Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1

Chapter 14 – Basic Elements and Phasors

Average Power & Power Factor Complex Numbers Math Operations with Complex Numbers

We know for any load v = V m sin(ωt + θ v ) i = I m sin(ωt + θ i ) Then the power is defined by Using the trigonometric identity Thus, sine function becomes

Putting above values in We have The average value of 2 nd 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.

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 | θ v – θ i |, where | | indicates that only the magnitude is important and the sign is immaterial, we have average power or power delivered or dissipated as

The above eq for average power can also be written as But we know V rms and I rms values as Thus average power in terms of v rms and i rms becomes, 13/12/2015 8

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

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.

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.

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

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.

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

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

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

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.

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

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

EXAMPLE - Sketch the following complex numbers in the complex plane: c. C = j20

EXAMPLE - Sketch the following complex numbers in the complex plane:

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

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

EXAMPLE - Convert the following from rectangular to polar form: C = j 3 Solution:

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

Let us first examine the symbol j associated with imaginary numbers. By definition,

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)

Addition Example (Rectangular) Add C 1 = 3 + j 6 and C 2 = -6 + j 3. Solution

Subtraction Example (Rectangular) Solution

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°.

Addition Example (Polar)

Subtraction Example (Polar)

Average Power & Power Factor Complex Numbers Math Operations with Complex Numbers

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