Lesson 16 Flashcards — Complex Math and Impedance#
Click a question to reveal the answer.
1. Why do we need complex numbers for AC circuits?
A sinusoid has two independent properties, amplitude and phase. A single real number cannot carry both; a complex number carries exactly two.
2. Which form is best for addition? For multiplication?
Rectangular (\(Re + jIm\)) for addition and subtraction. Phasor (polar) (\(A\angle\phi\)) for multiplication and division.
3. How do you convert rectangular to phasor (polar) form?
\(A = \sqrt{Re^2+Im^2}\) and \(\phi = \tan^{-1}(Im/Re)\). Check the quadrant — \(\tan^{-1}\) cannot distinguish quadrant II from IV.
4. How do you write a sinusoid as a phasor, and a phasor back as a sinusoid?
\(v(t) = V_m\cos(2\pi f t + \phi) \longleftrightarrow V = V_m\angle\phi\). The frequency never changes, so only magnitude and angle travel. A phasor carries whichever kind of magnitude the problem gives you — an amplitude or an RMS value — and the answer comes back in the same kind; \(\sqrt{2}\) only converts between them.
5. For a capacitor, does current lead or lag the voltage? For an inductor?
Capacitor: current leads by \(90^\circ\). Inductor: current lags by \(90^\circ\). Check it against the sign of \(Z_L = +j\omega L\).
6. Give the impedance of a resistor, capacitor, and inductor.
\(Z_R = R\), \(Z_C = 1/(j\omega C) = -j/(\omega C)\), \(Z_L = j\omega L\), with \(\omega = 2\pi f\).
7. What does a capacitor look like at DC? At high frequency?
At DC, \(|Z_C| \to \infty\) — an open circuit. At high frequency, \(|Z_C| \to 0\) — a short circuit.
8. What does an inductor look like at DC? At high frequency?
Exactly the reverse of a capacitor: a short at DC and an open at high frequency.
9. Once elements are expressed as impedances, what changes about circuit analysis?
Nothing. Series impedances add, parallel is product over sum, and Ohm's Law and the divider rules apply unchanged. Only the arithmetic is complex. One Lesson 4 habit does not carry over: two opposite reactances in parallel can be larger than either branch.
10. Why does impedance depend on frequency, and why does that matter?
Because \(Z_C\) and \(Z_L\) both contain \(\omega\). That frequency dependence is precisely what makes filters possible.
11. What must you check on your calculator before a complex conversion?
The angle unit — degrees versus radians. A calculator left in the wrong mode produces an answer that looks entirely reasonable and is wrong.
12. A 1 kΩ resistor is in series with a capacitor of impedance −j1592 Ω. What is Z_EQ in phasor form?
\(Z_\text{EQ} = 1000 - j1592 = 1880\angle{-}57.9^\circ\ \Omega\). The angle is negative, so the circuit is capacitive and the current leads: \(\phi = -57.9^\circ\) and \(\text{pf} = \cos\phi = 0.532\).