ECE Course Equations

ece-1109 ECE-1109 Introduction to ECE

Key Equations from ECE 1109 Notes

I. Semiconductor Physics and Diodes (Pillars 1 & 2)

These equations govern carrier concentration, junction formation, and diode operation.

Category	Equation	Meaning of Symbols	Applicable Conditions/Context
Mass Action Law (PYQ Focus)	$\mathbf{n}\cdot \mathbf{p}={\mathbf{n}}_{\mathbf{i}}^2$	$n$: free electron concentration; $p$: free hole concentration; ${\mathbf{n}}_{\mathbf{i}}$: intrinsic carrier concentration (density).	Thermal equilibrium condition.
Extrinsic Carrier (n-type)	${\mathbf{P}}_{\mathbf{n}}=\frac{{\mathbf{n}}_{\mathbf{i}}^2}{{\mathbf{N}}_{\mathbf{D}}}$	$P_n$: concentration of holes (minority carriers) in $n$-type material; $N_D$: concentration of donor ions.	$n$-type material, assuming $N_A=0$ and electron concentration is dominated by donor doping ($\mathbf{n}\approx {\mathbf{N}}_{\mathbf{D}}$).
Extrinsic Carrier (p-type)	${\mathbf{n}}_{\mathbf{p}}=\frac{{\mathbf{n}}_{\mathbf{i}}^2}{{\mathbf{N}}_{\mathbf{A}}}$	$n_p$: concentration of electrons (minority carriers) in $p$-type material ; $N_A$: concentration of acceptor ions.	$p$-type material, assuming $N_D=0$ and hole concentration is dominated by acceptor doping ($\mathbf{p}\approx {\mathbf{N}}_{\mathbf{A}}$).
Drift Current Density	$J=en{\mu }_{n}E+ep{\mu }_{p}E$	$J$: total drift current density; $e$: charge magnitude; ${\mu }_n,{\mu }_p$: electron and hole mobility, respectively; $E$: electric field .	Total current density due to the influence of an electric field (drift) .
P-N Junction Contact Potential (PYQ Derivation)	${\mathbf{V}}_0=\frac{\mathbf{KT}}{\mathbf{q}}\ln \frac{{\mathbf{N}}_{\mathbf{D}}{\mathbf{N}}_{\mathbf{A}}}{{\mathbf{n}}_{\mathbf{i}}^2}$	$V_0$: Contact potential/Built-in potential ($V_{bi}$); $K$: Boltzmann constant; $T$: Temperature (K); $q$: elementary charge; $N_D,N_A,n_i$: concentrations defined above .	P-N junction in thermal equilibrium .
Depletion Region Width (PYQ Derivation)	$\mathbf{W}=\sqrt{\frac{2ϵ({\mathbf{V}}_{\mathbf{bi}}-\mathbf{V})}{\mathbf{q}}\left(\frac{1}{{\mathbf{N}}_{\mathbf{A}}}+\frac{1}{{\mathbf{N}}_{\mathbf{D}}}\right)}$	$W$: width of depletion region; $ϵ$: electrical permittivity ; $V_{bi}$: built-in potential; $V$: applied voltage.	Applies to a P-N junction . $\mathbf{V}$ is positive for forward bias and negative for reverse bias .
Diode Current (Shockley Equation)	${\mathbf{I}}_{\mathbf{D}}={\mathbf{I}}_{\mathbf{s}}({\mathbf{e}}^{\frac{{\mathbf{V}}_{\mathbf{D}}}{\eta {\mathbf{V}}_{\mathbf{T}}}}-1)$	$I_D$: Diode current; $I_s$: Reverse saturation current; $V_D$: Applied diode voltage; $\eta$: Ideality factor; ${\mathbf{V}}_{\mathbf{T}}$: thermal voltage .	General equation describing the V-I characteristics of a P-N junction diode .
Thermal Voltage	${\mathbf{V}}_{\mathbf{T}}=\frac{\mathbf{kT}}{\mathbf{q}}$	$\mathbf{k}$: Boltzmann constant ($1.38\times 10^{-23}$ J/K) ; $T$: Absolute temperature in Kelvin ($T_K=T_C+273^{\circ }$) ; $q$: Elementary charge .	Applicable to all semiconductor theory equations requiring thermal voltage .
Dynamic (AC) Resistance ($r_d$) (PYQ Focus)	${\mathbf{r}}_{\mathbf{d}}=\frac{\mathbf{\Delta }{\mathbf{V}}_{\mathbf{d}}}{\mathbf{\Delta }{\mathbf{I}}_{\mathbf{d}}}=\frac{\eta {\mathbf{V}}_{\mathbf{T}}}{{\mathbf{I}}_{\mathbf{D}}}$	${\mathbf{r}}_{\mathbf{d}}$: dynamic AC resistance (defined by tangent line at Q-point ); $\Delta V_d/\Delta I_d$: change in voltage/current .	Diode operating in the forward bias region (active region) . The approximation ${\mathbf{r}}_{\mathbf{d}}\approx \frac{26\text{ mV}}{{\mathbf{I}}_{\mathbf{D}}}$ (for $\eta \approx 1$) is often used for numerical calculations .
DC (Static) Resistance	$R_D=\frac{V_D}{I_D}$	$R_D$: DC resistance .	Defined as a point on the characteristics (used in graphical analysis) .
Load Line Analysis	$I_D=-\frac{1}{R_L}{V}_D+\frac{V}{R_L}$	$V$: source voltage; $R_L$: load resistance.	Derived from KVL for a series diode circuit ($V=I_D{R}_L+V_D$). Used to find the Quiescent (Q) operating point .

II. Rectifier Circuits (PYQ Numericals Focus)

These formulas are essential for calculation problems involving half-wave and full-wave rectification.

Category	Equation	Meaning of Symbols	Applicable Conditions/Context
Peak Load Current	${\mathbf{I}}_{\mathbf{m}}=\frac{\sqrt{2}{\mathbf{V}}_{\mathbf{rms}}}{{\mathbf{R}}_{\mathbf{f}}+{\mathbf{R}}_{\mathbf{L}}}$	${\mathbf{I}}_{\mathbf{m}}$: Peak current; $V_{rms}$: RMS input voltage; $R_f$: Internal diode resistance; $R_L$: Load resistance .	Calculation of maximum current when the diode is conducting .
Half-Wave DC Current	${\mathbf{I}}_{\mathbf{dc}}=\frac{{\mathbf{I}}_{\mathbf{m}}}{\pi }$	${\mathbf{I}}_{\mathbf{dc}}$: DC (average) load current .	Half-wave rectification .
Half-Wave RMS Current	${\mathbf{I}}_{\mathbf{rms}}=\frac{{\mathbf{I}}_{\mathbf{m}}}{2}$	${\mathbf{I}}_{\mathbf{rms}}$: RMS load current .	Half-wave rectification .
Half-Wave DC Voltage	${\mathbf{V}}_{\mathbf{dc}}=\frac{{\mathbf{V}}_{\mathbf{m}}}{\pi }$	${\mathbf{V}}_{\mathbf{dc}}$: DC (average) load voltage .	Half-wave rectification (ideal diode assumption: $V_{dc}=0.318V_m$) .
Half-Wave PIV	$\mathbf{PIV}={\mathbf{V}}_{\mathbf{m}}$	$\mathbf{PIV}$: Peak Inverse Voltage .	Half-wave rectification .
Full-Wave Average Current	${\mathbf{I}}_{\mathbf{avg}}=\frac{2{\mathbf{I}}_{\mathbf{m}}}{\pi }$	${\mathbf{I}}_{\mathbf{avg}}$: Average load current .	Full-wave rectification (Center-tapped or Bridge) .
Full-Wave DC Voltage	${\mathbf{V}}_{\mathbf{dc}}=\frac{2{\mathbf{V}}_{\mathbf{m}}}{\pi }$	${\mathbf{V}}_{\mathbf{dc}}$: DC (average) load voltage .	Full-wave rectification (ideal diode assumption: $V_{dc}=0.636V_m$) .
Full-Wave RMS Current	${\mathbf{I}}_{\mathbf{rms}}=\frac{{\mathbf{I}}_{\mathbf{m}}}{\sqrt{2}}$	${\mathbf{I}}_{\mathbf{rms}}$: RMS load current .	Full-wave rectification .
General Efficiency	$\eta \%=\frac{{\mathbf{P}}_{\mathbf{out}}}{{\mathbf{P}}_{\mathbf{in}}}\times 100\%$	General efficiency formula. For Half-Wave (ideal): $\eta \approx 40.6\%$ .	Applicable generally.

III. Bipolar Junction Transistors (BJTs)

These equations define foundational current relationships and essential DC biasing calculations.

Category	Equation	Meaning of Symbols	Applicable Conditions/Context
Fundamental Current Relation	${\mathbf{I}}_{\mathbf{E}}={\mathbf{I}}_{\mathbf{C}}+{\mathbf{I}}_{\mathbf{B}}$	$I_E$: Emitter current; $I_C$: Collector current; $I_B$: Base current .	Always applicable in BJT operation .
DC Common-Base Gain	${\alpha }_{\mathbf{dc}}=\frac{{\mathbf{I}}_{\mathbf{C}}}{{\mathbf{I}}_{\mathbf{E}}}$	${\alpha }_{dc}$: DC common-base current gain .	Common Base configuration .
DC Common-Emitter Gain	${\beta }_{\mathbf{dc}}=\frac{{\mathbf{I}}_{\mathbf{C}}}{{\mathbf{I}}_{\mathbf{B}}}$	${\beta }_{dc}$: DC common-emitter current gain ($h_{FE}$) .	Common Emitter configuration .
$\alpha$ and $\beta$ Relation (1) (Key PYQ Derivation)	$\beta =\frac{\alpha }{1-\alpha }$	Derived from $I_E=I_C+I_B$ .	Necessary for converting between the two BJT gain parameters .
$\alpha$ and $\beta$ Relation (2)	$\alpha =\frac{\beta }{\beta +1}$	The inverse relationship .	Necessary for converting between the two BJT gain parameters .
Collector Current (General)	$I_C=\beta I_B+(\beta +1)I_{CBO}$	$I_{CBO}$: Reverse saturation current (Common Base) .	General CE collector current formula. ${\mathbf{I}}_{\mathbf{C}}\approx \beta {\mathbf{I}}_{\mathbf{B}}$ is the simplified approximation used in active region calculations .
DC Bias: Emitter Stabilized ($I_B$)	${\mathbf{I}}_{\mathbf{B}}=\frac{{\mathbf{V}}_{\mathbf{CC}}-{\mathbf{V}}_{\mathbf{BE}}}{{\mathbf{R}}_{\mathbf{B}}+(\beta +1){\mathbf{R}}_{\mathbf{E}}}$	$V_{CC}$: Supply voltage; $V_{BE}$: Base-Emitter voltage ($\approx 0.7$ V for Si) ; $R_B$: Base resistor; $R_E$: Emitter resistor .	Emitter-stabilized configuration (active region bias calculation) .
DC Bias: Voltage Divider ($V_B$ Approx)	${\mathbf{V}}_{\mathbf{B}}=\frac{{\mathbf{R}}_2{\mathbf{V}}_{\mathbf{CC}}}{{\mathbf{R}}_1+{\mathbf{R}}_2}$	$V_B$: DC base voltage; $R_1,R_2$: Resistors forming the voltage divider .	Valid if the condition $\beta {\mathbf{R}}_{\mathbf{E}}\ge 10{\mathbf{R}}_2$ is satisfied .
DC Bias: General Output KVL	${\mathbf{V}}_{\mathbf{CE}}={\mathbf{V}}_{\mathbf{CC}}-{\mathbf{I}}_{\mathbf{C}}({\mathbf{R}}_{\mathbf{C}}+{\mathbf{R}}_{\mathbf{E}})$	${\mathbf{V}}_{\mathbf{CE}}$: Collector-Emitter voltage; $R_C$: Collector resistor; $R_E$: Emitter resistor .	Applicable to most Common-Emitter (CE) DC biasing schemes (Emitter stabilized, Voltage-Divider) .
AC Emitter Resistance	${\mathbf{r}}_{\mathbf{e}}=\frac{26\text{ mV}}{{\mathbf{I}}_{\mathbf{E}}}$	$r_e$: AC emitter resistance (used in AC analysis and hybrid models); $I_E$: DC Emitter Current .	At room temperature (25 ${}^{\circ }$ C) .

IV. Field-Effect Transistors (FETs)

This section includes Shockley’s equation, the primary equation for JFET analysis used in numerical problems to find Q-points.

Category	Equation	Meaning of Symbols	Applicable Conditions/Context
JFET Drain Current (Shockley’s Equation) (Key PYQ Focus)	${\mathbf{I}}_{\mathbf{D}}={\mathbf{I}}_{\mathbf{DSS}}{\left(1-\frac{{\mathbf{V}}_{\mathbf{GS}}}{{\mathbf{V}}_{\mathbf{P}}}\right)}^2$	${\mathbf{I}}_{\mathbf{D}}$: Drain current; ${\mathbf{I}}_{\mathbf{DSS}}$: Drain saturation current (when $V_{GS}=0$) ; ${\mathbf{V}}_{\mathbf{GS}}$: Gate-source voltage ; ${\mathbf{V}}_{\mathbf{P}}$: Pinch-off voltage .	JFET operating in the constant current region (saturation) . Used extensively for finding Q-points graphically or mathematically .
JFET Drain-Source Voltage (Self-Bias)	${\mathbf{V}}_{\mathbf{DS}}={\mathbf{V}}_{\mathbf{DD}}-{\mathbf{I}}_{\mathbf{D}}({\mathbf{R}}_{\mathbf{S}}+{\mathbf{R}}_{\mathbf{D}})$	$V_{DD}$: Drain supply voltage; $R_S,R_D$: Source and Drain resistors .	Self-bias configuration (Derived from KVL) .
MOSFET (Enhancement-Type) Drain Current	${\mathbf{I}}_{\mathbf{D}}=\mathbf{k}({\mathbf{V}}_{\mathbf{GS}}-{\mathbf{V}}_{\mathbf{GS}(\mathbf{Th})}{)}^2$	$k$: Transconductance parameter; ${\mathbf{V}}_{\mathbf{GS}(\mathbf{Th})}$: Threshold voltage .	Enhancement-type MOSFET (E-MOSFET) .

V. Zener Diode Constraint (Numerical Design)

This constraint is crucial for Zener regulator problems, particularly when determining the limiting series resistance ($R$).

Category	Equation	Meaning of Symbols	Applicable Conditions/Context
Current Limiting Resistor Constraint	$\mathbf{R}\le \frac{{\mathbf{V}}_{\mathbf{i}}-{\mathbf{V}}_{\mathbf{Z}}}{{\mathbf{I}}_{\mathbf{Z}}+{\mathbf{I}}_{\mathbf{L}}}$	$R$: Series resistance; $V_i$: Input voltage; $V_Z$: Zener voltage; $I_Z$: Zener current; $I_L$: Load current .	Determining the maximum value of the series resistance $R$ to maintain the Zener diode in the breakdown (regulation) region, where ${\mathbf{I}}_{\mathbf{R}}\ge {\mathbf{I}}_{\mathbf{Z}}+{\mathbf{I}}_{\mathbf{L}}$ .