Capacitors

Capacitors

Stores energy in electric field between two plates.

Passes AC, blocks DC. "High pass filter"

Capacitance is the ratio of charge to voltage, measured in Farads. $$ C = \frac{Q}{V} $$

The energy stored in a capacitor is $$ E = \frac{1}{2}CV^2 $$

$$ \frac{d}{dt}Q = \frac{d}{dt}CV $$

$$ I_C = C\frac{dV}{dt} $$

Inductors

Stores energy in magnetic field around a coil of wire due to current flowing through it. Units are Henrys (H).

Passes DC, blocks AC. "Low pass filter"

Energy stored in an inductor is $$ E = \frac{1}{2}LI^2 $$

$$ V_L = L\frac{dI}{dt} $$

Complex Numbers Review

$$e = 2.71828...$$

$$ j \cdot j = -1 \to j = \sqrt{-1} $$

$$ e^{jq} = \cos(q) + j\sin(q) $$

Complex Numbers for AC Signals

• Pretend signals are complex during analysis, then take real part at the end.
• Multiplying by real number is scaling (magnitude/amplitude)
• Multiplying by imaginary number is phase shift
• Multiplying by complex number is scaling and phase shift

EX:

$$ cos(2\pi ft + \phi) = e^{j\phi}e^{j2\pi ft} $$

Impedance

The AC version of resistance.

$$ Z_{cap} = \frac{1}{j\omega C} $$

$$ Z_{ind} = j\omega L $$

$$ Z_{res} = R $$

Phasors

Complex numbers that represent the amplitude and phase of a sinusoidal signal. Can be used to represent AC signals and greatly simplify analysis by