EEE Course Formulas & Equations

eee-1109 EEE-1109 Basic Electrical Engineering

1. DC Circuit Analysis and Network Theorems

This section covers laws for simplifying resistive networks and calculating electrical quantities in DC systems.

Topic	Formula / Equation	Meaning of Symbols	Concept Note
Series Voltage Sources	$I=\frac{E_1+E_2}{R}$ (Same Pol)$I=\frac{E_2-E_1}{R}$ (Diff Pol)	$I$: Current$E$: Source Voltage$R$: Resistance	Voltage sources in series add up if polarities match; the larger source dominates if they oppose.
Parallel Current Sources	$I_{total}=I_1+I_2$ (Same Dir)$I_{total}=I_2-I_1$ (Diff Dir)	$I_{total}$: Net source current$I_{1,2}$: Individual sources	Current sources in parallel can be algebraically summed based on their direction arrows.
Delta ($\Delta$) to Wye (Y)	$R_1=\frac{R_b{R}_c}{R_a+R_b+R_c}$	$R_1$: Y-resistor at node 1$R_{b,c}$: Adjacent $\Delta$ resistors	Used to simplify bridge circuits or complex networks where series/parallel rules don’t apply.
Wye (Y) to Delta ($\Delta$)	$R_a=\frac{R_1{R}_2+R_2{R}_3+R_3{R}_1}{R_1}$	$R_a$: $\Delta$-resistor opposite node 1$R_{1,2,3}$: Y-resistors	Converts a T-network into a $\pi$-network for circuit simplification.
Millman’s Theorem	$V=\frac{\sum \frac{V_n}{R_n}}{\sum \frac{1}{R_n}}$	$V$: Voltage across parallel branches$G_n=1/R_n$: Conductance	Calculates the common voltage across multiple parallel voltage sources with internal resistance.
Thevenin’s Theorem	$V_{th},R_{th}$	$V_{th}$: Open circuit voltage$R_{th}$: Equivalent resistance	Any linear circuit can be replaced by a single voltage source in series with a resistor.
Norton’s Theorem	$I_N,R_N$	$I_N$: Short circuit current$R_N$: Equivalent resistance	Any linear circuit can be replaced by a single current source in parallel with a resistor.
Max Power Transfer	$P_{max}=\frac{V_{th}^2}{4R_{th}}$	$P_{max}$: Max Power$R_L$: Load Resistance	Maximum power is transferred to the load only when load resistance equals source resistance ($R_L=R_{th}$).

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2. AC Fundamentals: Sinusoids and Phasors

These equations describe alternating current/voltage behavior in time and frequency domains.

Topic	Formula / Equation	Meaning of Symbols	Concept Note
Sinusoidal Voltage	$v(t)=V_m\sin (\omega t+\varphi )$	$V_m$: Amplitude$\omega$: Angular freq$\varphi$: Phase angle	Generators naturally produce sine waves due to rotational motion; they change smoothly with no sudden drops.
Angular Frequency	$\omega =2\pi f=\frac{2\pi }{T}$	$f$: Frequency (Hz)$T$: Period (s)	Relates the speed of rotation (rad/s) to the frequency of the wave cycle.
RMS Value	$I_{rms}=\frac{I_m}{\sqrt{2}}$	$I_{rms}$: Effective Value$I_m$: Peak Value	Represents the equivalent DC current that produces the same heat in a resistor.
Average Value	$I_{avg}=\frac{2I_m}{\pi }$	$I_{avg}$: Mean over half-cycle	The arithmetic mean of the current over one positive half-cycle.
Form Factor	${\text{K}}_f=\frac{I_{rms}}{I_{avg}}\approx 1.11$	Ratio of RMS to Average	Indicates the “spikiness” of the waveform; 1.11 is specific to pure sine waves.
Crest Factor	${\text{K}}_p=\frac{V_m}{V_{rms}}=\sqrt{2}$	Ratio of Peak to RMS	Indicates how extreme the peaks are compared to the effective value.
Phasor Transform	$V_m\cos (\omega t+\theta )\Rightarrow V_m\angle \theta$	Polar Form: $r\angle \theta$	A complex number representing amplitude and phase, simplifying AC calculus to algebra.
Euler’s Identity	$e^{\pm j\theta }=\cos \theta \pm j\sin \theta$	$j$: Imaginary unit	Connects trigonometric functions to complex exponentials.

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3. AC Circuit Components, Power, and Resonance

Formulas regarding impedance, power consumption, and energy storage in AC systems.

Topic	Formula / Equation	Meaning of Symbols	Concept Note
Impedance (Z)	$Z = R + jX =	Z	\angle \theta$
Inductive Reactance	$X_L=\omega L=2\pi fL$	$L$: Inductance (Henry)	Opposition to current proportional to frequency; Inductor voltage leads current by $90^{\circ }$.
Capacitive Reactance	$X_C=\frac{1}{\omega C}$	$C$: Capacitance (Farad)	Opposition decreases as frequency increases; Capacitor current leads voltage by $90^{\circ }$.
Admittance	$Y=\frac{1}{Z}=G+jB$	$G$: Conductance$B$: Susceptance	Reciprocal of impedance; useful for parallel circuit analysis.
Active Power	$P=V_{rms}{I}_{rms}\cos \theta$	$\theta$: Phase diff ($V,I$)Unit: Watts	The actual power consumed by the resistive part of the circuit.
Power Factor	$pf=\cos \theta =\frac{R}{Z}$	Z	}$
Series Resonance	$f_m=\frac{1}{2\pi \sqrt{LC}}$	$f_m$: Resonant Freq	The frequency where $X_L=X_C$, making impedance purely resistive and minimum.
Energy Stored	W_L = \frac{1}{2} L I_m^2$$W_C = \frac{1}{2} C V_m^2	$W$: Energy (Joules)	Inductors store energy in magnetic fields; capacitors store energy in electric fields.

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4. Polyphase Circuits (Three-Phase)

Equations for analyzing balanced and unbalanced three-phase power systems.

Topic	Formula / Equation	Meaning of Symbols	Concept Note
Star (Y) Relationships	V_{line} = \sqrt{3} V_{ph} \angle 30^\circ$$I_{line} = I_{ph}	$V_L,V_{ph}$: Line/Phase Voltages	In a balanced Star connection, line voltage is $\sqrt{3}$ times phase voltage.
Three-Phase Power	$P=\sqrt{3}{V}_L{I}_L\cos \theta$	$V_L,I_L$: Line quantities	Total active power delivered to a balanced three-phase load.
Two-Wattmeter Power	$P_{total}=W_a+W_b$	$W_{a,b}$: Wattmeter readings	Algebraic sum of two wattmeters gives total power in a 3-wire system.
Reactive Power (2-W)	$Q=\sqrt{3}(W_a-W_b)$	$Q$: Reactive Power (VAR)	The difference in wattmeter readings relates to the reactive power in the system.

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5. Magnetism and Magnetic Circuits

Formulas covering magnetic fields, forces, materials, and circuit calculations.

Topic	Formula / Equation	Meaning of Symbols	Concept Note
Magnetic Force (Motor)	$F=BLI$	$F$: Force (N)$B$: Flux Density	A current-carrying conductor in a magnetic field experiences a mechanical force.
Flux Density (B)	$B=\frac{\Phi }{A}=\mu H$	$\Phi$: Flux (Webers)Unit: Tesla or Wb/m${}^2$	The concentration of magnetic flux lines per unit area.
Permeability	$\mu ={\mu }_0{\mu }_r$	${\mu }_0=4\pi \times 10^{-7}$	A measure of how easily a material allows magnetic flux to pass through it.
Magnetomotive Force	$\text{MMF}=NI=Hl$	Unit: Ampere-turns (AT)	The driving force (analogous to Voltage/EMF) that produces magnetic flux.
Ampere’s Law	$H=\frac{NI}{l}=\frac{I}{2\pi r}$	$H$: Field Intensity (AT/m)	Relates magnetic field strength to the current enclosed by the path.
Reluctance ($\mathcal{R}$)	$\mathcal{R}=\frac{l}{\mu A}$	Unit: AT/Wb	Analogous to Resistance; opposes the creation of magnetic flux.
Hopkinson’s Law	$\Phi =\frac{\text{MMF}}{\mathcal{R}}$	”Ohm’s Law” for Magnetism	Flux equals MMF divided by Reluctance, similar to $I=V/R$.
Susceptibility ($K$)	$K=\frac{I}{H}$	$I$: Magnetisation Intensity$H$: Magnetising Force	The ratio of the intensity of magnetization to the magnetizing force.
Series Magnetic Circuit	$\text{Total AT}=H_1{l}_1+H_2{l}_2+\dots$	Kirchhoff’s Law analogy	Total MMF required is the sum of MMF drops across each section of the circuit.
Self Inductance	$L=\frac{N\varphi }{I}=\frac{N^2}{\mathcal{R}}$	$L$: Inductance (Henry)	Property of a coil to oppose changes in current flowing through it.
Mutual Inductance	$M=k\sqrt{L_1{L}_2}$	$k$: Coupling coefficient	A measure of the ability of one inductor to induce a voltage in a nearby inductor.