Alternating Current, Circuits & Transformers Explained | Chapter 31 of University Physics

In Chapter 31, we explore alternating current (AC) fundamentals, from sinusoidal sources and phasors to reactive circuit elements, resonance, and the operation of transformers—key to power distribution and modern electronics.

Watch the full video summary here for in-depth walkthroughs and phasor demonstrations.

Phasors & Sinusoidal Sources

AC voltages and currents vary as v(t) = V_max cos(ωt), where ω = 2πf. Phasors represent these sinusoids as rotating vectors in the complex plane, simplifying circuit analysis:

• RMS values: V_rms = V_max/√2, I_rms = I_max/√2
• US AC frequency: 60 Hz (ω ≈ 377 rad/s); Europe: 50 Hz (ω ≈ 314 rad/s).

Resistance & Reactance

• Resistor (R): V_R = I R, voltage and current in phase.
• Inductor (L): V_L = I X_L, X_L = ωL, voltage leads current by 90°.
• Capacitor (C): V_C = I X_C, X_C = 1/(ωC), voltage lags current by 90°.

L-R-C Series Circuits & Impedance

In series, current is common and total impedance is:

Z = √[R² + (X_L – X_C)²], with phase angle φ = tan^–1((X_L – X_C)/R). The RMS voltage satisfies:

V_rms = I_rms Z, and the phasor sum of V_R, V_L, V_C completes the vector diagram.

Power & Power Factor

• Resistive load: P_avg = I_rms²R = V_rmsI_rms.
• Pure L or C: P_avg = 0 (energy alternately stored and returned).
• General circuit: P_avg = V_rmsI_rms cos φ, where cos φ is the power factor.

Power factor correction uses capacitors or inductors to bring cos φ closer to 1, improving efficiency.

Resonance in L-R-C

At resonance, X_L = X_C, so Z = R and current is maximal. The resonant angular frequency is:

ω₀ = 1/√(LC). Here, voltage and current are in phase (φ = 0) and the power factor is unity.

Transformers & Energy Transfer

Transformers use mutual inductance between primary (N₁) and secondary (N₂) coils on a shared core. The ideal turns ratio relates voltages:

V₂/V₁ = N₂/N₁. Power conservation (neglecting losses) gives V₁I₁ = V₂I₂. Impedance seen by the primary is transformed as:

Z_primary = Z_load (N₁/N₂)².

Conclusion

Chapter 31 equips you to analyze AC circuits using phasors, compute reactances and impedance, understand resonance behavior, and apply transformer principles for efficient power transmission. These concepts are foundational to electrical engineering and power systems.

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