AC Circuits: True, Reactive & Instantaneous Power

• In an AC circuit, power is classified into different types, unlike power in a DC circuit.

True Power and Reactive Power

• True power, as in DC circuits, is power dissipated by resistance as heat when electrical energy converts.
• Reactive power is energy delivered alternatively to a purely reactive component (capacitive or inductive) and returned to the AC source.
• Reactive power is measured in VARS (Volt-Amps Reactive).

Power in Pure Resistive Circuits

• In purely resistive circuits, sinusoidal voltage applied will be in phase with the sinusoidal current.
• Radiated heat when current flows through a resistor represents energy or power loss.

Instantaneous Power

• The instantaneous power curve is obtained by multiplying the instantaneous current by the instantaneous voltage.
• The instantaneous power waveform occurs at twice the frequency of basic voltage and current sine waves.
• Power in a purely resistive AC circuit is always positive because it dissipates regardless of current flow direction.

True Power

• Average power is calculated by multiplying the RMS voltage value by the RMS current value.
• The power dissipated by a resistor equals the power dissipated if DC voltage of equal magnitude to the AC RMS voltage were applied.
• True or real power refers to the average energy loss or power dissipation in AC circuits, measured in watts (W).
• True power is only dissipated when electrical energy converts into another form like heat, light, or mechanical power.
• True power can be calculated using:
  • TP = VR × IR
  • TP = IR² × R
  • TP = VR²/R
  • TPR is the true power dissipated by the resistor in watts (W).
  • IR is the current flowing through the resistor in amps (A).
  • R is the resistor's value in ohms (Ω).

Power in a Series Parallel Resistor Circuit

• Series Parallel Resistances Circuit is a resistance circuit with series and parallel-connected resistors.

Series Parallel Resistance Circuit

• The equivalent resistance of the parallel network (RE) is calculated as: 1/Rpar = 1/R2 + 1/R3
• Total circuit resistance RT can be calculated by: RT = Rpar + R1
• Total true power TPT dissipated by the resistive AC circuit can be calculated by: TPT = V²/R
• Total circuit current IT can be calculated by: IT = V/R

Series Parallel Resistance Circuit - Component Power

• True power dissipated by R1 (TPR1) can be calculated by: TPR1 = IR1² × R1

Series Parallel Resistance Circuit - Voltage

• Voltage drop across resistor R1 (VR1) is: VR1 = IR1 × R1
• Voltage dropped across each branch of the parallel network can be determined using Kirchoff's voltage laws.

Series Parallel Resistance Circuit - Total Power

• Total power dissipated by resistors R2 (PTR2) and R3 (PTR3) can be calculated by: PT = (V²/R2) + (V²/R3)
• In a purely resistive AC circuit, total true power dissipated by the circuit equals the sum of true powers dissipated by all individual resistors.

Series Parallel Resistance Circuit - Power Summary

• Only true power uses (dissipates) by a circuit when the applied voltage is in phase with the total circuit current, indicating a purely resistive circuit.
• True power is defined as the power dissipated by a resistor as heat when electrical energy converts.
• True power (PT) formulas:
  • TP = V × I
  • TP = V²/R
  • TP = I² × R
• The total power dissipated in a purely resistive AC circuit equals the sum of the powers dissipated by all individual resistors.

Reactive Power in Inductive Circuits

• In a purely resistive AC circuit, true power dissipates.
• In an AC circuit with purely inductive components.
• Current through the ideal inductor lags applied voltage by 90°.
• All inductors ideally contain only inductance without resistance.

Pure Inductive AC Circuits - Instantaneous Power

• The instantaneous power graph of an ideal inductor is obtained by multiplying the instantaneous current by the instantaneous voltage.
• Reactive circuit instantaneous power - pure inductance with V leading I by 90 degrees.
• When a negative instantaneous voltage (-ve) is multiplied by a positive instantaneous current (+ve), the resulting instantaneous power will be negative.
• A negative instantaneous voltage (-ve) by a negative instantaneous current (-ve) gives an instantaneous power that is positive.
• For one complete cycle of the input voltage, the instantaneous power curve has a zero average value, indicating an equal amount of positive and negative power.
• The positive portion of the power curve represents energy (power) delivered from the source to the inductor, used to build a magnetic field around the coil.
• The area below the zero axis line (negative power) is returned to the source when the magnetic field collapses.
• The inductor alternately consumes and supplies power.

Reactance Power

• In an ideal inductor, power supplied from the inductor back to the voltage generator equals power supplied to the inductor.
• In a purely inductive circuit, average true power (Tp) is zero, meaning a wattmeter connected to the circuit will read zero, and the ideal inductor never warms.
• The source must deliver power for a quarter cycle, and will be returned during the next quarter cycle.

Pure Inductive Circuits - Reactive Power

• The power initially used by the inductor is the reactive power.
• Reactive power measured in volt-amp reactive (VAR).
• Reactive power supplied to an inductive circuit is lagging because the current lags the applied voltage (CIVIL).
• Reactive power of inductive circuit formula
  • PRL = VL × IL
  • PRL is the inductive reactive power in volt-amps reactive (VAR).
  • VL is the RMS voltage dropped across the inductor.
  • IL is the RMS current flowing through the inductor.
• Other calculation formulas
  • PRL = IR² × XL
  • PRL = VL²/XL
• The equations for reactive power are similar to the equations for true power, but use inductance instead of resistance.

Series Parallel Inductor Circuit - Reactive Power

• Use the formula XL = 2πfL to calculate inductive reactance of an inductor.
• If Inductor L3 is the same value as inductor L2, its inductive reactance will be equal.

Series Parallel Inductive Cicuit - Total Inductive Reactance

• The equivalent inductive reactance of the parallel circuit (Xpar) is the sum of the Inverse values of the Inductance.
• Total inductive reactance (XT) of the series parallel inductive reference circuit is the sum of the Inductance.
• The rounded off values of XL2 and XL3 can be calculated using the full calculated value.

Series Parallel Inductive Cicuit - Circuit Current

• Total circuit current is calculated: IT = VR/XT

Series Parallel Inductive Cicuit - Total Reactive Power

• The reactive power of an Inductor: RPL = IL1 2 X XL1

Kirchhoff's Current Law

• The current in each branch of a parallel network is 140 mA.
• If the current and inductive reactance of inductor L2 are known, reactive power (RPL2) can be calculated: RPL₂ = IL22 × XL2
• The source's reactive power: RPT = IT × VA
• Total, all three inductors together: PRT = PRL1 + PRL2 + PRL3
• Because reactive power delivered to inductors is in-phase with each other, the values can be added together.
• Reactive power is out of phase to true power by 90°.

Reactive Power in Capacitive Circuits

• The purely capacitive AC circuit is on the power supplied to the circuit.
• The current through the capacitor will lead the applied voltage by 90°.

Pure Capacitive Circuits - Instantaneous Power

• The instantaneous power curve is calculated for a capacitive circuit by multiplying the instantaneous voltage by the instantaneous current, resulting in an instantaneous power waveform.
• The power curve for a capacitive circuit is twice the frequency of current and voltage waveforms. During one input voltage cycle, the power curve has an average value of zero.
• The capacitor consumes and supplies power.

Pure Capacitive Circuits - Power Consumption

• The power consumption equals the power given back to the source, so average true power in a purely capacitive AC circuit is zero.
• The reactive power must be supplied because reactive power (RP) is drawn.
• A capacitor is leading because current leads the voltage (CIVIL).

Ohm's Law

• Reactive power for a capacitor: PRC = VC × Ic
• The capacitance and RMS current flow calculation:
  • PRC = I2 × XC
  • PRC = VC²/XC

Series Parallel Capacitor Circuit - Reactive Power

• Capacitive reactance of the series parallel AC circuit.

Series Parallel Capacitive Circuit - Capacitive Reactance

• To determine reactive power, capacitive reactance must be calculated.
• Formula: Xc = 1/(2πfC)
• Equivalent capacitive reactance of the parallel network (Xpar) of the reference circuit is: 1/Xpar= 1/Xc2+ 1/Xc3
• The Capacitive reactance of the series parallel reference circuit by: XT = XC1 + Xpar

Series Parallel Capacitive Circuit - Total Current

• Calculation of total current: IT = VA /Xт

Series Parallel Capacitive Circuit - Reactive Power

• Formula for Reactive Power: PRC1 = IC12 × XC1

Kirchoff's Current Law

• Total reactive Power supplied: RPT = RPC1 + RPC2 + RPC3
• Because reactive power delivered to capacitors is in phase with the capacitive values, they can be added.
• Capacitive reactive power/ Inductive reactive power is out of phase by 180 degrees.

Reactive Power Summary

• Reactive power is the energy alternatively delivered to a reactive component/circuit (capacitive or inductive), and is then returned to the source.
• Reactive power, in VAR, is the product of voltage and current at 90° out of phase. -Formula:
  • RPL = VL × IL
  • RPL = VL²/XL
  • RPL = IL² × XL
• Reactive Inductive Circuit Formula:
  • RPC = VC × Ic
  • RPC = Ic² × Xc
  • RPC = VC²/Xc
• Each total sum component:
  • The total inductive reactive power of a reactive circuit is equal to the sum of the reactive powers used by the inductive components.
  • The circuit is equivalent to the capacitive components

Apparent Power

• The power the component dissipates is called true power. Dissipated by a resistor while voltage/current is in phase. The Voltage of Power Type (90°) that relates to component power that delivers/returns by the inductor and reactive power. Apparent power measures AC circuit which in the total circuit current and of of phase voltage's by less than (90°). Formula AP = VA × Іт is the abbreviation.

Apparent Power - Inductive Resistive Circuits

• Use a voltage or phaser diagram to show a series of LR circuits. Current lags.
• Amount of circuit power will depend on the source. total/ applied circuits are out of phase to each other with (90°) apparent power.

Formula for Apparent Power

• Equation- AP = VA × Іт (Va); formula for AP (apparent Power) in a drawing's power range. Equation -AP = IT² × Z by multiplying circuit by It's impedance (Ohm's). Resistance formula- VA = √V+V