Presentation on theme: "Complex Numbers for AC Circuits Topics Covered in Chapter 24 24-1: Positive and Negative Numbers 24-2: The j Operator 24-3: Definition of a Complex Number."— Presentation transcript:
24-1: Positive and Negative Numbers Our common use of numbers as either positive or negative represents only two special cases. In their more general form, numbers have both quantity and phase angle.
24-2: The j Operator The operator of a number can be any angle between 0° and 360°. Since the angle of 90° is important in ac circuits, the factor j is used to indicate 90°. In Fig. 24-2, the number 5 means 5 units at 0°, the number −5 is at 180°, and j5 indicates the number 5 at the 90° angle. Fig. 24-2
24-3: Definition of a Complex Number The combination of a real and an imaginary term is called a complex number. Usually, the real number is written first. As an example, 3 + j4 is a complex number including 3 units on the real axis added to 4 units 90° out of phase on the j axis. Complex numbers must be added as phasors.
24-4: How Complex Numbers Are Applied to AC Circuits Applications of complex numbers are a question of using a real term for 0°, +j for 90°, and −j for −90°, to denote phase angles. An angle of 0° or a real number without any j operator is used for resistance R. An angle of 90° or +j is used for inductive reactance X L. An angle of −90° or −j is used for X C.
24-5: Impedance in Complex Form The rectangular form of complex numbers is a convenient way to state the impedance of series resistance and reactance. The general form of stating impedance is Z = R ± jX. If one term is zero, substitute 0 for this term to keep Z in its general form. This procedure is not required, but there is usually less confusion when the same form is used for all types of Z.
24-6: Operations with Complex Numbers Real numbers and j terms cannot be combined directly because they are 90° out of phase. For addition or subtraction, add or subtract the real and j terms separately. To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term. To multiply or divide a real number by a real number, just multiply or divide the real numbers, as in arithmetic.
24-6: Operations with Complex Numbers To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel. To multiply complex numbers, follow the rules of algebra for multiplying two factors, each having two terms. To divide complex numbers, the denominator must first be converted to a real number without any j term. Converting the denominator to a real number without any j term is called rationalization.
24-7: Magnitude and Angle of a Complex Number In electrical terms, the complex impedance (4 + j3) means 4 Ω of resistance and 3 Ω of inductive reactance with a leading phase angle of 90°. The magnitude of Z is the resultant of 5 Ω. Finding the square root of the sum of the squares is vector or phasor addition of two terms in quadrature, 90° out of phase. The phase angle of the resultant is the angle whose tangent is 0.75. This angle equals 37°
24-8: Polar Form for Complex Numbers Calculating the magnitude and phase angle of a complex number is actually converting to an angular form in polar coordinates. The rectangular form 4 + j3 is equal to 5 in polar form. In polar coordinates, the distance from the center is the magnitude of the phasor Z. Its phase angle Θ is counterclockwise from the 0° axis. To convert any complex number to polar form, Find the magnitude by phasor addition of the j term and real term. Find the angle whose tangent is the j term divided by the real term.
24-8: Polar Form for Complex Numbers Phasors Expressed in Polar Form Magnitude is followed by the angle. 0 means no rotation. Positive angles provide CCW rotation. Negative angles provide CW rotation. 6 6 6 8.49 6 6 4.24
24-9: Converting Polar to Rectangular Form Complex numbers in polar form are convenient for multiplication and division, but cannot be added or subtracted if their angles are different because the real and imaginary parts that make up the magnitude are different. When complex numbers in polar form are to be added or subtracted, they must be converted into rectangular form.
24-10: Complex Numbers in Series AC Circuits Refer to Fig. 24-10 (next slide). Although a circuit like this with only series resistances and reactances can be solved graphically with phasor arrows, the complex numbers show more details of the phase angles. The total Z T in Fig. 24-10 (a) is the sum of the impedances: Z T = 2 + j4 − j12 = 6 − j8 Convert Z T to polar and divide into V T to determine I.
24-11: Complex Numbers in Parallel AC Circuits A useful application is converting a parallel circuit to an equivalent series circuit. See Fig. 24-11 (next slide), with a 10-Ω X L in parallel with a 10-Ω R. In complex notation, R is 10 + j0 and X l is 0 + j10. Their combined parallel impedance Z T equals the product divided by the sum. Z T in polar form is
24-14: Parallel Circuit with Three Complex Branches Because the circuit in Fig. 24-14 (next slide) has more than two complex impedances in parallel, use the method of branch currents. Convert each branch impedance to polar form. Convert the individual branch currents from polar to rectangular form so they can be added for I T. Convert I T from rectangular to polar form. Z T can remain in polar form with its magnitude and phase angle or can be converted to rectangular form for its resistive and reactive components.