GATE Analog Electronics formula sheet

Exponential model of the junction below breakdown (series resistance neglected). $I_S$ = reverse saturation current, $n$ = ideality factor (1 to 2). $V_T=25.9$ mV at 300 K; these notes round it to 25 mV, and 26 mV is also common, so take the value the question gives. For $V_D\gg n{V}_T$, $I_D\approx I_S{e}^{V_D/(n{V}_T)}$; for $V_D\ll -V_T$, $I_D\approx -I_S$.

Valid when both currents are much larger than $I_S$. With $n=1$ and $V_T=25$ mV, a tenfold change in current changes $V_D$ by 57.6 mV (about 60 mV with 26 mV).

Forward-active region only. With leakage included, $I_C=\beta I_B+(\beta +1)I_{CO}$. That is the form used to define the stability factor $S$.

Forward active with $V_{BE}\gg V_T$ and the Early effect neglected. $I_S$ is the saturation current, proportional to the emitter area, and $V_T$ is the thermal voltage (25 mV or 26 mV at room temperature; use the question's value). The constant-drop model replaces this curve by a fixed $V_{BE}$. Each decade of collector current changes $V_{BE}$ by $V_T\ln 10$: about 58 mV with $V_T=25$ mV, or 60 mV with $V_T=26$ mV. If an Early voltage $V_A$ is given, multiply $I_C$ by $(1+V_{CE}/V_A)$; the output resistance is then $r_o=V_A/I_C$.

Active region, DC collector current $I_C$, Early voltage $V_A$. $r_e$ is the resistance looking into the emitter with the base at AC ground (T-model), and $r_{\pi }=(\beta +1)r_e$. Example: at $I_C=1$ mA with $V_T=25$ mV, $g_m=40$ mA/V.

NMOS in saturation, square law. The $r_o$ expression uses $I_D(1+\lambda V_{DS})\approx I_D$. The gate draws no current, so the resistance looking into the gate is infinite. The body-effect transconductance $g_{mb}$ matters only when a signal appears between source and body, as in CG and CD stages with the body tied to a rail. Its value depends on the DC source-body voltage through $\chi =\gamma /(2\sqrt{2{\varphi }_F+V_{SB}})$. Neglect it unless the question gives $\gamma$, $\chi$ or $g_{mb}$.

Convention: $x_i=x_s-x_f$, $x_f=\beta x_o$. The feedback is negative when $A\beta >0$ in the mid-band (in general, when $|1+A\beta |>1$).

$T$ is the gain once round the loop. Every benefit of negative feedback, and the loss of gain, is by the factor $D$.

The input difference goes to zero only when negative feedback holds the output at a finite value inside $\pm V_{sat}$.

The input goes through $R_1$ to the virtual-ground node, so the source sees only $R_1$.

R in series from the input to the + terminal, C from the + terminal to ground; $R_1$ (− terminal to ground) and $R_f$ (output to − terminal) set the passband gain and do not affect $f_c$. $|H(jf)|=A_0/\sqrt{1+(f/f_c{)}^2}$ and $\angle H=-{\tan }^{-1}(f/f_c)$. With $R_f=0$ (voltage follower), $A_0=1$.

R and C swap places: C in series to the + terminal, R from the + terminal to ground. $|H(jf)|=A_0(f/f_c)/\sqrt{1+(f/f_c{)}^2}$, phase $90^{\circ }-{\tan }^{-1}(f/f_c)$. A real op-amp's finite gain-bandwidth product eventually turns every active high-pass into a band-pass; the ideal model ignores this.

Open-circuit noise of resistance $R$ at absolute temperature $T$ over noise bandwidth $B$; $k=1.38\times 10^{-23}\text{J/K}$. Model: a noiseless $R$ in series with this voltage source. One-sided spectral density $4kTR$ in ${\text{V}}^2/\text{Hz}$.

Short-circuit noise current: a noiseless $R$ in parallel with this current source. Use it for resistors in parallel or for a node driven by a current source.