Unit 1: Fundamentals of D.C. Circuits

1. Introduction to Direct Current (D.C.) Circuits

An electric circuit is an interconnection of electrical elements designed to transfer energy or process electrical signals. In a Direct Current (D.C.) circuit, driving voltages and currents are unidirectional and time-invariant in steady state. Any initial transient responses associated with reactive components (inductors and capacitors) die out asymptotically, leaving purely constant voltages and currents governed by algebraic rather than differential equations.

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+-------------------------------------------------------------------------+
|                           D.C. CIRCUIT FOUNDATION                       |
|                                                                         |
|   Independent / Dependent Sources ---> Interconnected Network ---> Load |
|                 |                                    |                  |
|          Energy Delivered                    Energy Dissipated /        |
|          (P_gen = sum(V*I))                   Stored in Equilibrium     |
+-------------------------------------------------------------------------+

2. Fundamental Circuit Variables and Passive Sign Convention

2.1 Electric Charge ($q$) and Electric Current ($i$)

• Electric Charge ($q$): The fundamental physical property of matter that causes it to experience a force when placed in an electromagnetic field. The standard SI unit is the Coulomb (C). One electron possesses an elementary charge of e pprox -1.6022 imes 10^{-19}\text{ C}. Thus, $1\text{ C}$ represents the aggregate charge of approximately $6.242 imes 10^{18}$ electrons.
• Electric Current ($i$): The time rate of transfer of electric charge through a specified boundary or cross-sectional area:
  $i(t) = \frac{dq(t)}{dt}$
  In steady-state direct current:
  $I = \frac{\Delta Q}{\Delta t}$
  The SI unit of current is the Ampere (A), where $1\text{ A} = 1\text{ C/s}$. By universal convention established by Benjamin Franklin, conventional current is defined as the direction of positive charge motion, which is diametrically opposite to the actual drift velocity of conduction electrons in metallic conductors.

2.2 Electric Potential and Voltage ($v$)

• Voltage (Potential Difference, $v$): The work required to move a unit positive test charge between two designated points through an electrostatic field:
  $v_{ab} = \frac{dw}{dq}$
  where $v_{ab}$ denotes the potential at point $a$ with respect to point $b$. The SI unit of voltage is the Volt (V), where $1\text{ V} = 1\text{ J/C}$.
• If moving a positive charge from point $b$ to point $a$ requires net positive external work, point $a$ is at a higher potential than point $b$ ($v_a > v_b$ and $v_{ab} > 0$).

2.3 Electric Power ($p$) and Energy ($w$)

• Instantaneous Power ($p$): The time rate of absorbing or transferring energy:
  $p(t) = \frac{dw}{dt} = \left(\frac{dw}{dq}\right)\left(\frac{dq}{dt}\right) = v(t) \cdot i(t)$
  The SI unit of power is the Watt (W), where $1\text{ W} = 1\text{ J/s} = 1\text{ V} \cdot \text{A}$.
• Total Energy ($w$): The time integral of power over a specified interval:
  $w = \int_{t_0}^{t_1} p(t) \, dt = \int_{t_0}^{t_1} v(t) i(t) \, dt$
  The commercial billing unit is the Kilowatt-Hour (kWh):
  $1\text{ kWh} = (10^3\text{ W}) \times (3600\text{ s}) = 3.6 \times 10^6\text{ J} = 3.6\text{ MJ}$

2.4 Passive Sign Convention (PSC)

The Passive Sign Convention (PSC) is a universally binding referencing standard:

• Absorbing Element: If the reference current arrow enters the positive (+) terminal of an element, the power absorbed is given by:
  $p = +v \cdot i$
  • If $p > 0$, the element absorbs/dissipates electric power (e.g., resistors, charging batteries).
  • If $p < 0$, the element delivers power to the remainder of the network.
• Delivering Element (Active Convention): If reference current leaves the positive (+) terminal (or enters the negative terminal), power generated/delivered is:
  $p_{\text{del}} = +v \cdot i$
• Law of Conservation of Energy (Tellegen's Principle): In any lumped electric network at any instant, the algebraic sum of power absorbed across all branches is identically zero:
  $\sum_{k=1}^{B} p_k = 0 \iff \sum p_{\text{delivered}} = \sum p_{\text{absorbed}}$

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Passive Sign Convention:
     +    i --->    -            -    i --->    +
     o-----[ Z ]----o            o-----[ Z ]----o
        p = + v * i                 p = - v * i
    (Absorbing model)            (Delivering model)

3. Passive Circuit Elements in D.C. Networks

3.1 Linear Resistance and Temperature Dependence

• Resistance quantifies opposition to the drift of free charge carriers. By Pouillet's Law:
  $R = \rho \frac{L}{A}$
  where $\rho$ is material resistivity ($\Omega\cdot\text{m}$), $L$ is conductor length ($\text{m}$), and $A$ is uniform cross-sectional area ($\text{m}^2$).
• Conductance ($G$): The reciprocal of resistance:
  $G = \frac{1}{R} = \frac{\sigma A}{L}$
  where $\sigma = 1/\rho$ is electrical conductivity ($\text{S/m}$), measured in Siemens (S) or mhos ($\mho$).
• Temperature Coefficient of Resistance ($\alpha$):
  For metallic conductors, thermal agitation of the crystal lattice increases carrier scattering, leading to a positive temperature coefficient:
  $R(T) = R_0 \left[ 1 + \alpha_0 (T - T_0) \right]$
  For semiconductors and intrinsic electrolytes, thermal energy elevates covalent valence electrons across the bandgap into the conduction band, giving a negative temperature coefficient (NTC).

3.2 Inductance in D.C. Steady State

• An inductor stores energy within its self-induced magnetic field. By Faraday's Law of Electromagnetic Induction and Lenz's Law:
  $v_L(t) = L \frac{di(t)}{dt}$
• In a constant D.C. regime, currents are invariant over time:
  $\frac{di}{dt} = 0 \implies v_L = 0\text{ V}$
• Consequently, an ideal inductor with zero internal winding resistance behaves identically to an ideal short circuit ($0\,\Omega$) in D.C. steady state.

3.3 Capacitance in D.C. Steady State

• A capacitor stores electric potential energy within the electrostatic field between separated conductive plates:
  $i_C(t) = C \frac{dv(t)}{dt}$
• In a constant D.C. regime, voltages across elements reach constant values:
  $\frac{dv}{dt} = 0 \implies i_C = 0\text{ A}$
• Therefore, an ideal capacitor with infinite dielectric resistance behaves as an ideal open circuit ($\infty\,\Omega$) in D.C. steady state.

4. Fundamental Circuit Laws

4.1 Ohm's Law

George Simon Ohm formulated in 1827 that for a metallic conductor maintained under constant physical conditions (especially temperature and mechanical stress), the electric current through the conductor is directly proportional to the potential difference applied across its terminals:
$V = I \cdot R$

• Limitations of Ohm's Law:
  1. Non-linear Devices: Does not apply to semiconductor junctions (diodes, transistors), vacuum tubes, or electric arcs, where current-voltage characteristics are exponential or non-linear.
  2. Non-bilateral Elements: Fails for elements whose conduction properties depend upon polarity (e.g., rectifiers).
  3. Varying Temperature: In incandescent filaments or thermistors, self-heating causes resistance to drift substantially as current flows.

4.2 Topological Definitions in Graph Theory of Networks

• Node: A junction or terminal point where two or more circuit elements are joined together.
• Essential Node: A node connecting three or more branches.
• Branch: A single path formed by one or more elements connected in series between two nodes.
• Loop: Any closed planar or non-planar conducting path in which no node is encountered more than once.
• Mesh: A primitive loop that does not enclose any other loops or branches within its interior (applicable strictly to planar circuits).
• Topological Relationship: For any connected planar graph with $B$ branches and $N$ nodes:
  $M = B - N + 1$
  where $M$ is the number of independent mesh loops (independent KVL equations), and $N - 1$ is the number of independent non-reference node equations (independent KCL equations).

4.3 Kirchhoff's Current Law (KCL)

• Statement: Based on the fundamental physical principle of the Conservation of Electric Charge, the algebraic sum of all electric currents entering any closed boundary, node, or supernode is identically zero at every instant of time:
  $\sum_{k=1}^{n} I_k = 0 \iff \sum I_{\text{entering}} = \sum I_{\text{leaving}}$
• Because charge cannot accumulate at a geometric point node ($dq/dt = 0$), incoming flux must match outgoing flux exactly.

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Kirchhoff's Current Law (KCL):
            I1 (in)       I2 (out)
              \           /
               \         /
                v       ^
            -----> Node <-----
                ^       |
               /        v
             I3 (in)   I4 (out)

      I1 + I3 = I2 + I4  ===>  sum(I_in) - sum(I_out) = 0

4.4 Kirchhoff's Voltage Law (KVL)

• Statement: Based on the fundamental physical principle of the Conservation of Energy, the algebraic sum of electrical potential differences (voltages) traced along any closed loop or mesh in a circuit is identically zero:
  $\sum_{k=1}^{m} V_k = 0 \iff \sum V_{\text{rises}} = \sum V_{\text{drops}}$
• Tracing around an electrostatic path, a unit charge returns to its starting potential, yielding zero net mechanical work: $\oint \vec{E} \cdot d\vec{l} = 0$.

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Kirchhoff's Voltage Law (KVL):
               +  V1  -
          o-----[ R1 ]-----o
          |                |
      +   |                |   +
     Vs  (~) Loop ->       [ ] V2
      -   |                |   -
          |                |
          o-----[ R3 ]-----o
               -  V3  +

      + Vs - V1 - V2 - V3 = 0  ===>  Vs = V1 + V2 + V3

5. Network Simplification & Reduction Techniques

5.1 Series and Parallel Resistor Combinations

• Series Connection: Resistors connected end-to-end such that they carry the identical current:
  $R_{\text{eq}} = R_1 + R_2 + R_3 + \dots + R_n = \sum_{i=1}^{n} R_i$
• Parallel Connection: Resistors connected across common pair of nodes such that they experience identical potential differences:
  $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \iff G_{\text{eq}} = \sum_{i=1}^{n} G_i$
  For two resistors in parallel:
  $R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$

5.2 Voltage Division Rule (VDR)

For $n$ resistors connected in series across total voltage $V_{\text{total}}$, the voltage drop across the $k$-th resistor $R_k$ is directly proportional to its resistance:
$V_k = V_{\text{total}} \cdot \frac{R_k}{\sum_{j=1}^{n} R_j} = V_{\text{total}} \cdot \frac{R_k}{R_{\text{eq}}}$

5.3 Current Division Rule (CDR)

For two parallel resistors $R_1$ and $R_2$ fed by total incoming current $I_{\text{total}}$:
$I_1 = I_{\text{total}} \cdot \frac{R_2}{R_1 + R_2}, \qquad I_2 = I_{\text{total}} \cdot \frac{R_1}{R_1 + R_2}$
In general, for $n$ parallel branches with conductances $G_k = 1/R_k$:
$I_k = I_{\text{total}} \cdot \frac{G_k}{\sum_{j=1}^{n} G_j}$

5.4 Star-Delta (Y-$\Delta$) and Delta-Star ($\Delta$-Y) Transformations

Bridge networks and three-phase balanced/unbalanced topologies often contain configurations that cannot be simplified via pure series or parallel reductions.

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       DELTA (Delta) NETWORK                    STAR (Y) NETWORK
                A                                      A
               / \                                     |
              /   \                                    |
            R_ca  R_ab                                 R_A
            /       \                                  |
           /         \                                 |
          C-----R_bc---B                         C-----N-----B
                                                    R_C   R_B

Delta to Star ($\Delta \to \text{Y}$) Transformation:

Each resistance connected to a terminal node in the star equivalent equals the product of the two adjacent delta branch resistances divided by the perimeter sum of all three delta resistances:
$R_A = \frac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}}$
$R_B = \frac{R_{ab} R_{bc}}{R_{ab} + R_{bc} + R_{ca}}$
$R_C = \frac{R_{bc} R_{ca}}{R_{ab} + R_{bc} + R_{ca}}$
Special Case: If $R_{ab} = R_{bc} = R_{ca} = R_\Delta$, then $R_A = R_B = R_C = R_{\text{Y}} = \frac{R_\Delta}{3}$.

Star to Delta ($\text{Y} \to \Delta$) Transformation:

Each delta branch resistance equals the sum of the two adjacent star resistances plus the product of those two divided by the opposite star resistance:
$R_{ab} = R_A + R_B + \frac{R_A R_B}{R_C} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_C}$
$R_{bc} = R_B + R_C + \frac{R_B R_C}{R_A} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_A}$
$R_{ca} = R_C + R_A + \frac{R_C R_A}{R_B} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_B}$
Special Case: If $R_A = R_B = R_C = R_{\text{Y}}$, then $R_\Delta = 3 R_{\text{Y}}$.

6. Energy Sources and Systematic Analysis Techniques

6.1 Independent and Dependent Sources

• Ideal Independent Voltage Source: Maintains a fixed terminal voltage $v(t) = V_s$ regardless of the output current supplied. Its internal source resistance is $0\,\Omega$.
• Practical Voltage Source: Modelled as an ideal voltage source $V_s$ in series with internal resistance $R_{se}$. Terminal voltage decreases under load:
  $V_L = V_s - I_L R_{se}$
• Ideal Independent Current Source: Maintains a prescribed current $i(t) = I_s$ regardless of the voltage across its terminals. Its internal source resistance is $\infty\,\Omega$.
• Practical Current Source: Modelled as an ideal current source $I_s$ in parallel with internal resistance $R_{sh}$. Terminal current supplied to load:
  $I_L = I_s - \frac{V_L}{R_{sh}}$
• Dependent (Controlled) Sources: Output is governed by a voltage or current existing elsewhere in the network:
  1. VCVS: Voltage-Controlled Voltage Source ($v_x = \mu v_1$, dimensionless gain $\mu$).
  2. VCCS: Voltage-Controlled Current Source ($i_x = g_m v_1$, transconductance $g_m$ in S).
  3. CCVS: Current-Controlled Voltage Source ($v_x = r_m i_1$, transresistance $r_m$ in $\Omega$).
  4. CCCS: Current-Controlled Current Source ($i_x = \beta i_1$, dimensionless gain $\beta$).

6.2 Source Transformation (Thévenin-Norton Equivalence)

A practical voltage source ($V_s$ in series with $R_s$) is electrically identical at its exterior terminals to a practical current source ($I_s$ in parallel with $R_s$), provided:
$I_s = \frac{V_s}{R_s} \quad \text{and} \quad R_p = R_s$
The transformation preserves terminal voltages and currents for external loads, though internal internal power dissipation differs.

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       Source Transformation:
          o---+---[ Rs ]---o A           o---+-------o A
              |                              |       |
             (~) Vs                         (^) Is  [ ] Rp = Rs
              |                              |       |
          o---+------------o B           o---+-------o B
              Vs = Is * Rs                   Is = Vs / Rs

6.3 Mesh Current Analysis (Planar Networks)

• Mesh analysis assigns fictitious circulating currents $I_1, I_2, \dots, I_M$ to each primitive mesh.
• Matrix Formulation: $[R] [I] = [V]$ R_{11} & -R_{12} & \dots & -R_{1M} \\ -R_{21} & R_{22} & \dots & -R_{2M} \\ \vdots & \vdots & \ddots & \vdots \\ -R_{M1} & -R_{M2} & \dots & R_{MM} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \\ \vdots \\ I_M \end{bmatrix} = \begin{bmatrix} V_{s1} \\ V_{s2} \\ \vdots \\ V_{sM} \end{bmatrix}$$ where $R_{kk}$ is the self-resistance of mesh $k$ (sum of all resistors in mesh $k$), $R_{kj}$ ($k \neq j$) is the mutual resistance shared between meshes $k$ and $j$, and $V_{sk}$ is the algebraic sum of driving voltage rises around mesh $k$.
• Supermesh Formation: When a branch contains a current source (dependent or independent) shared between two adjacent meshes, the current source branch is temporarily removed (open-circuited), and a single combined KVL equation is written around the outer perimeter loop, supplemented by a constraint equation:
  $I_{\text{meshA}} - I_{\text{meshB}} = I_{\text{source}}$

6.4 Nodal Voltage Analysis

• Nodal analysis designates one essential node as the reference datum (0 V Ground) and solves for the unknown potentials $V_1, V_2, \dots, V_{N-1}$ at all remaining essential nodes.
• Matrix Formulation: $[G] [V] = [I]$ G_{11} & -G_{12} & \dots & -G_{1(N-1)} \\ -G_{21} & G_{22} & \dots & -G_{2(N-1)} \\ \vdots & \vdots & \ddots & \vdots \\ -G_{(N-1)1} & -G_{(N-1)2} & \dots & G_{(N-1)(N-1)} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ \vdots \\ V_{N-1} \end{bmatrix} = \begin{bmatrix} I_{s1} \\ I_{s2} \\ \vdots \\ I_{s(N-1)} \end{bmatrix}$$ where $G_{kk}$ is the self-conductance at node $k$ (sum of conductances connected to node $k$), $G_{kj}$ is the mutual conductance connected directly between nodes $k$ and $j$, and $I_{sk}$ is the algebraic sum of independent source currents injected directly into node $k$.
• Supernode Formation: If an ideal voltage source is connected directly between two non-reference nodes without any series resistor, the two nodes are enclosed in a supernode boundary. KCL is applied to the generalized supernode surface, supplemented by the constraint relation:
  $V_A - V_B = V_{\text{source}}$

7. Fundamental Network Theorems for D.C. Circuits

7.1 Thévenin's Theorem

• Formal Statement: Any linear, bilateral, two-terminal network containing resistive elements and energy sources can be replaced at its designated output terminals $(A-B)$ by an electrically equivalent circuit comprising a single ideal voltage source $V_{\text{th}}$ in series with an equivalent resistance $R_{\text{th}}$.
  • $V_{\text{th}}$ (Thévenin Voltage): The open-circuit voltage across terminals $A-B$ ($V_{\text{th}} = V_{oc}$).
  • $R_{\text{th}}$ (Thévenin Resistance): The input resistance looking into terminals $A-B$ with all independent sources deactivated (independent voltage sources replaced with short circuits, $0\,\text{V}$; independent current sources replaced with open circuits, $0\,\text{A}$).
• If dependent sources exist, $R_{\text{th}}$ must be computed by applying an external test excitation source ($v_{\text{test}}$ or $i_{\text{test}}$) at terminals $A-B$:
  $R_{\text{th}} = \frac{v_{\text{test}}}{i_{\text{test}}} = \frac{V_{oc}}{I_{sc}}$
• Load Current and Voltage:
  $I_L = \frac{V_{\text{th}}}{R_{\text{th}} + R_L}, \qquad V_L = I_L R_L = V_{\text{th}} \left( \frac{R_L}{R_{\text{th}} + R_L} \right)$

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       Thévenin Equivalent Circuit:
            o-----+----[ Rth ]----o A
                  |               |
                 (~) Vth         [ ] RL
                  |               |
            o-----+---------------o B

7.2 Norton's Theorem

• Formal Statement: Any linear, bilateral, two-terminal network can be replaced at terminals $(A-B)$ by an equivalent circuit comprising a single ideal current source $I_N$ connected in parallel with an equivalent resistance $R_N$.
  • $I_N$ (Norton Current): The short-circuit current flowing through a zero-resistance conductor placed across terminals $A-B$ ($I_N = I_{sc}$).
  • $R_N$ (Norton Resistance): The equivalent resistance looking back into terminals $A-B$ with all independent sources turned off. By duality:
    $R_N = R_{\text{th}}$
• Relationship between Thévenin and Norton Equivalents:
  $V_{\text{th}} = I_N \cdot R_N \iff I_N = \frac{V_{\text{th}}}{R_{\text{th}}}$

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       Norton Equivalent Circuit:
            o-----+---------------+----o A
                  |               |    |
                 (^) IN          [ ]  [ ] RL
                  |              RN    |
            o-----+---------------+----o B

7.3 Maximum Power Transfer Theorem (MPTT)

• Statement: An active linear D.C. network delivers maximum power to an adjustable resistive load $R_L$ when the resistance of the load equals the Thévenin equivalent resistance of the driving network looking into the load terminals:
  $R_L = R_{\text{th}}$
• Mathematical Derivation:
  $P_L = I_L^2 R_L = \left( \frac{V_{\text{th}}}{R_{\text{th}} + R_L} \right)^2 R_L = \frac{V_{\text{th}}^2 R_L}{(R_{\text{th}} + R_L)^2}$
  To maximize $P_L$ with respect to the variable load parameter $R_L$, set $\frac{dP_L}{dR_L} = 0$:
  $\frac{dP_L}{dR_L} = V_{\text{th}}^2 \left[ \frac{(R_{\text{th}} + R_L)^2 - 2 R_L (R_{\text{th}} + R_L)}{(R_{\text{th}} + R_L)^4} \right] = 0$
  $(R_{\text{th}} + R_L) - 2 R_L = 0 \implies R_{\text{th}} = R_L$
• Maximum Power Transferred:
  $P_{L,\text{max}} = \frac{V_{\text{th}}^2 R_{\text{th}}}{(2 R_{\text{th}})^2} = \frac{V_{\text{th}}^2}{4 R_{\text{th}}}$
• Transmission Efficiency at Maximum Power Transfer:
  $P_{\text{source}} = I_L^2 (R_{\text{th}} + R_L) = I_L^2 (2 R_{\text{th}})$
  $\eta = \frac{P_{\text{load}}}{P_{\text{total}}} = \frac{I_L^2 R_{\text{th}}}{I_L^2 (2 R_{\text{th}})} = 50\%$
  Crucial Engineering Insight: Maximum power transfer does not coincide with maximum efficiency. In power utility grids, matching $R_L = R_{\text{th}}$ would destroy generating systems and cause $50\%$ energy wastage in transmission lines. MPTT is therefore applied primarily in telecommunications, RF systems, and sensor instrumentation where signal power transfer is paramount.

7.4 Superposition Theorem

• Statement: In any linear, active, bilateral network containing two or more independent sources, the voltage across or current through any branch element is the algebraic sum of the individual voltages or currents produced by each independent source operating alone, with all other independent sources disabled.
• Deactivation Rules:
  • Independent Voltage Sources are set to $0\,\text{V} \implies$ replaced by a Short Circuit.
  • Independent Current Sources are set to $0\,\text{A} \implies$ replaced by an Open Circuit.
  • Dependent (controlled) sources must NEVER be deactivated; their control variables must remain intact throughout all partial sub-circuit solutions.
• Limitation of Superposition:
  Superposition cannot be applied directly to calculate electric power, because power is a quadratic (non-linear) function of current and voltage:
  $P = I^2 R = (I_1 + I_2)^2 R = I_1^2 R + I_2^2 R + 2 I_1 I_2 R \neq P_1 + P_2$
  To determine total power, one must first find the net total current or voltage via superposition, and subsequently compute $P = I_{\text{net}}^2 R$.

8. Summary Comparison of Network Theorems

9. Common Pitfalls & Exam Traps

1. Attempting Direct Power Superposition: Calculating total power by summing powers from individual source calculations ($P_{\text{tot}} \neq P_1 + P_2$). Always sum the branch currents first ($I_{\text{tot}} = I_1 + I_2$) and then evaluate $P = I_{\text{tot}}^2 R$.
2. Deactivating Controlled Sources in Thevenin/Superposition: Turning off dependent sources like independent ones is an invalid error. Dependent sources must remain energized during all steps.
3. Confusion in Current Division Formula: For two parallel resistors, the numerator must be the opposite branch resistor ($I_1 = I_{\text{total}} \frac{R_2}{R_1 + R_2}$).
4. Neglecting Internal Resistances of Practical Sources: An ideal voltage source has zero series resistance; an ideal current source has infinite parallel resistance. Mixing these up distorts source transformation.
5. Ignoring Inductor and Capacitor D.C. Steady-State Equivalents: In steady D.C., inductors act as short circuits ($V_L = 0$) and capacitors act as open circuits ($I_C = 0$). Failing to substitute these leads to erroneous circuit topologies.

10. Quick Reference Formula Cheat-Sheet

• Pouillet's Resistance Law: $R = \rho \frac{L}{A}$
• Temperature Dependence: $R_T = R_0 [1 + \alpha_0 (T - T_0)]$
• Ohm's Law: $V = I R \iff I = G V$
• Power Dissipation: $P = V I = I^2 R = \frac{V^2}{R}$
• Energy Dissipated / Billed: $W = P \cdot t \quad (1\text{ kWh} = 3.6 \times 10^6\text{ J})$
• Series Resistance: $R_{\text{eq}} = \sum_{i=1}^n R_i$
• Parallel Resistance (2 branches): $R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$
• Voltage Division: $V_k = V_s \left( \frac{R_k}{\sum R} \right)$
• Current Division: $I_1 = I_s \left( \frac{R_2}{R_1 + R_2} \right)$
• Delta to Star Conversion: $R_A = \frac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}}$
• Star to Delta Conversion: $R_{ab} = R_A + R_B + \frac{R_A R_B}{R_C}$
• Mesh Analysis Count: $M = B - N + 1$
• Thévenin-Norton Relation: $V_{\text{th}} = I_N R_N, \quad R_N = R_{\text{th}}$
• Maximum Power Transfer: $R_L = R_{\text{th}} \implies P_{\text{max}} = \frac{V_{\text{th}}^2}{4 R_{\text{th}}}, \quad \eta = 50\%$