Lesson 6 Flashcards — AC Circuit Analysis

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1. Write the general equation of an AC signal and name each term.

\(v(t) = V_\text{bias} + A\cos(2\pi f t + \phi)\) — DC offset, amplitude, frequency, and phase shift. The period is \(T = 1/f\).

2. How do you compute a phase shift from a time offset?

\(\phi = (2\pi/T)\,\Delta t\), or in degrees \(\phi^\circ = (360^\circ/T)\,\Delta t\). Measure \(\Delta t\) from the origin leftward to the nearest positive peak — that direction is the convention, and measuring rightward instead flips the sign. So a peak left of the origin gives \(\phi > 0\) and the wave leads; a peak to the right gives \(\phi < 0\) and it lags.

3. How do you convert between sine and cosine form?

\(\sin(x) = \cos(x - 90^\circ)\). Problems appear in both forms — convert to cosine first and stay there.

4. Why is nearly all generated and transmitted power AC?

Generators produce it naturally by spinning a coil in a magnetic field, and transformers — which do nothing with DC — are what make long-distance transmission affordable. So the wall outlet is AC too, and every device you own converts it back to DC itself (Lesson 10). DC is what devices want; AC is how power gets to them.

5. For a resistor, what is the phase relationship between voltage and current?

They are in phase — peaking together and crossing zero together. Every Block 1 analysis tool therefore applies to AC resistive circuits unchanged.

6. What is the RMS value of a signal, physically?

The DC voltage that would deliver the same average power to the same resistor.

7. Give the RMS value of a sinusoid.

\(V_\text{rms} = V_\text{pk}/\sqrt{2}\). This factor holds only for sinusoids.

8. A signal has a DC bias on top of a sinusoid. What is its RMS value, and what power does it deliver to a resistor?

Use the general form \(V_\text{rms} = \sqrt{V_\text{bias}^2 + V_\text{pk}^2/2}\) — the \(V_\text{pk}/\sqrt{2}\) you already know is just its bias-free case. Equivalently, the DC and AC powers add in a resistor: \(P = V_\text{bias}^2/R + \left(V_\text{pk}/\sqrt{2}\right)^2/R\), which is the same number as \(V_\text{rms}^2/R\). The bias is not part of the \(\sqrt{2}\) relationship — handle it separately.

9. A US wall outlet is '120 volts'. What is the peak voltage?

\(V_\text{pk} = 120\sqrt{2} \approx 170\ \text{V}\). Every AC rating is RMS unless stated otherwise.

10. At what frequency does instantaneous power in a resistor oscillate?

At twice the signal frequency, and it is never negative. It swings between 0 and \(V_\text{pk}^2/R\), averaging \(V_\text{pk}^2/2R\).

11. Once you are working in RMS, what does the power equation look like?

Exactly like the DC one: \(P = V_\text{rms}I_\text{rms} = I_\text{rms}^2R = V_\text{rms}^2/R\).