AC Circuit Power Analysis Chapter 11

Chapter 11 AC Circuit Power Analysis

• This chapter analyzes different power representations in AC circuits.
• Objectives include calculating real power, reactive power, apparent power, and power factor for circuits or elements.
• Power representations include instantaneous power, average power, apparent power, complex power, and reactive power.
• Additional related measures include effective (RMS) current and voltage, and power factor.

Introduction

• The chapter examines various representations of power in alternating current (AC) circuits.
• Instantaneous power, average power, apparent power, complex power, and reactive power are discussed.
• Effective (root-mean-square, or RMS) current and voltage, along with power factor, are also key concepts.

Instantaneous Power

• Instantaneous power is the power at any given moment, calculated as p(t) = v(t)i(t).
• The power supplied versus time is compared to the power absorbed by the components in a circuit.

Power from Sinusoidal Source

• If the source voltage is sinusoidal, its waveform p(t) is described involving constant and double frequency.

Calculating Average Power

• The average power over an arbitrary interval (t₁ to t₂) is calculated via the integral of p(t) over the interval divided by the change in time, P =(1/(t₂−t₁))∫t₁^t₂p(t)dt .
• For periodic functions with period T, average power calculation occurs over one period. P = (1/T)∫₀^T p(t) dt .

Average Power: Sinusoidal Steady State

• For sinusoidal steady-state conditions where v(t) and i(t) are sinusoids, the average power P= (1/2)VₘIₘcos(θ- φ).

Average Power for Elements

• Average power absorbed by a pure resistor R is PR = (1/2)Vᵢ²ₘ/R= (1/2)Iᵢ²ₘR.
• In purely reactive elements, the average power absorbed is zero due to the 90° phase difference between current and voltage.

Example: Average Power

• An example demonstrates calculating average power delivered to an impedance. Only the resistive component contributes to average power.

Example: Average Power(2)

• Another example demonstrates calculation of average power absorbed by individual elements in a circuit.

Maximum Power Transfer

• Maximum power is transferred to a load impedance Z₁ when Z₁ is the complex conjugate of the source impedance Zth.
• This occurs when the resistance of Z₁ is equal to the resistance of Zth, and the reactance of Z₁ is equal to the negative of the reactance of Zth.

Maximum Power Transfer Derivation

• The derivation of maximum power transfer demonstrates the condition for maximum power transfer.

Effective Values of Current and Voltage

• Effective current(or voltage) in a circuit is equivalent to the DC current that generates the same average power dissipation in a resistor.
• If the current or voltage is a continuous function with period T, then Iₛ = [(1/T) ∫₀^T i²(t)dt].^(1/2) .
• This calculation of the effective values is often called the Root-Mean-Square (RMS).

Effective (RMS) for Sine Wave

• For sine waves, the effective value (RMS) is √(1/2) times the peak value.
• RMS values can be used in power calculations.

Apparent Power & Power Factor

• Apparent power (S) is calculated as the product of the RMS voltage and current.
• It's expressed in volt-amperes (VA).
• Power factor is the ratio of average power to apparent power, expressed as PF = P/S= cos(θ - φ).

Apparent Power & Power Factor (2)

• The power factor is used to determine the ratio of average power and apparent power
• For purely resistive loads, the power factor is 1.
• For purely reactive loads, the power factor is 0.
• Generally, the power factor is a value between zero and 1.

Power Factor: Lagging & Leading

• Power factor considers whether the current leads or lags the voltage. A lagging power factor implies current lags the voltage. A leading power factor means the current leads.

Example: Power Factor

• A problem demonstrates calculating average power to multiple loads, the apparent power supplied by the source, and the power factor for those loads.

This quiz focuses on the power representations in alternating current (AC) circuits as outlined in Chapter 11. Participants will explore concepts such as real power, reactive power, apparent power, and power factor. The quiz will also cover effective (RMS) current and voltage, enhancing understanding of AC circuit dynamics.