Formulario

Serie / parallelo / partitori

$R_s = \sum R_k$, $\;G_p = \sum G_k$, $\;R_1\parallel R_2 = \frac{R_1R_2}{R_1+R_2}$

$v_k = v\frac{R_k}{\sum R}$, $\;i_1 = i\frac{R_2}{R_1+R_2}$

Stella ↔ triangolo

$R_a = \frac{R_{ab}R_{ca}}{R_{ab}+R_{bc}+R_{ca}}$

$R_{ab} = \frac{R_aR_b+R_bR_c+R_cR_a}{R_c}$

Millman

$v = \dfrac{\sum e_kG_k + \sum a_j}{\sum G_k}$

Thevenin / Norton

$v = v_{eq} + R_{eq}i$ · $i = G_{eq}v - i_{eq}$

$v_{eq} = v_{c.a.}$, $i_{eq} = i_{c.c.}$, $R_{eq} = v_{c.a.}/i_{c.c.}$

$R_{eq}$: spegni solo gli indipendenti

AO ideale · BJT

$v_+ = v_-$, $i_\pm = 0$

$v_{be} = h_{ie}i_b + h_{re}v_{ce}$, $i_c = h_{fe}i_b + h_{oe}v_{ce}$

Trasformatore · giratore

$v_1 = nv_2$, $i_2 = -ni_1$ ($R \to n^2R$)

$v_1 = -ri_2$, $v_2 = ri_1$ ($R\to r^2/R$)

Max potenza (DC)

$R_L = R_{eq}$, $P_{max} = \dfrac{v_{eq}^2}{4R_{eq}}$

Topologia

$n-1$ KI indip., $l-n+1$ KV indip.; Tellegen $\sum v_ki_k = 0$

Fasori e impedenze

$v = \mathrm{Re}[\bar Ve^{j\omega t}]$, $V_{eff} = V_M/\sqrt2$

$Z_R = R$, $Z_L = j\omega L$, $Z_C = \frac{1}{j\omega C}$

$\omega = 2\pi f$ (50 Hz → 314 rad/s)

Potenze (efficaci)

$\bar S = \bar V\bar I^* = P + jQ$

$P = VI\cos\varphi = RI^2$, $Q = VI\sin\varphi = XI^2$

$A = VI = \sqrt{P^2+Q^2}$, $\cos\varphi = P/A$

Elementi

$Q_L = X_LI^2 = V^2/X_L$

$Q_C = -\omega CV^2$

$p_{max} = P + A$, $p_{min} = P - A$

$I_a = I\cos\varphi$, $I_r = I\sin\varphi$

Rifasamento

$C = \dfrac{Q - P\tan\varphi'}{\omega V^2}$

$\tan(\arccos0.95) = 0.3287$ · $\tan(\arccos0.9) = 0.4843$

$I' = \dfrac{P}{V\cos\varphi'}$

Boucherot · energia

$P = \sum P_k$, $Q = \sum Q_k$

$Q = 2\omega(W_M - W_E)$, $W_M = \tfrac12LI^2$, $W_E = \tfrac12CV^2$

Max trasferimento

$\bar Z_u = \bar Z_s^*$

$P_d = \dfrac{V_{sM}^2}{8R_s} = \dfrac{V_s^2}{4R_s}$, $\eta = 50\%$

1° ordine

$\tau = R_{eq}C$, $\;\tau = L/R_{eq}$

$x(t) = x_\infty + (x_0 - x_\infty)e^{-t/\tau}$

Risonanza

$\omega_0 = 1/\sqrt{LC}$, $B = \omega_0/Q$

$Q_s = \omega_0L/R$, $\;Q_p = \omega_0CR$

Trasformate

$\delta \to 1$, $u \to \frac1s$, $t \to \frac1{s^2}$, $e^{-at}\to\frac1{s+a}$, $te^{-at}\to\frac1{(s+a)^2}$

$\sin\omega t \to \frac{\omega}{s^2+\omega^2}$, $\cos\omega t\to\frac{s}{s^2+\omega^2}$

$e^{-at}\sin\omega t \to \frac{\omega}{(s+a)^2+\omega^2}$, $e^{-at}\cos\omega t \to \frac{s+a}{(s+a)^2+\omega^2}$

Proprietà

$\dot f \to sF - f(0^-)$, $\;\ddot f \to s^2F - sf(0^-) - \dot f(0^-)$

$\int f \to F/s$, $\;f(t-T)u(t-T) \to e^{-sT}F$

$f(0^+) = \lim_{s\to\infty}sF$, $\;f(\infty) = \lim_{s\to0}sF$

Equivalenti di Laplace

L: $V = sLI - Li(0^-)$ · $I = \frac{V}{sL} + \frac{i(0^-)}{s}$

C: $V = \frac{I}{sC} + \frac{v(0^-)}{s}$ · $I = sCV - Cv(0^-)$

2° ordine

serie $\alpha = \frac{R}{2L}$, parallelo $\alpha = \frac{1}{2RC}$, $\omega_0 = \frac1{\sqrt{LC}}$

$s_{1,2} = -\alpha\pm\sqrt{\alpha^2-\omega_0^2}$, $\omega_d = \sqrt{\omega_0^2-\alpha^2}$

$R_c = 2\sqrt{L/C}$, $\;G_c = 2\sqrt{C/L}$

Costanti ($\Delta = x_0 - x_\infty$)

Sovra: $k_1 = \frac{\dot x_0 - s_2\Delta}{s_1-s_2}$, $k_2 = \frac{s_1\Delta - \dot x_0}{s_1-s_2}$

Critico: $k_1 = \Delta$, $k_2 = \dot x_0 + \alpha\Delta$

Sotto: $k_1 = \Delta$, $k_2 = \frac{\dot x_0 + \alpha\Delta}{\omega_d}$

Derivata iniziale

$\dot i_L(0^+) = v_L(0^+)/L$

$\dot v_C(0^+) = i_C(0^+)/C$

Fratti semplici

$A = \lim_{s\to p}(s-p)F = N(p)/D'(p)$

Coppia complessa: $2e^{-\alpha t}(\mathrm{Re}B\cos\omega t - \mathrm{Im}B\sin\omega t)$

Due porte

z: $v = Zi$ · y: $i = Yv$ · h: $\binom{v_1}{i_2} = H\binom{i_1}{v_2}$

ABCD: $v_1 = Av_2 - Bi_2$, $i_1 = Cv_2 - Di_2$

Reciproca: $z_{12}=z_{21}$, $h_{12}=-h_{21}$, $AD-BC=1$

Serie ΣZ · parallelo ΣY · cascata ΠT

$Z_{in} = z_{11} - \frac{z_{12}z_{21}}{z_{22}+Z_L}$