UL/FE/UNI-ELT/NEL/Enačbe

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Objavljene enačbe

Osnove nelinearnih elementov

Četveropoli

$\underline{y}_{mn} = g_{mn} + \mbox{j}b_{mn}$

$\underline{z}_{mn} = r_{mn} + \mbox{j}x_{mn}$

$\underline{h}_{mn} = \mbox{Re}[g_{mn}] + \mbox{j }\mbox{Im}[b_{mn}]$

$\underline{I}_1 = \underline{y}_11 \underline{U}_1 + \underline{y}_12 \underline{U}_2$

$\underline{I}_2 = \underline{y}_21 \underline{U}_1 + \underline{y}_22 \underline{U}_2$

Polprevodnik

$n = N_C e^{-\frac{E_C - E_F}{kT}} = n_i e^{\frac{E_F - E_{F_i}}{qU_T}} = n_i e^{\frac{V_{F_i} - V_F}{U_T}}$

$p = N_V e^{-\frac{E_F - E_V}{kT}} = n_i e^{\frac{E_{F_i} - E_F}{qU_T}} = n_i e^{\frac{V_F - V_{F_i}}{U_T}}$

$J_n = q\mu_n n E + q D_n \frac{dn}{dx}$

$J_p = q\mu_p p E - q D_p \frac{dp}{dx}$

$\frac{\partial n}{\partial t} = - \frac{n - n_0}{\tau_n} + \frac{1}{q} \frac{\partial j_n}{\partial x}$

$\frac{\partial p}{\partial t} = - \frac{p - p_0}{\tau_p} - \frac{1}{q} \frac{\partial j_p}{\partial x}$

ρ = q (p - n + N D - N A)

$L = \sqrt{Dt}$

$-\frac{d^2V}{d x^2} = \frac{dE}{dx} = \frac{\rho}{\epsilon}$

$\frac{D}{\mu} = \frac{kT}{q} = U_T$

pn dioda

$I = AJ = Aq\left( \frac{D_p p_{n0}}{L_p} + \frac{D_n n_{p0}}{L_n} \right)\left(e^{\frac{U}{U_T}} -1 \right) = Aq n_i^2\left( \frac{D_p}{L_p N_D} + \frac{D_n}{L_n N_A} \right)\left(e^{\frac{U}{U_T}} -1 \right)=$

$= I_s\left(e^{\frac{U}{U_T}} -1 \right)$

$U_D = V_{F_{in}} - V_{F_{ip}} = U_T \mbox{ln}\frac{p_{p0}}{p_{n0}} = U_T \mbox{ln}\frac{n_{p0}}{n_{n0}} = U_{F_n} + U_{F_p} = U_T \mbox{ln}\frac{p_{p0} n_{n0}}{n_i} \approx$

$\approx U_T \mbox{ln}\frac{N_A N_D}{n_i} = U_p + U_n = \frac{q}{2 \epsilon}(N_Ax_p^2 + N_Dx_n^2)$

$x_n = \sqrt{\frac{2 \epsilon}{q}\frac{N_A}{N_D}\frac{(U_D + U_R)}{(N_A + N_D)}}$

$x_p = \sqrt{\frac{2 \epsilon}{q}\frac{N_D}{N_A}\frac{(U_D + U_R)}{(N_A + N_D)}}$

$D = x_n + x_p = \sqrt{\frac{2 \epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)(U_D + U_R)}$

Malosignalni model pn diode

$C_T = \frac{\epsilon A}{D}$

N A > > N D

$\underline{y} = \frac{\underline{I}}{\underline{U}} = \frac{\underline{I}}{\underline{U}_T}\sqrt{1 + j\omega\tau_p} = g + jb$

$\left.g\right|_{NF} = \frac{I}{U_T}$

$\left.g\right|_{VF} = \left.g\right|_{NF} \sqrt{\frac{\omega\tau_p}{2}}$

$\left.C_d\right|_{NF} = \frac{\left.g\right|_{NF} \tau_p}{2}$

$\left.C_d\right|_{VF} = \frac{\left.g\right|_{VF}}{\omega}$

Ebers-Mollov model bipolarnega tranzistorja (pnp)

I E = I F - α R I R

I C = - α F I F + I R

$I_F = I_{ES}\left( e^{\frac{U_{EB}}{U_T} - 1} \right)$

$I_R = I_{CS}\left( e^{\frac{U_{CB}}{U_T} - 1} \right)$

$I_E = I_{E0}\left( e^{\frac{U_{EB}}{U_T} - 1} \right) - \alpha_R I_C$

$I_C = - \alpha_F I_E I_{C0}\left( e^{\frac{U_{CB}}{U_T} - 1}\right)$

I E 0 = I E S (1 - α F α R)

I C 0 = I C S (1 - α F α R)

α F I E S = α R I C S

Malosignalni model bipolarnega tranzistorja

$g_m = \alpha_0 g_e = \frac{I_C}{U_T}$

$g_e = \frac{I_E}{U_T}$

$r_{be} = \frac{\beta}{g_m}$

$\left.\underline{A}_i\right|_{\underline{U}_{CB}=0} = \left.- \frac{\underline{I}_C}{\underline{I}_E}\right|_{\underline{U}_{CB}=0} = \frac{\alpha_0}{1 + \frac{j\omega}{\omega_\alpha}} = \frac{\alpha_0}{1 + \frac{j f}{f_\alpha}} = \underline{\alpha}(f)$

Spojni FET (n-kanal)

$I_{DS} = I_{DsS}\left( 1 - \frac{U_{GS}}{U_P} \right)^2$

$U_P = U_D - \frac{qN_DD^2}{8\epsilon}$

U D S s a t = U G S - U P

MOS tranzistor (n-kanal)

$I_D = \frac{C_0W\mu_n}{L}\left[ (U_{GS} - U_T)U_{DS} - \frac{U_{DS}^2}{2} \right]$

$I_{DS} = \frac{C_0W\mu_n}{2 L}U_T^2\left( 1 - \frac{U_{GS}}{U_T} \right)^2$

$U_{DSsat} = U_{GS} - U_T\,\!$

Štirislojna dioda

$I = \frac{M I_{C0}}{1-M\alpha_1 - M\alpha_2}$

Tiristor

$I = \frac{M I_{C0} + M\alpha_2I_G}{1-M\alpha_1 - M\alpha_2}$

Fotodioda

$I = - I_S\left( e^{-\frac{U}{nU_T}} - 1 \right) + I_L$

Fototranzistor

$-I_C = -I_B\frac{\alpha_F}{1 - \alpha_F} + \frac{I_{C0} + I_L}{1 - \alpha_F}$

Nelinearni elementi

Polprevodnik v termičnem ravnovesju

$f_{FD}(E) = \frac{1}{1 + e^\frac{E - E_F}{kT}}$

$f_{MB}(E) = e^{-\frac{E - E_F}{kT}}$

$f_{D}(E) = \frac{1}{1 + \frac{1}{2}e^\frac{E_D - E_F}{kT}}$

$f_{A}(E) = \frac{1}{1 + 2e^\frac{E_A - E_F}{kT}}$

$1 - \left( \frac{1}{1+x} \right) = \frac{1}{1 + 1/x}$

Polprevodnik izven termičnega ravnovesja

$\frac{\partial n}{\partial t} = G - R + \frac{1}{q} \frac{\partial j_n}{\partial x}$

$\frac{\partial p}{\partial t} = G - R - \frac{1}{q} \frac{\partial j_p}{\partial x}$

Relaksacije

$\tau_R = \frac{\epsilon}{\sigma}$

SRH generacije in rekombinacije

$f = \frac{\sigma_n n + \sigma_p p_1}{\sigma_n (n + n_1) + \sigma_p(p+p_1)}$

$n_1 = N_C e^{-\frac{E_C - E_t}{kT}} = n_i e^\frac{E_t - E_{Fi}}{kT}$

$p_1 = N_V e^{-\frac{E_t - E_V}{kT}} = n_i e^\frac{E_{Fi} - E_t}{kT}$

$G-R = G_n-R_n=G_p-R_p = \frac{n_i^2 - pn}{\tau_{p0}(n+n_1) + \tau_{n0}(p + p_1)}$

$\tau_{p0} = \frac{1}{\sigma_p \nu_{th}N_t}$

$\tau_{n0} = \frac{1}{\sigma_n \nu_{th}N_t}$

Za $\Delta n <\!\!< n_0$ , $\Delta p <\!\!< p_0$ in $Δ p = Δ p$ :

$G-R = G_n-R_n=G_p-R_p = \frac{-\Delta n}{\tau_{p0}(n_0+n_1)/(n_0 + p_0) + \tau_{n0}(p_0 + p_1)/(n_0 + p_0)} = -\frac{\Delta n}{\tau}$

Polprevodnik pri osvetlitvi

$E_{ph} = \mbox{h}\frac{c}{\lambda}$

$J_{ph} = \Phi_{ph}E_{ph}\,\!$

$\Phi_{ph}(x) = \Phi_{ph}(0)e^{-\alpha_{ph} x}$

G L = α p h Φ p h

Planarni npn tranzistor

$-J_{nE} = q D_n n_{B0}(0) \left\{ \left( \frac{\ln K}{2w} + \frac{b}{\mbox{th}(bw)} \right) \left[ e^\frac{U_{BE}}{U_T} -1 \right] - \frac{b\sqrt{K}}{\mbox{sh}(bw)} \left[e^\frac{U_{BC}}{U_T} - 1 \right] \right\}$

$-J_{nC} = q D_n n_{B0}(0) \left\{ \frac{b\sqrt{K}}{\mbox{sh}(bw)} \left[e^\frac{U_{BE}}{U_T} - 1 \right] - \left( \frac{b}{\mbox{th}(bw)} - \frac{\ln K}{2w} \right)K \left[ e^\frac{U_{BC}}{U_T} -1 \right]\right\}$

$b = \sqrt{\left( \frac{\ln K}{2w} \right)^2 + \frac{1}{L_n^2}}$

$K = \frac{N_A(0)}{N_A(w)}$

Gummel-Poonov model npn tranzistorja

$-I_E = \frac{I_S}{q_B}\left( e^\frac{U_{BE}}{U_T} - e^\frac{U_{BC}}{U_T} \right) + \frac{I_S}{\beta_F}\left( e^\frac{U_{BE}}{U_T} - 1 \right) + I_1\left( e^{\frac{U_{BE}}{n_E U_T}} - 1 \right)$

$-I_C = \frac{I_S}{q_B}\left( e^\frac{U_{BE}}{U_T} - e^\frac{U_{BC}}{U_T} \right) + \frac{I_S}{\beta_R}\left( e^\frac{U_{BC}}{U_T} - 1 \right) + I_2\left( e^{\frac{U_{BC}}{n_C U_T}} - 1 \right)$

$q_B = \frac{q_1}{2}\left(1 + \sqrt{1 + \frac{4q_2}{q_1^2}}\right)$

$q_1 = \frac{1}{\left( 1-\frac{U_{BC}}{|U_A|}-\frac{U_{BE}}{|U_B|} \right)}$

$q_2 = \frac{I_S}{I_{KF}} \left( e^\frac{U_{BE}}{U_T} -1 \right) + \frac{I_S}{I_{KR}} \left( e^\frac{U_{BC}}{U_T} -1 \right)$

Sončna celica

$I = I_L - I_S\left( e^\frac{u}{nU_T} - 1\right)$

$U_{mp} \approx U_OC - U_T \ln$

MOS struktura

$U_G = U_FB + \Phi_S + \Phi_{ox}\,\!$

$U_T = U_FB + 2\Phi_B + \Phi_{ox}\,\!$

$\Phi_{ox} = \pm\frac{q N_{sub} x_{ox}}{\epsilon_{ox}} D$

$\Phi_S = \pm\frac{q N_{sub}}{2 \epsilon_S} D^2$

$\Phi_{ox} = -\frac{Q_s + Q_{SS}}{C_{ox}} = -\frac{\sigma_s + \sigma_{SS}}{\epsilon_{ox}/x_{ox}}$

$Q_S = \pm A \sqrt{4 q \epsilon_S N_{sub} |\phi_B|}$

Schottky-eva dioda

$I_S = AT^2 e^{-\frac{E_B}{kT}}$

Hallova sonda (p-tip)

$U_H = R_HwJ_xB_z = R_H \frac{1}{d}I_xB_z = K_B I_x B_z$

$R_H = \frac{r_H}{qp}$

$K_B = \frac{R_H}{d}$

$r_H = \frac{\mu_H}{\mu_p}$

Svetloba

	Radiometrične veličine	Fotometrične veličine
Sevanje:	$P_{ph} \quad[\mbox{W}]$	$P_v \quad[\mbox{lm}]$
Občutljivost očesa:	$P_{ph\lambda}(\lambda) = dP_{ph}/d\lambda\,\!$	$P_{v\lambda}(\lambda) = K (\lambda)\cdot P_{ph\lambda}(\lambda)$
Gostota sevanja vira:	$I_{ph} = dP_{ph}/d\Omega \quad [\mbox{W}/\mbox{sr}]$	$I_v = dP_v/d\Omega \quad [\mbox{cd} = \mbox{lm}/\mbox{sr}]$
Gostota vpadnega sevanja:	$J_{ph} = dP_{ph}/dA \quad [\mbox{W}/\mbox{m}^2]$	$J_v = dP_v/dA \quad [\mbox{lx} = \mbox{lm}/\mbox{m}^2]$

Prostorski kot

Ω = A / r 2

Ω = 2π(1 - cosθ)

Tabela fizikalnih konstant

Boltzmannova konstanta: $k = 1.38\cdot 10^{-23} \mbox{J}/\mbox{K}$
absolutna vrednost naboja elektrona: $q = 1.6\cdot 10^{-19} \mbox{As}$
Planckova konstanta: $h = 6.625\cdot 10^{-34} \mbox{Ws}^2$
masa elektrona: $m = 9.11\cdot 10^{-31} \mbox{kg}$
termična napetost: $U_T = \frac{kT}{q} = 25.66 \mbox{eV} \qquad (T = 297.8\mbox{K} = 24.8^\circ\mbox{C})$
intrinsična koncentracija (Si): $n_i = 10^{10}\mbox{cm}^3 \qquad (T = 297.8\mbox{K} = 24.8^\circ\mbox{C})$
dielektrična konstanta: $\epsilon_0 = 8.854\cdot10^{-12}\mbox{As}/\mbox{Vm} = 8.854\cdot10^{-14}\mbox{As}/\mbox{Vcm}$
relativna dielektrična konstanta Si: $\epsilon_r(\mbox{Si}) = 11.7 \approx 12 \qquad \epsilon_0 \epsilon_{rSi} = 10^{-12} \mbox{As}/\mbox{Vcm}$
relativna dielektrična konstanta Si0₂: $\epsilon_r(\mbox{SiO}_2) = 3.85 \approx 4$