Chapter 14. The Basic Elements and Phasors

Chapter 14. The Basic Elements and Phasors

1. 14.2 DERIVATIVE

   • The derivative of a sine wave is a cosine wave. The peak value of the cosine wave is directly related to the frequency of the original waveform. The higher the frequency, the steeper the slope at the horizontal axis and the greater the value of dx/dt.

2. 14.4 FREQUENCY RESPONSE OF THE BASIC ELEMENTS

   • Although we would like to think of every element as ideal, it is important to realize that every commercial element available today will not respond in an ideal fashion for the full range of possible frequencies. That is, each element is such that for a particular range of frequencies, it performs in an essentially ideal manner. However, there is always a range of frequencies in which the performance varies from the ideal. Fortunately, the designer is aware of these limitations and will take them into account in the design.

3. 14.5 AVERAGE POWER AND POWER FACTOR

   • The fact that the power curve is always above the horizontal axis reveals that power is being delivered to the load at each instant of time of the applied sinusoidal voltage. This power level is called the average or real power level. It establishes a particular level of power transfer for the full cycle, so that we do not have to determine the level of power to apply to a quantity that varies in a sinusoidal nature.

4. 14.10 MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS

   • Complex numbers lend themselves readily to the basic mathematical operations of addition, subtraction, multiplication, and division. A few basic rules and definitions must be understood before considering these operations. j = raiz(-1)
     • Reciprocal
     • Complex Conjugate
     • Addition
     • Multiplication
     • Division

5. 14.11 CALCULATOR AND COMPUTER METHODS WITH COMPLEX NUMBERS

   • The process of converting from one form to another or working through lengthy operations with complex numbers can be time-consuming and often frustrating if one lost minus sign or decimal point invalidates the solution. Fortunately, technologists of today have calculators and computer methods that make the process measurably easier with higher degrees of reliability and accuracy.
     • Mathcad
     • Calculators

6. 14.12 PHASORS

   • Phasor algebra for sinusoidal quantities is applicable only for waveforms having the same frequency.

7. 14.3 RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT

   • Capacitor: for a capacitor, iC leads YC by 90°, or YC lags iC by 90°.
   • Resistor: for a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law.
   • Inductor: for an inductor, YL leads iL by 90°, or iL lags YL by 90°.

8. 14.9 CONVERSION BETWEEN FORMS

   • Polar to Rectangular
   • Rectangular to Polar

9. 14.13 COMPUTER ANALYSIS

   • PSpice, Multisim

10. 14.6 COMPLEX NUMBERS

    • Complex numbers that, when related to the sinusoidal ac waveform, results in a technique for finding the algebraic sum of sinusoidal waveforms that is quick, direct, and accurate. In the following chapters, the technique is extended to permit the analysis of sinusoidal ac networks in a manner very similar to that applied to dc networks. The methods and theorems as described for dc networks can then be applied to sinusoidal ac networks with little difficulty. A complex number represents a point in a two-dimensional plane located with reference to two distinct axes.

11. 14.7 RECTANGULAR FORM

    • The format for the rectangular form is C = X+jY

12. 14.8 POLAR FORM

    • The format for the polar form is C = < teta