accheatsheet.mdown

% AC Cheatsheet % Diego Trapero

AC Cheatsheet

Sinusoidal signal

A sinusoidal signal in AC takes the form

$$s(t) = \sqrt{2} A \cdot \cos{\omega t + \varphi}$$

where

• $A \equiv A_{RMS}$ is the RMS value of the signal
• $\omega$ is the angular frecuency, $\omega = 2 \pi f$ $[rad/s]$
• Frecuency, $f = \omega / 2\pi$
• Period, $T = 1/f$
• Amplitude of the signal is $A/\sqrt{2}$

Note: In AC the sinusoidal function amplitude is taken as $A/\sqrt{2}$, so $A$ is the RMS value of the signal. This way, we can directly use the RMS values of the signals to make calculations with power.

Average and RMS values

• Average value: $S_{AVG} = 0$
• RMS value: $S_{RMS} = A$

Phasors and complex numbers

Phasor

$$s(t) = \sqrt{2} A \cdot \cos{\omega t + \varphi} \to \mathbf{S} = A \angle \varphi $$

In physics and engineering, a phase vector, or phasor, is a representation of a sinusoidal function whose amplitude, frequency, and phase are time-invariant

Euler's formula indicates that sinusoids can be represented mathematically by the sum of two complex-valued functions:

$$A\cdot \cos{\omega t + \theta} = A \cdot \frac{e^{i(\omega t + \theta)} + e^{-i(\omega t + \theta)}}{2}$$

or by the real part of one of the functions:

$$ A\cdot \cos{\omega t + \theta} = \operatorname{Re} \left{ A\cdot e^{i(\omega t + \theta)}\right} \\ = \operatorname{Re} \left{ A e^{i\theta} \cdot e^{i\omega t}\right}. $$

The term phasor can refer to either $A e^{i\theta} e^{i\omega t},$ or just the complex constant, $A e^{i\theta},$. In the latter case, it is understood to be a shorthand notation, encoding the amplitude and phase of an underlying sinusoid.

An even more compact shorthand is angle notation: $A \angle \theta.,$

Generalized Ohm Law, complex impedance

The Generalized Ohm Law:

$$ \mathbf{V} = \mathbf{Z} \mathbf{I} $$

where $\mathbf{Z}$ is

• $\mathbf{Z} = R$ for resistors
• $\mathbf{Z} = \frac{1}{j \omega C} = \frac{-j}{\omega C}$ for capacitors
• $\mathbf{Z} = j \omega L$ for inductors

AC Power

Power in a two-terminal component

• Voltage: $v(t) = \sqrt{2} V \cdot \cos{\omega t}$ ($\varphi = 0$ because it's the reference)
• Current: $i(t) = \sqrt{2} I \cdot \cos{\omega t - \varphi}$

$$ \begin{circuitikz} \draw (1,1) togeneric, i>_=$i(t)$, v^<=$v(t)$ (0,1) -- (1,1) (0,-1) -- (1,-1) (0,1) node[ocirc]{} (0,-1) node[ocirc]{} ; \end{circuitikz} $$

Instantaneous power

1. Instantaneous power definition

\begin{align*} p(t) &= v(t) i(t) \ &= \sqrt{2} V \cos{\omega t} \cdot \sqrt{2} I \cos{\omega t - \varphi} \ &= 2 VI \cos{\omega t} \cos{\omega t + \varphi} \ \end{align*}

2. Transform into sum of cosines using the identity $\cos{a \cdot b} = \frac{1}{2} [ \cos{a+b} + \cos{a-b} ]$

\begin{align*} p(t) &= \cancel{2} VI \cdot \cancel{\frac{1}{2}} [ \cos{\cancel{\omega t} - \cancel{\omega t} + \varphi} + \cos{\omega t + \omega t - \varphi} ] \ &= VI [ \cos{\varphi} + \cos{2 \omega t - \varphi} ] \ &= VI \cos{\varphi} + VI \cos{2 \omega t - \varphi} \ \end{align*}

The instantaneous power in an AC component is:

$$ p(t) = VI \cos{\varphi} + VI \cos{2 \omega t - \varphi} $$

We observe

• Power in the component has a constant term $VI \cos{\varphi}$ and a fluctuating term that varies with time $VI \cos{2 \omega t - \varphi}$.

Average power

The instantaneous power is $p(t) = VI \cos{\varphi} + VI \cos{2 \omega t - \varphi}$, so we can calculate the average power using the definition:

$\bar{p(t)} = integral VI \cos{\varphi} + VI \cos{2 \omega t - \varphi}$

Active and reactive power

• Active power: $P = VI \cos{\varphi}$
• Reactive power: $Q = VI \sin{\varphi}$

Complex and apparent power

• Complex power is the complex number $\mathbf{S} = \mathbf{V} \mathbf{I}^* = P + Qj$
• Apparent power is the module of the complex power, $S = VI$

Power factor

$$PF = \frac{P}{S}$$

$$PF = \cos{\varphi}$$

AC power in a resistor

$$P = VI$$ $$Q = 0$$

Power in a inductor

Power in a capacitor

Three-phase systems

Delta-Wye transform

$$Z_{\Delta} = 3 Z_{Y}$$