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Lesson 8.7 • Competency 8.6Complete Lesson
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Alternating Current (AC) & Transformers

AC generation principles, sinusoidal voltage V(t) = V₀ sin(ωt), Root-Mean-Square (RMS) definitions V_rms = V₀/√2 and I_rms = I₀/√2, transformer action V_s/V_p = N_s/N_p, step-up/step-down conversions, efficiency η, and transformer power losses.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Define sinusoidal AC peak and RMS values, deriving V_rms = V₀/√2 and I_rms = I₀/√2 from average Joule heating.
  • Derive the ideal transformer equation V_s / V_p = N_s / N_p = I_p / I_s from Faraday’s law of mutual flux linkage.
  • Analyze power losses in practical transformers (copper I²R, eddy currents, magnetic hysteresis, flux leakage) and calculate electrical efficiency η = (P_out / P_in) * 100%.

Core Theory & Concepts

Alternating Current (AC) reverses its direction periodically with a continuous sinusoidal time variation. The paramount engineering advantage of AC over DC is that AC voltages can be stepped up to hundreds of thousands of volts for long-distance transmission with negligible resistive loss, and then stepped down safely for consumer use using electrical transformers.

RMS Voltage and Current

For a sinusoidal waveform, the effective heating value is related to the peak value by: V_rms = V₀ / √2 ≈ 0.707 V₀, and I_rms = I₀ / √2.

V_{\text{rms}} = \frac{V_0}{\sqrt{2}}, \quad I_{\text{rms}} = \frac{I_0}{\sqrt{2}}

Ideal Transformer Relations

Voltages are directly proportional to turns, while currents are inversely proportional: V_s / V_p = N_s / N_p = I_p / I_s.

\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}

Transformer Efficiency

Efficiency η is the ratio of useful secondary power output to primary power input: η = (V_s I_s / V_p I_p) * 100%.

\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% = \frac{V_s I_s}{V_p I_p} \times 100\%

Common Misconceptions

Misconception

A step-up transformer creates electrical power from nothing, violating energy conservation.

Physical Clarification:A step-up transformer increases voltage at the expense of an exactly proportional reduction in current (P = VI). In fact, because of real core and copper losses, power output is always slightly less than power input (efficiency < 100%).

Syllabus Topic Outline

Competency 8.6
  • 1Generation of alternating sinusoidal EMF by rotating coils in uniform B-fields
  • 2Mathematical waveform: V(t) = V₀ sin(2πft) and peak vs RMS values
  • 3Thermal definition of RMS current (P_avg = I_rms² R)
  • 4Construction and mutual induction principle of electrical transformers
  • 5Transformer turn ratio equations: step-up vs step-down configurations
  • 6Physical origins of transformer power loss and efficiency calculations
  • 7High-voltage national grid electrical power transmission