Comprehensive Study Notes on DC Circuits, Magnetism, AC Circuits, and Electrical Machines

DC Circuits and Resistance

• Resistance ($R$): The opposition to current flow, measured in Ohms ($\Omega$).
• Factors Affecting Resistance:   - Length ($L$): Resistance is directly proportional to length.   - Cross-Sectional Area ($A$): Resistance is inversely proportional to area.   - Nature of Material (Resistivity, $\rho$): Measured at a constant temperature (usually $20^{\circ}C$).   - Resistivity Units: MKS ($\Omega \cdot m$) or English ($\Omega \cdot cm$).   - Formula: $R = \frac{\rho L}{A}$.
• Conductivity ($\sigma$): The reciprocal of resistivity ($\sigma = \frac{1}{\rho}$).
• Wire Drawing Process (Constant Volume):   - When a wire is drawn, volume ($V = AL$) remains constant.   - Relationship between Initial ($R_1$) and Final ($R_2$) resistance:     - In terms of length: $\frac{R_2}{R_1} = (\frac{L_2}{L_1})^2$.     - In terms of area: $\frac{R_2}{R_1} = (\frac{A_1}{A_2})^2$.     - In terms of diameter ($d$): $\frac{R_2}{R_1} = (\frac{d_1}{d_2})^4$.
• Units Conversion:   - $1 \, \text{inch} = 1000 \, \text{mils}$.   - $1 \, \text{MCM} = 1000 \, \text{CM}$ (Circular Mils).   - Area in Circular Mils ($A_{CM}$): $A = d^2 \, \text{mils}$.

Temperature Relations

• Inferred Absolute Zero Temperature ($T$):   - For Copper ($Cu$): $T = 234.5^{\circ}C$.   - For Aluminum ($Al$): $T = 236^{\circ}C$.
• Resistance-Temperature Formula (using similar triangles):   - $\frac{R_2}{R_1} = \frac{|T| + t_2}{|T| + t_1}$.
• Temperature Coefficient of Resistance ($\alpha$):   - $\alpha_t = \frac{1}{|T| + t}$.   - Relationship: $R_2 = R_1[1 + \alpha_1(t_2 - t_1)]$.

Ohm's Law and Basic Circuits

• Ohm's Law: $V = IR$.
• Series Resistor Circuits:   - Total Resistance: $R_T = R_1 + R_2 + \dots + R_n$.   - Current: $I_T = I_1 = I_2 = \dots = I_n$.   - Total Voltage: $V_T = V_1 + V_2 + \dots + V_n$.   - Voltage Divider Theorem (VDT): $V_n = V_T \frac{R_n}{R_T}$.
• Conductance ($G$):   - Ability to allow current flow, measured in Mhos ($\mho$) or Siemens ($S$).   - $G = \frac{1}{R}$.
• Parallel Resistor Circuits:   - Total Conductance: $G_T = G_1 + G_2 + \dots + G_n$.   - Total Resistance: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$.   - Voltage: $V_T = V_1 = V_2 = \dots = V_n$.   - Total Current: $I_T = I_1 + I_2 + \dots + I_n$.   - Current Divider Theorem (CDT):     - Using Conductance: $I_n = I_T \frac{G_n}{G_T}$.     - For two resistors in parallel: $I_1 = I_T \frac{R_2}{R_1 + R_2}$ and $I_2 = I_T \frac{R_1}{R_1 + R_2}$.

Kirchhoff's Laws

• Kirchhoff's Current Law (KCL): Law of Conservation of Charge (L.O.C.C.). At any node, $\sum I_{\text{entering}} = \sum I_{\text{leaving}}$.
• Kirchhoff's Voltage Law (KVL): Law of Conservation of Energy (L.O.C.E.). In any closed loop, $\sum V = 0$.   - Supply voltage is positive ($+$).   - Voltage drop across resistors ($IR$) is negative (-$ $).\n\n# Electrical Power and Work\n\n- **Electrical Power (P$)**: Measured in Watts ($W$).$1 \, \text{Watt} = 1 \, \text{Joule/sec}.\n  - Formulas: P = IV = I^2R = \frac{V^2}{R}.\n- **Electrical Work (W$)**: Measured in Joules ($J$).$1 \, \text{Joule} = 1 \, \text{Watt-sec}.\n  - Formula: W = P \times t.\n- **Thermal Efficiency (\eta_T)**:\n  - \eta_T = \frac{\text{Output (Heat utilized)}}{\text{Input (Electrical Energy)}} = \frac{Q}{W}.\n  - Heat energy (Q$):$Q = mc\Delta t$, where$c = 4.186 \, \text{J/g}^{\circ}C.\n\n# Magnetic Circuits\n\n- **Hopkinson's Law**: \Phi = \frac{\mathcal{F}}{\mathcal{R}}$, where$\Phi$is Flux,$\mathcal{F}$is MMF ($NI$), and$\mathcal{R} is Reluctance.\n- **Magnetic Flux (\Phi)**:\n  - MKS: Weber (Wb$).$1 \, Wb = 10^8 \, \text{lines}.\n  - CGS: Maxwell (Mx) or Lines.\n- **Reluctance (\mathcal{R}$)**:$\mathcal{R} = \frac{L}{\mu A}.\n- **Permeability (\mu$)**:$\mu = \mu_0 \mu_r.\n- **Magnetic Field Strength (H$)**:$H = \frac{NI}{L}$. Measured in Amp-turns/meter ($At/m).\n- **Magnetic Flux Density (B$)**:$B = \frac{\Phi}{A}$. Measured in Tesla ($T$) or Gauss ($G$).$10^4 \, G = 1 \, T.\n- **Magnetic Force (F)**:\n  - On a conductor: F = BIL \sin(\theta).\n  - Between two parallel conductors: F = \frac{\mu I_1 I_2 L}{2\pi d}.\n  - Between two poles: F = \frac{m_1 m_2}{4\pi \mu d^2}$(MKS) or$F = \frac{m_1 m_2}{d^2} (CGS).\n- **Induction**:\n  - **Faraday's Law**: e = -N \frac{d\Phi}{dt}.\n  - **Dynamic Induction**: e = BLv \sin(\theta).\n  - **Self-Inductance (L$)**:$L = \frac{N\Phi}{i} = \frac{\mu N^2 A}{L}$. Measured in Henry ($H).\n  - **Mutual Inductance (M$)**:$M = k \sqrt{L_1 L_2} = \frac{N_2 \Phi_{12}}{i_1}.\n- **Combined Inductance**:\n  - Series Aiding: L_T = L_1 + L_2 + 2M.\n  - Series Opposing: L_T = L_1 + L_2 - 2M.\n  - Parallel Aiding: L_T = \frac{L_1 L_2 - M^2}{L_1 + L_2 - 2M}.\n  - Parallel Opposing: L_T = \frac{L_1 L_2 - M^2}{L_1 + L_2 + 2M}.\n\n# Electrostatics and Capacitance\n\n- **Coulomb's Law**: F = \frac{Q_1 Q_2}{4\pi \epsilon d^2}.\n- **Permittivity (\epsilon)**:\n  - \epsilon = \epsilon_0 \epsilon_r.\n  - \epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}.\n- **Electric Field Strength (E$)**:$E = \frac{F}{Q} = \frac{Q}{4\pi \epsilon d^2} = \frac{V}{d}.\n- **Electric Potential (V$)**:$V = \frac{Q}{4\pi \epsilon d}.\n- **Capacitance (C$)**:$C = \frac{Q}{V}$. Measured in Farads ($F).\n  - Isolated Sphere: C = 4\pi \epsilon R.\n  - Two-Plate Capacitor: C = \frac{\epsilon A}{d}.\n  - Multiplate Capacitor: C = (n-1) \frac{\epsilon A}{d}.\n  - Cylindrical Capacitor: C = \frac{2\pi \epsilon L}{\ln(r_2/r_1)}.\n- **Capacitor Circuits**:\n  - Series: \frac{1}{C_T} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$. Note:$Q_T = Q_1 = Q_2.\n  - Parallel: C_T = C_1 + C_2 + \dots + C_n$. Note:$V_T = V_1 = V_2.\n\n# DC Machines: Generators and Motors\n\n- **Generated EMF (E_g$)**:$E_g = \frac{P Z \Phi N}{60 a}.\n  - P$= Number of poles,$Z$= Number of conductors,$N = speed in rpm.\n  - a$= Parallel paths ($a=P$for Lap winding,$a=2 for Wave winding).\n- **DC Generator Types**:\n  - **Separately Excited**: Field winding is independent of armature.\n  - **Self-Excited (Shunt)**: Field winding in parallel with armature. I_a = I_L + I_f.\n  - **Compound (Long Shunt)**: E_g = V_t + I_a(R_a + R_s).\n  - **Compound (Short Shunt)**: I_s = I_L$,$I_a = I_f + I_s.\n- **Efficiency (\eta$)**:$\eta = \frac{P_{\text{out}}}{P_{\text{in}}}.\n  - Maximum efficiency occurs when Variable Losses (Armature Copper Loss) equal Constant Losses (Stray Power Losses + Field Copper Loss).\n- **DC Motors**:\n  - Back EMF (E_b$):$E_b = V_t - I_a R_a.\n  - Speed: N \propto \frac{E_b}{\Phi}$. Speed Regulation ($SR\%$) =$\frac{S_{NL} - S_{FL}}{S_{FL}} \times 100\%.\n  - Torque (T$):$T_d = \frac{P Z \Phi I_a}{2\pi a}$.$T \propto \Phi I_a.\n\n# AC Circuits: Fundamentals and Impedance\n\n- **Sinusoidal Waveform**: v(t) = V_m \sin(\omega t).\n  - \omega = 2\pi f (Angular frequency in rad/s).\n  - RMS Value (V_{rms}$): For sine wave,$V_{rms} = \frac{V_m}{\sqrt{2}} = 0.707V_m.\n  - Average Value (V_{ave}$): For sine wave,$V_{ave} = \frac{2V_m}{\pi} = 0.637V_m.\n  - Form Factor (FF$) =$\frac{V_{rms}}{V_{ave}}$. Peak Factor =$\frac{V_{max}}{V_{rms}}.\n- **Impedance (Z$)**: Measured in Ohms ($\Omega$).$Z = R + jX.\n  - **Pure Resistor**: Current in phase with voltage (\theta = 0^{\circ}).\n  - **Pure Inductor**: Current lags voltage by 90^{\circ}$.$X_L = 2\pi fL.\n  - **Pure Capacitor**: Current leads voltage by 90^{\circ}$.$X_C = \frac{1}{2\pi fC}.\n- **Power in AC**:\n  - Real/True Power (P$):$P = VI \cos(\phi) (Watts).\n  - Reactive Power (Q$):$Q = VI \sin(\phi) (VAR).\n  - Apparent Power (S$):$S = VI$(VA).$S = \sqrt{P^2 + Q^2}.\n  - Power Factor (pf$) =$\cos(\phi) = \frac{P}{S}.\n- **Resonance**:\n  - **Series**: X_L = X_C$. Impedance is minimum ($Z=R), current is maximum.\n  - **Parallel**: Impedance is maximum, current is minimum.\n  - Resonant Frequency (f_r$):$f_r = \frac{1}{2\pi \sqrt{LC}}.\n  - Quality Factor (Q$):$Q = \frac{1}{R} \sqrt{\frac{L}{C}}.\n  - Bandwidth (BW$):$BW = f_2 - f_1 = \frac{f_r}{Q}.\n\n# Three-Phase Systems\n\n- **Wye (Star) Connection**:\n  - Line Voltage: V_L = \sqrt{3} V_{\phi}.\n  - Line Current: I_L = I_{\phi}.\n- **Delta (Mesh) Connection**:\n  - Line Voltage: V_L = V_{\phi}.\n  - Line Current: I_L = \sqrt{3} I_{\phi}.\n- **Three-Phase Power**: P_T = \sqrt{3} V_L I_L \cos(\phi) = 3 V_{\phi} I_{\phi} \cos(\phi).\n- **Two-Wattmeter Method**: P_T = W_1 + W_2$.$\tan(\phi) = \sqrt{3} \frac{W_1 - W_2}{W_1 + W_2}.\n\n# Transformers\n\n- **Transformation Ratio (a$)**:$a = \frac{N_P}{N_S} = \frac{E_P}{E_S} = \frac{I_S}{I_P}.\n- **EMF Equation**: E = 4.44 f N \Phi_m.\n- **Equivalent Circuit Referred to Primary**:\n  - Resistance: R_{ep} = R_P + a^2 R_S.\n  - Reactance: X_{ep} = X_P + a^2 X_S.\n- **Voltage Regulation (VR\%$)**:$\frac{V_{NL} - V_{FL}}{V_{FL}} \times 100\%.\n- **Open Circuit Test**: Used to determine core losses (P_{core}).\n- **Short Circuit Test**: Used to determine copper losses (P_{cu}) and impedance parameters.\n- **Maximum Efficiency**: Occurs when P_{cu} = P_{core}.\n- **Autotransformers**: Use a single winding. Power transferred via both induction and conduction.\n- **Instrument Transformers**: Current Transformer (CT) and Potential Transformer (PT) for measurement.\n\n# Power Systems and Illumination\n\n- **Transmission Line Parameters**:\n  - Inductance (L$):$L = 2 \times 10^{-7} \ln(\frac{GMD}{GMR}) \, \text{H/m}.\n  - Capacitance (C$):$C = \frac{2\pi \epsilon}{\ln(GMD/r)} \, \text{F/m}.\n- **Power Plant Terminologies**:\n  - **Load Factor**: \frac{\text{Average Demand}}{\text{Maximum Demand}}.\n  - **Capacity Factor**: \frac{\text{Actual Energy Produced}}{\text{Max Energy Plant can Produce}}.\n  - **Diversity Factor**: \frac{\sum \text{Individual Max Demands}}{\text{Actual Simultaneous Max Demand}}.\n- **Illumination**:\n  - **Luminous Flux (\Phi$)**: Measured in Lumens ($lm).\n  - **Illumination (E$)**: Measured in Lux ($lx$) or foot-candles.$E = \frac{\Phi}{A}.\n  - **Inverse Square Law**: E = \frac{I}{d^2}.\n  - **Lambert's Cosine Law**: E = \frac{I \cos(\theta)}{d^2}.\n  - **Coefficient of Utilization (CU)** and **Maintenance Factor (MF)**: Used in practical calculations for total flux \Phi_T = \frac{E \times A}{CU \times MF}$$.