PX157 - E2 - complex impedences

for DC, using ohm's law: $V = R I$
for AC, generalizing ohm's law:

\tilde{V} = \tilde{Z} \tilde{I}

$\tilde{Z} =$ complex impedance (units: $Ω$ )
$\tilde{Y} = \frac{1}{\tilde{Z}} =$ complex admittance (units: siemens ( $S$ ))

resistors

source:

V (t) = V_{0} \cos (ω t)

current:

I (t) = \frac{V (t)}{R} = \frac{V_{0}}{R} \cos (ω t) = I_{0} \cos (ω t)

for the resistor:

{\tilde{Z}}_{R} = R

capacitors

charge on capacitor: $Q = C V$
current: $I = \frac{d Q}{d t} = C \frac{d V}{d t}$
if $V (t) = V_{0} \cos (ω t) :$

I (t) = - C V_{0} ω \sin (ω t)

$V (t)$ and $I (t)$ are out of phase
complex notation:

\begin{aligned} \tilde{V} & = V_{0} e^{j ω t} \\ \tilde{I} & = j ω C \tilde{V} \\ {\tilde{Z}}_{C} & = \frac{1}{j ω C} \end{aligned}

\tilde{I} = ω C V_{0} e^{j ω t} e^{j \frac{π}{2}}

$\tilde{I}$ leads to $\tilde{V}$ by $\frac{π}{2} :$

I (t) = ω C V_{0} \cos (ω t + \frac{π}{2})

inductors

voltage across the inductor:

V = L \frac{d I}{d t}

if $V (t) = V_{0} \cos (ω t) :$

I (t) = \frac{V_{0}}{ω L} \sin (ω t)

$V (t)$ and $I (t)$ are out of phase
complex notation:

\begin{aligned} \tilde{I} & = I_{0} e^{j ω t} \\ \tilde{V} & = L \frac{d \tilde{I}}{d t} \\ = j ω L \tilde{I} \end{aligned}

comparing with $\tilde{V} = \tilde{Z} \tilde{I} :$

{\tilde{Z}}_{L} = j ω L

since $I = \frac{1}{j ω L} \tilde{V} :$

I = \frac{- j}{ω L} \tilde{V} = \frac{1}{ω L} \tilde{V} e^{- j \frac{π}{2}}

the current lags behind the voltage by $\frac{π}{2}$
imp note: distinguish ' $L$ ' from ' $C$ ' in writing