BEEE Cheatsheet

Cheatsheet Content

### D.C. Circuits #### Introduction to D.C. Circuits - **Modern electron theory:** Explains current flow as movement of electrons. - **Electric potential & potential difference:** - **Electric Potential:** Work done per unit charge to bring a charge from infinity to a point. Unit: Volt (V). - **Potential Difference:** Work done per unit charge to move a charge between two points. Also measured in Volts. - **Resistance (R):** Opposition to current flow. Unit: Ohm ($\Omega$). - $R = \rho \frac{L}{A}$ where $\rho$ is resistivity, $L$ is length, $A$ is cross-sectional area. - **Conductance (G):** Reciprocal of resistance. Unit: Siemens (S). $G = \frac{1}{R}$. #### Effect of Temperature on Resistance - Resistance of most conductors increases with temperature. - **Formula:** $R_t = R_0 (1 + \alpha_0 t)$ - $R_t$: Resistance at temperature $t$ - $R_0$: Resistance at reference temperature $0^\circ C$ - $\alpha_0$: Temperature coefficient of resistance at $0^\circ C$ - **Computation of $\alpha$ at different temperatures:** $\alpha_t = \frac{\alpha_0}{1 + \alpha_0 t}$ #### Ohm's Law - States that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature and other physical conditions remain constant. - **Formula:** $V = IR$ - $V$: Voltage (Volts) - $I$: Current (Amperes) - $R$: Resistance (Ohms) #### Resistor Combinations - **Series:** Resistors are connected end-to-end. - Total Resistance: $R_{eq} = R_1 + R_2 + R_3 + ...$ - Current is same through all resistors. - Voltage divides across resistors. - **Parallel:** Resistors are connected across the same two points. - Total Resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$ - Voltage is same across all resistors. - Current divides among resistors. - **Star-Delta Transformation:** Used to simplify complex networks. #### Kirchhoff's Laws (KVL & KCL for Resistive Circuits) - **Kirchhoff's Current Law (KCL):** The algebraic sum of currents entering a node (or a junction) is equal to the algebraic sum of currents leaving that node. - $\sum I_{in} = \sum I_{out}$ - **Kirchhoff's Voltage Law (KVL):** The algebraic sum of all voltages around any closed loop in a circuit is equal to zero. - $\sum V = 0$ ### A.C. Circuits #### Single Phase A.C. Circuits - **Generation of sinusoidal voltage:** An alternating voltage is generated by rotating a coil in a magnetic field or vice-versa. - $v(t) = V_m \sin(\omega t + \phi)$ - **Definitions:** - **Average Value:** The average of all instantaneous values over one complete cycle (for a sine wave, average value is 0). For a half-cycle, $V_{avg} = \frac{2V_m}{\pi}$. - **Root Mean Square (RMS) Value:** The effective value of an AC current or voltage, which produces the same heating effect as an equivalent DC current or voltage. - $V_{RMS} = \frac{V_m}{\sqrt{2}}$ - $I_{RMS} = \frac{I_m}{\sqrt{2}}$ - **Form Factor:** Ratio of RMS value to Average value. $FF = \frac{V_{RMS}}{V_{avg}} = \frac{\pi}{2\sqrt{2}} \approx 1.11$ (for sine wave). - **Peak Factor (Crest Factor):** Ratio of Peak value to RMS value. $PF = \frac{V_m}{V_{RMS}} = \sqrt{2} \approx 1.414$ (for sine wave). #### Analysis of Purely Resistive, Inductive and Capacitive Circuits - **Purely Resistive Circuit (R):** - Voltage and current are in phase. - Power factor = 1 (unity). - $P = V_{RMS} I_{RMS}$ - **Purely Inductive Circuit (L):** - Current lags voltage by $90^\circ$. - Inductive Reactance: $X_L = \omega L = 2\pi f L$. - Power factor = 0 (lagging). - No real power consumption. - **Purely Capacitive Circuit (C):** - Current leads voltage by $90^\circ$. - Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$. - Power factor = 0 (leading). - No real power consumption. #### Series Circuit Analysis (R-L, R-C, R-L-C) - **R-L Series Circuit:** - Impedance: $Z = \sqrt{R^2 + X_L^2}$ - Phase angle: $\phi = \arctan(\frac{X_L}{R})$ (current lags voltage) - **R-C Series Circuit:** - Impedance: $Z = \sqrt{R^2 + X_C^2}$ - Phase angle: $\phi = \arctan(\frac{-X_C}{R})$ (current leads voltage) - **R-L-C Series Circuit:** - Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - Phase angle: $\phi = \arctan(\frac{X_L - X_C}{R})$ - **Resonance:** Occurs when $X_L = X_C$, so $Z=R$ and $\phi=0$. #### Concepts of Power in A.C. Circuits - **Real Power (P):** Actual power consumed by the circuit, measured in Watts (W). - $P = V_{RMS} I_{RMS} \cos\phi$ - **Reactive Power (Q):** Power that oscillates between the source and reactive components (inductors and capacitors), measured in Volt-Ampere Reactive (VAR). - $Q = V_{RMS} I_{RMS} \sin\phi$ - **Apparent Power (S):** Total power supplied by the source, measured in Volt-Amperes (VA). - $S = V_{RMS} I_{RMS} = \sqrt{P^2 + Q^2}$ - **Power Factor (PF):** Ratio of real power to apparent power. - $PF = \cos\phi = \frac{R}{Z}$ - Ranges from 0 to 1. A higher power factor indicates more efficient use of power. ### Basics of Three Phase Circuits - **Necessity and Advantages of three phase systems:** - More efficient power generation and transmission. - Constant power output (less vibration in motors). - Requires less conductor material for the same power compared to single-phase. - Self-starting for induction motors. - **Generation of three phase power:** Three separate sinusoidal voltages, each displaced by $120^\circ$ from each other, are generated. - **Relationship between line and phase values of balanced three phase circuit:** - **Star (Y) Connection:** - Line Voltage $V_L = \sqrt{3} V_P$ (Phase Voltage) - Line Current $I_L = I_P$ (Phase Current) - **Delta ($\Delta$) Connection:** - Line Voltage $V_L = V_P$ (Phase Voltage) - Line Current $I_L = \sqrt{3} I_P$ (Phase Current) ### Electromagnetism #### Magnetic Circuit - **Definition:** The path followed by magnetic flux. Analogous to an electric circuit. - **Key terms:** - **Magnetomotive Force (MMF):** Analogous to EMF in electric circuits. $MMF = NI$ (Ampere-turns). - **Magnetic Flux ($\Phi$):** Total number of magnetic field lines. Unit: Weber (Wb). - **Reluctance ($\mathcal{R}$):** Opposition to magnetic flux. $\mathcal{R} = \frac{l}{\mu A}$ (Ampere-turns/Weber). - **Magnetic Field Intensity (H):** MMF per unit length. $H = \frac{NI}{l}$ (Ampere-turns/meter). - **Magnetic Flux Density (B):** Flux per unit area. $B = \frac{\Phi}{A}$ (Tesla, T). - **Permeability ($\mu$):** Ability of a material to support the formation of a magnetic field. $\mu = \mu_0 \mu_r$. #### Comparison Between Electric And Magnetic Circuits | Feature | Electric Circuit | Magnetic Circuit | | :--------------- | :---------------------- | :----------------------- | | Driving Force | Electromotive Force (EMF) | Magnetomotive Force (MMF) | | Flow | Current (I) | Magnetic Flux ($\Phi$) | | Opposition | Resistance (R) | Reluctance ($\mathcal{R}$) | | Path | Conductor | Magnetic Material | | O's Law Analogue | $I = \frac{EMF}{R}$ | $\Phi = \frac{MMF}{\mathcal{R}}$ | #### Faraday's Law of Electromagnetic Induction - **First Law:** Whenever a conductor cuts magnetic flux, an EMF is induced in that conductor. - **Second Law:** The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linkage. - $e = -N \frac{d\Phi}{dt}$ (where N is number of turns, $\Phi$ is magnetic flux, and the negative sign indicates Lenz's Law). #### Lenz's Law - States that the direction of the induced current (or EMF) is always such as to oppose the cause producing it. #### Electromagnetic Induction - The process by which an electromotive force (EMF) is induced in a conductor when it is exposed to a changing magnetic field. #### Fleming's Right Hand Rule - Used to determine the direction of induced current when a conductor moves in a magnetic field. - **Thumb:** Direction of Motion of conductor. - **Forefinger:** Direction of Magnetic Field (North to South). - **Middle Finger:** Direction of Induced Current. #### Fleming's Left Hand Rule - Used to determine the direction of force on a current-carrying conductor placed in a magnetic field. - **Thumb:** Direction of Force/Motion. - **Forefinger:** Direction of Magnetic Field (North to South). - **Middle Finger:** Direction of Current.