EGN 3373 Introduction to Electrical Systems I A Systems Approach to Electrical Engineering Graphics Adapted from “Physical, Earth, and Space Science”, Tom Hsu, cpoScience.

Let’s Look at Power in AC Circuits Emphasis is placed on Power in AC circuits, and to facilitate the discussion the following voltage and current signals will be used along with impedance Z.

Power in AC Circuits Based on the definition for the average power: For a Resistive Load the phase angle is zero and therefore the average power becomes: The residential voltage of 110 V is actually the RMS value of the sinusoid signal delivered by the power company, which means the maximum value of the sinusoid is approximately 155 volts.

Effective RMS Values In addition, effective or RMS (root mean square) values of a periodic signal are defined in terms of the average power delivered to a resistive load. The value of an AC voltage is continually oscillating from the positive peak to the negative peak through zero. This makes the voltage value less than the peak value during most of the cycle. We use the root mean square (rms) value for sine waves (V and I) to represent the effective value of a varying voltage or current. V RMS is 0.7 of the peak voltage V peak

Effective RMS Values In addition, effective or RMS (root mean square) values of a periodic signal are defined in terms of the average power delivered to a resistive load. Applying a periodic voltage v(t) with period T to a resistive load R, then power delivered is defined as P(t) = v 2 (t) / R P avg = E T /T = V rms 2 / R Furthermore, the average power delivered to the resistance is the energy delivered in one cycle divided by the period T where

Power in AC Circuits I, V, and P for a Resistive Load

Power in AC Circuits I, V, and P for a Capacitive Load

Power in AC Circuits I, V, and P for a Inductive Load

Three-Phase Systems Why not DC? No easy way to change voltage levels Complicated rotating machine design and construction Why not single-phase AC? Pulsating torque Efficiency issue: kVA per MCM of copper

Two-Phase Power

AC Power v(t) = V m cos (  t +  v ) i(t) = I m cos (  t +  i ) = V m I m cos (  t +  v ) cos (  t +  i ) s(t) = (V m I m )/2 [cos (  v   i ) + cos (2  t +  v +  i )] s(t) = v(t) x i(t)

AC Power s(t) = (V m I m )/2 [cos (  v   i ) + cos (2  t +  v +  i )] 2  t +  v +  i = 2(  t +  v ) – (  v –  i ) cos (  –  ) = cos  cos  + sin  sin 

AC Power Constant component of real power Oscillating component of real power Reactive power s(t) = P + P cos [2(  t +  v )] + Q sin [2(  t +  v )] Let P = V rms I rms cos (  v –  i ) and Q = V rms I rms sin (  v –  i )

AC Power

s(t) = P + P cos [2(  t +  v )] + Q sin [2(  t +  v )] P = V rms I rms cos (  v –  i ) Q = V rms I rms sin (  v –  i ) = V rms I rms [cos (  v –  i ) + j sin (  v –  i )] S = P + j Q P = P + P cos [2(  t +  v )] Q = Q sin [2(  t +  v )]

AC Power = V rms I rms [cos (  v –  i ) + j sin (  v –  i )] = V rms I rms e j(  v ) e j(–  i ) = V rms  v I rms –  i S = P + j Q S = V I* S = V rms I rms e j(  v –  i ) S = V rms e j(  v ) I rms e j(–  i ) Complex Conjugate

Power Triangle Total or Apparent Power S kVA Real Power P kW Reactive Power Q kVAR  cos   Power Factor This is the only “useful” work done. This component of the power furnishes the energy required to establish and maintain electric and magnetic fields. Our electrical system must handle this component. THEREFORE, VARS REDUCE THE “USEFULNESS” OF THE TOTAL ELECTRIC POWER.

Power Triangle  Power triangle in first quadrant (positive  ) indicates INDUCTIVE load.

Power Triangle  Power triangle in fourth quadrant (negative  ) indicates CAPACITIVE load.

Power Triangle  Power triangle and impedance triangle are SIMILAR TRIANGLES.  P Q S Z X R R = resistance X = reactance Z = impedance

Three-Phase Systems Three-phase systems Most common: 3-wire delta and 4-wire wye Mathematical representation Time domain (oscilloscope view) Frequency domain (phasor diagram)

Three-Phase Systems

Delta-connected loads Line-to-line voltage only No neutral, so best for balanced loads Ideal for motors Wye-connected loads Line-to-line AND line-to-neutral voltage Better for imbalanced loads, particularly mix of three-phase and single-phase

Mathematical Relationships P = |V|  |I|  cos  Q = |V|  |I|  sin  S = V  I* * denotes COMPLEX CONJUGATE Recommendation: For three-phase systems, use LINE-TO-LINE voltages and THREE-PHASE POWER values. Alternatively, LINE-TO-NEUTRAL voltages and POWER PER PHASE.

A Simple Derivation

Power in AC Circuits Based on the definition for the average power:

Power in AC Circuits Based on the definition for the average power:

EXAMPLES