Power Factor: True vs Apparent Power

Power Factor & Description

• Power factor (PF) represents the ratio of true power to apparent power.

Power factor ratio

• True power is adjacent to the angle theta (θ) in a phasor diagram while apparent power is the hypotenuse.
• Power Factor = True Power / Apparent Power = Adjacent / Hypotenuse = Cos θ
• TP = AP × Cos θ
• TP = Vapp × IT × Cos θ

Inductive Circuits

• Current lags the applied voltage, the power factor is referred to as a 'lagging' power factor.

Significance of Power Factor value of 1

• A power factor of 1 indicates no reactive power in an AC circuit.
• All power delivered is dissipated as true power, signifying a purely resistive circuit.

Significance of Power Factor value of 0

• A power factor of 0 means no true power is dissipated in an AC circuit.
• All delivered power returns to the source as reactive power.
• The cosine of the angle between true and apparent power equals zero.
• This angle matches the phase angle between total circuit current & applied voltage.

Apparent Power Triangle for AC Inductive Circuits

• Power relationships in AC inductive circuits can be represented by a right triangle - power triangle.
• True power is drawn horizontally, representing current and voltage that are in phase.
• Reactive power is drawn vertically downward, representing the product of voltage across an ideal inductor.
• Current lags the voltage by 90°.
• Apparent power is the hypotenuse and is the result of true and reactive power.
• The angle θ represents a lagging power factor (Cos θ).
• Phasor diagrams are always in reference to voltage.

Phasor and Power Triangle Differences

• Power triangle always references applied voltage, phasor diagrams reference current in series circuits and voltage in parallel circuits.

Calculating TP in Series LR Circuit

• TP = IR² × R
• Example: TP = 5² × 1000 = 25,000 W

Calculating RP in Series LR Circuit

• RPL = IR² × XL
• Example: RP₁ = 5² × 600 = 15 000 VAR

Calculating AP in Series LR Circuit

• AP = √ TP² + RP2
• Example: AP = √25, 0002 + 15,0002 = 29.15 kVA

Calculating Power Factor in Series LR Circuit

• Power Factor = True Power / Apparent Power = Cos θ
• Power Factor = 25 000 / 29 150 = Cos θ
• Power Factor = 0.86 lagging

Phase Angle in Series LR Circuit

• To calculate the relationship between circuit current and applied voltage, use:
• Power Factor = Cos θ, e.g., 0.86 = Cos θ
• θ = Cos-1 0.86
• θ = 30.96 degrees
• Total circuit current lags the applied voltage.

Apparent Power in Parallel LR Circuits

• Parallel AC circuits use voltage drop across the parallel network as the horizontal reference phasor in a phasor diagram.
• Current through the resistor is in phase with the applied voltage
• True power will be dissipated.
• Current through the inductor lags the applied voltage by 90°, this will create reactive power.
• Resultant source power is the apparent power.

Apparent Power Formulas in Parallel LR circuits

• AP = VAPP × IT
• AP = IT² × Z
• AP = V App²/ Z
• Z = (RXXL) / √R2 + X (L) 2
• IT = √ IL² + IR2

Power Triangle for a Parallel LR Circuit

• The power triangle for a parallel LR circuit is the same as in series
• Current in parallel/series LR circuits lags the voltage (CIVIL)
• Ratio of apparent to true power will show a lagging power factor, and use shown formulas for series circuits. True Power / Apparent Power

Parallel LR Circuits Formulas

• TP = AP × Cos θ
• TP = VT × IT × Cos θ

Series Capacitive Resistive Circuits

• The current flowing through the resistor is in phase with the applied voltage, which means that some true power will be dissipated
• The voltage dropped across the capacitor lags the total circuit current by 90°
• Power delivered to/returned by capacitor relates to the reactive power.
• The source delivers the resultant power, also known as apparent power

Instantaneous Power Curve in Series CR Circuit

• There's more positive power than negative.
• The undissipated power turns into heat energy by the resistor.
• Phasor diagram: applied voltage lags the total current by 45°.
• This phase difference indicates equal capacitive reactance and resistance.
• Delivering power to a CR circuit depends on the phase difference between the applied voltage/circuit current.
• If the phase angle increases from 45° (more capacitive), the power returning to the source will also increase
• If the phase angle decreases from 45° (more resistive), the power returned to the source will decrease.

Apparent Power Formulas for Series CR Circuit

• AP = VAPP × IT
• AP = IT² × Z
• AP = V App² / Z
• Z = √ R2 + X (C) 2
• VA = √VR² + VC²

Power Phasor Diagram for Series CR Circuit

• True power represents the horizontal reference, because VR and IR are in phase.
• Capacitive reactive power is 90° clockwise from the TP/voltage across a capacitor lags behind current by 90°.

Series CR Circuit formulas

• AP = √ TP2 + RP2
• Power Factor = Cos θ = True Power / Apparent Power
• Apparent power delivered to a circuit equals the product of the applied voltage and the total circuit current.
• The current/capacitive circuits LEAD the applied voltage (CIVIL), we refer to it for capacitive circuits as a leading power factor.

Power Triangle of a Series CR Circuit

• CR circuit power triangle differs from LR circuits, it is drawn ABOVE the horizontal
• Reactive power drawn vertically upwards because current leads the voltage that is dropped across it by 90°.
• Apparent power combines two power types and is represented via the right power triangle's hypotenuse.
• The angle θ appears above the horizontal to represent a leading power factor (cos θ).

Example of TP, RP and AP in a Series CR Circuit

• AP = VAPP × IT
• AP = 240 × 0.5 = 120 VA

Determining TP of a Series CR Circuit

• Power Factor = Cos θ = True Power/ AP
• TP = Power Factor × AP(0.21 × 120)
• TP = 25.2 W

Determining RP of a Series CR Circuit

• AP = √ TP2 + RP2

Calculating capacitor values

RP = Ic² × Xc Xc = RP / Ic Xc = 1 / 2πfc

Parallel CR circuit diagram

• Use the voltage dropped across the parallel network as the horizontal referencing phasor for the parallel CR circuit
• Use phase relationships to determine the current for the circuit components and complete the phasor diagram.

CR circuits rules

• True power is dissipated from a resistor/ reactive power provided is given to a capacitor.
• A source will deliver power into a parallel CR circuit.

Apparent Power Formulas in a Parallel CR Circuit

• AP = VA × Іт
• AP = IT² × Z
• AP - V App² / Z
• Z = (RXXc) / √R2 + X (c) 2
• Iт = √ IR² + Ic2

Power Factor Formula in a Parallel CR Circuit

• Power Factor = Cos θ Power factor = Cos θ = True Power / Apparent Power
• in capacitive circuits, total current leads applied voltage which leads to the power factor for them as leading power factor.

Power Triangle of Parallel CR Circuit

• AP will be shown above TP.
• Regardless of differences in using current/series circuits as a reference to using relationships in an ideal capacitor for phase, the voltage across that remains = same/leads that by 90.

Parallel power factor in the formula

• Power Factor = Cos θ leading to the equal/applied leading current total
• Apparent/True powers and overall are equal. The differences are due to rounding.
• Total leading of current in capacitive circuit
• Apparent power in terms of total leading.

Power in series LCR circuit

• In power in series LCR circuits energy transferred (during quarter cycle) is delivered to a capacitor while it is returned by an inductor.
• The current will lead the voltage.

Power Triangle

• There is is a relationship between Power that is based on power triangle.
• The relationship between the power is described with 3 key reactive powers which are:
• Reactive power - inductor(drawn vertical above the below for lagging)
• Drawn Vertical for capacitance (RPC)
• Equivalent reactive power - Differences. A reactive power is indicated as higher (i.e above) that than RPL.

Power in LCR circuits and formulas

• LCR circuits use Pythagoras.
• AP = √TP2 + RPEq - in LCR with similar power factor. L/C will lead to application.
• With inductive - similar reactive- triangle drawn in horizontal is an example
• Power LCR factor comes from L & C circuit Power Factor = Cose = ΤΡ / AP

Power Factor Formula

• True power = applied voltage x current = Power Factor
• The power is positive and the Power Factors have a proportional relationship: True, apparent, voltage and the application of voltage that is reactive.

Types of Power (in Summary)

• True Power (TP): Power is moved from the circuit by resisting when converting into other forms
• Reactive power: Reactive power is stored in a capacitor and inductors, where it is than moved over to the source
• Apparent power (AP): Power appears to use AC circuits which reacts to the capacitors in the circuits.
• Positive power is an effective flow of current while negative power is a non-effective flow.