ELECTRIC CIRCUIT ANALYSIS - I

ELECTRIC CIRCUIT ANALYSIS - I Chapter 14 – Basic Elements and Phasors Lecture 17 by MoeenGhiyas

TODAY’S lesson Chapter 14 – Basic Elements and Phasors

Today’s Lesson Contents • Average Power & Power Factor • Complex Numbers • Math Operations with Complex Numbers

AVERAGE POWER AND POWER FACTOR • We know for any load v = Vm sin(ωt + θv) i = Im sin(ωt + θi) • Then the power is defined by • Using the trigonometric identity • Thus, sine function becomes

AVERAGE POWER AND POWER FACTOR • Putting above values in • We have • The average value of 2nd term is zero over one cycle, producing no net transfer of energy in any one direction. • The first term is constant (not time dependent) is referred to as the average power or power delivered or dissipated by the load.

AVERAGE POWER AND POWER FACTOR

AVERAGE POWER AND POWER FACTOR • Since cos(–α) = cos α, • the magnitude of average power delivered is independent of whether v leads i or i leads v. • Ths, defining θ as equal to | θv – θi |, where | | indicates that only the magnitude is important and the sign is immaterial, we have average power or power delivered or dissipated as

AVERAGE POWER AND POWER FACTOR • The above eq for average power can also be written as • But we know Vrms and Irms values as • Thus average power in terms of vrms and irms becomes,

AVERAGE POWER AND POWER FACTOR • For resistive load, • We know v and i are in phase, then |θv - θi| = θ = 0°, • And cos 0° = 1, so that • becomes • or

AVERAGE POWER AND POWER FACTOR • For inductive load ( or network), • We know v leads i, then |θv - θi| = θ = 90°, • And cos 90° = 0, so that • Becomes • Thus, the average power or power dissipated by the ideal inductor (no associated resistance) is zero watts.

AVERAGE POWER AND POWER FACTOR • For capacitive load ( or network), • We know v lags i, then |θv - θi| = |–θ| = 90°, • And cos 90° = 0, so that • Becomes • Thus, the average power or power dissipated by the ideal capacitor is also zero watts.

AVERAGE POWER AND POWER FACTOR • Power Factor • In the equation, • the factor that has significant control over the delivered power level is cos θ. • No matter how large the voltage or current, if cos θ = 0, the power is zero; if cos θ = 1, the power delivered is a maximum. • Since it has such control, the expression was given the name power factor and is defined by • For situations where the load is a combination of resistive and reactive elements, the power factor will vary between 0 and 1

AVERAGE POWER AND POWER FACTOR • In terms of the average power, we know power factor is • The terms leading and lagging are often written in conjunction with power factor and defined by the current through load. • If the current leads voltage across a load, the load has a leading power factor. If the current lags voltage across the load, the load has a lagging power factor. • In other words, capacitive networks have leading power factors, and inductive networks have lagging power factors.

AVERAGE POWER AND POWER FACTOR • EXAMPLE - Determine the average power delivered to network having the following input voltage and current: v = 150 sin(ωt – 70°) and i = 3 sin(ωt – 50°) • Solution

AVERAGE POWER AND POWER FACTOR • EXAMPLE - Determine the power factors of the following loads, and indicate whether they are leading or lagging: • Solution:

COMPLEX NUMBERS • Application of complex numbers result in a technique for finding the algebraic sum of sinusoidal waveforms • A complex number represents a point in a two-dimensional plane located with reference to two distinct axes. • The horizontal axis is called the real axis, while the vertical axis is called the imaginary axis. The symbol j (or sometimes i) is used to denote the imaginary component.

COMPLEX NUMBERS • Two forms are used to represent a complex number: rectangular and polar. • Rectangular Form • Polar Form

COMPLEX NUMBERS • Polar Form • θ is always measured counter-clockwise (CCW) from the positive real axis. • Angles measured in the clockwise direction from the positive real axis must have a negative sign

COMPLEX NUMBERS • EXAMPLE - Sketch the following complex numbers in the complex plane: a. C = 3 +j 4 b. C = 0 -j 6

COMPLEX NUMBERS • EXAMPLE - Sketch the following complex numbers in the complex plane: c. C = -10 - j20

COMPLEX NUMBERS • EXAMPLE - Sketch the following complex numbers in the complex plane:

COMPLEX NUMBERS - Conversion • Rectangular to Polar • Polar to Rectangular Angle determined to be associated carefully with the magnitude of the vector as per the quadrant in which complex number lies

COMPLEX NUMBERS • EXAMPLE - Convert the following from polar to rectangular form: • Solution:

COMPLEX NUMBERS • EXAMPLE - Convert the following from rectangular to polar form: C = - 6 + j 3 • Solution:

COMPLEX NUMBERS • EXAMPLE - Convert the following from polar to rectangular form: • Solution

Math Operations with Complex Numbers • Let us first examine the symbol j associated with imaginary numbers. By definition,

Math Operations with Complex Numbers • The conjugate or complex conjugate is found • by changing sign of imaginary part in rectangular form • or by using the negative of the angle of the polar form. • Rectangular form, • Polar form, (conjugate) (conjugate)

Math Operations with Complex Numbers • Addition Example (Rectangular) • Add C1 = 3 + j 6 and C2 = -6 + j 3. • Solution

Math Operations with Complex Numbers • Subtraction Example (Rectangular) • Solution

Math Operations with Complex Numbers • Imp Note • Addition or subtraction cannot be performed in polar form unless the complex numbers have the same angle θ or unless they differ only by multiples of 180°.

Math Operations with Complex Numbers • Addition Example (Polar)

Math Operations with Complex Numbers • Subtraction Example (Polar)

Summary / Conclusion • Average Power & Power Factor • Complex Numbers • Math Operations with Complex Numbers

ELECTRIC CIRCUIT ANALYSIS - I