as 10030° • RMS value is used in phasor form so that power calculations are correct • Above voltage would be written as 70.730°

  13. Voltages and Currents as Complex Numbers • We can represent a source by its phasor equivalent from the start • Phasor representation contains information we need except for angular velocity

  14. Voltages and Currents as Complex Numbers • By doing this, we have transformed from the time domain to the phasor domain • KVL and KCL • Apply in both time domain and phasor domain

  15. Summing AC Voltages and Currents • To add or subtract waveforms in time domain is very tedious • Convert to phasors and add as complex numbers • Once waveforms are added • Corresponding time equation of resultant waveform can be determined

  16. Important Notes • Until now, we have used peak values when writing voltages and current in phasor form • It is more common to write them as RMS values

  17. Important Notes • To add or subtract sinusoidal voltages or currents • Convert to phasor form, add or subtract, then convert back to sinusoidal form • Quantities expressed as phasors • Are in phasor domain or frequency domain

  18. R,L, and C Circuits with Sinusoidal Excitation • R, L, and C circuit elements • Have different electrical properties • Differences result in different voltage-current relationships • When a circuit is connected to a sinusoidal source • All currents and voltages will be sinusoidal

  19. R,L, and C Circuits with Sinusoidal Excitation These sine waves will have the same frequency as the source • Only difference is their magnitudes and angles

  20. Resistance and Sinusoidal AC In a purely resistive circuit • Ohm’s Law applies • Current is proportional to the voltage

  21. Resistance and Sinusoidal AC • Current variations follow voltage variations • Each reaching their peak values at the same time • Voltage and current of a resistor are in phase

  22. Inductive Circuit • Voltage of an inductor • Proportional to rate of change of current • Voltage is greatest when the rate of change (or the slope) of the current is greatest • Voltage and current are not in phase

  23. Inductive Circuit Voltage leads the current by 90°across an inductor

  24. Inductive Reactance • XL, represents the opposition that inductance presents to current in an ac circuit • XLis frequency-dependent • XL = V/I and has units of ohms XL = L = 2fL

  25. Capacitive Circuits • Current is proportional to rate of change of voltage • Current is greatest when rate of change of voltage is greatest • So voltage and current are out of phase

  26. Capacitive Circuits For a capacitor • Current leads the voltage by 90°

  27. Capacitive Reactance • XC, represents opposition that capacitance presents to current in an ac circuit • XC is frequency-dependent • As frequency increases, XC decreases

  28. Capacitive Reactance • XC= V/I and has units of ohms

  29. Impedance The opposition that a circuit element presents to current is impedance, Z • Z = V/I, is in units of ohms • Z in phasor form is Z •  is the phase difference between voltage and current

  30. Resistance • For a resistor, the voltage and current are in phase • If the voltage has a phase angle, the current has the same angle • The impedance of a resistor is equal to R0°

  31. Inductance • For an inductor • Voltage leads current by 90° • If voltage has an angle of 0° • Current has an angle of -90° • The impedance of an inductor • XL90°

  32. Capacitance • For a capacitor • Current leads the voltage by 90° • If the voltage has an angle of 0° • Current has an angle of 90° • Impedance of a capacitor • XC-90°

  33. Capacitance • Mnemonic for remembering phase • Remember ELI the ICE man • Inductive circuit (L) • Voltage (E) leads current (I) • A capacitive circuit (C) • Current (I) leads voltage (E)

  34. Introduction • Any steady-state voltage or current in a linear circuit with a sinusoidal source is a sinusoid. • This is a consequence of the nature of particular solutions for sinusoidal forcing functions. • All steady-state voltages and currents have the same frequency as the source.

  35. Introduction (cont.) • In order to find a steady-state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency). • Usually, an AC steady-state voltage or current is given by the particular solution to a differential equation.

  36. The Good News! • We do not have to find this differential equation from the circuit, nor do we have to solve it. • Instead, we use the concepts of phasorsand complex impedances. • Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problems.

  37. Phasors • A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current. • Remember, for AC steady-state analysis, this is all we need---we already know the frequency of any voltage or current.

  38. Complex Impedance • Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor). • Impedance is a complex number. • Impedance depends on frequency.

  39. Complex Impedance (cont.) • Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage, and voltage from current.

  40. Sinusoids • Period: T • Time necessary to go through one cycle • Frequency: f = 1/T • Cycles per second (Hz) • Angular frequency (rads/sec): w = 2pf • Amplitude: VM

  41. Example What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid?

  42. Phase

  43. Leading and Lagging Phase x1(t) leads x2(t) by q- x2(t) lags x1(t) by q- On the preceding plot, which signals lead and which signals lag?

  44. Phasors • A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current:

  45. Phasors (cont.) • Time Domain: • Frequency Domain:

  46. Summary of Phasors • Phasor (frequency domain) is a complex number: X = zq = x + jy • Sinusoid is a time function: x(t) = z cos(wt + q)

  47. Complex Numbers • x is the real part • y is the imaginary part • z is the magnitude • q is the phase imaginary axis y z q real axis x

  48. More Complex Numbers • Polar Coordinates: A = z q • Rectangular Coordinates: A = x + jy

  49. Are You a Technology “Have”? • There is a good chance that your calculator will convert from rectangular to polar, and from polar to rectangular. • Convert to polar: 3 + j4 and -3 - j4 • Convert to rectangular: 2 45 & -2 45

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