AC Analysis

Signal x (time function) is alternating for

x (t) = X_{m} \cos (ω t + ϕ), X_{m} \geq_{0}, ω > 0

$X_{m}$ - amplitude
$ω$ - pulsation
$ϕ$ - phase shift
$f = \frac{ω}{2 π}$ - frequency
$T = \frac{1}{ω}$ - period

Important

AC circuit has solutions where alternating signals have the same pulsation

Example

$i (t) = C u_{C}^{'} (t)$
$e (t) = \cos (ω t) [V]$

$u_{c} (t) = U_{c_{m}} \cos (ω t + ϕ)$
$u_{c}^{'} (t) = - U_{c_{m}} ω \sin (ω t + ϕ)$

$e (t) = u_{c} (t) + R i (t) = u_{c} (t) + R C u^{'} (c) (t)$
$\cos (ω t) [V] = U_{c_{m}} \cos (ω t + ϕ) - R C U_{c_{m}} ω \sin (ω t + ϕ) = U_{c_{m}} \sqrt{1 + (ω R C)^{2}} \cos (ω t + ϕ)$

U_{c_{m}} = \frac{1 [V]}{\sqrt{1 + (ω R C)^{2}}}

AC signal notation

$2 V \cos (ω t) = 2 V e^{j_{0}}$
$3 V \cos (ω t + \frac{π}{3}) = 3 V e^{j π / 3}$
$1 m A \cos (ω t - \frac{π}{4}) = 1 m A e^{- j π / 4}$

Phasors

Alternating signal $x (t) = X_{m} \cos (ω t + ϕ)$
has a phasor $X = X_{m} e^{j ϕ} = X_{m} (\cos ϕ + j \sin ϕ)$

Components

Voltage source

I_{m} \cos (ω t + ϕ) \to I_{m} e^{j ϕ}

Current source

E_{m} \cos (ω t + ϕ) \to E_{m} e^{j ϕ}

Resistor

$u (t) = U_{m} \cos (ω t + ϕ_{u}) = R i (t) = R I_{m} \cos (ω t + ϕ_{i})$

U = U_{m} e^{j ϕ_{u}}

I = I_{m} e^{j ϕ_{i}}

Inductors

$u (t) = U_{m} \cos (ω t + ϕ_{u}) = L i^{'} (t) = ω L I_{m} \cos (ω t + ϕ_{i} + \frac{π}{2})$

U = U_{m} e^{j ϕ_{u}}

I = I_{m} e^{j ϕ_{i}}

U = ω L I e^{j \frac{π}{2}} = j ω L I

Capacitors

$u (t) = U_{m} \cos (ω t + ϕ_{i}) = C u^{'} (t) = ω C U_{m} \cos (ω t + ϕ_{u} + \frac{π}{2})$

U = U_{m} e^{j ϕ_{u}}

I = I_{m} e^{j ϕ_{i}}

I = ω C u e^{j \frac{π}{2}} = j ω C u

U = \frac{1}{j ω C} I

Devices described in time domain by linear differential equations (inductors, capacitors, ...) are governed by linear algebraic equations in the phasor domain

Impedance

AC analysis tools

KCL, KVL

Nodal method

Superposition rule

CDF, VDF

Thevenin and Norton equivalents

Example

Math helper

$a r g (z_{1} z_{2}) = a r g (z_{1}) + a r g (z_{2})$
$a r g (\frac{z_{1}}{z_{2}}) = a r g (z_{1}) - a r g (z_{2})$
$| a + j b | = \sqrt{a^{2} + b^{2}}$

a r g (a + j b) = {\begin{cases} \frac{π}{2} & a = 0 \cap b > 0 \\ - \frac{π}{2} & a = 0 \cap b < 0 \\ \arctan (\frac{b}{a}) & a > 0 \\ \arctan (\frac{b}{a}) + \frac{π}{2} & a < 0 \end{cases}