Appendix A — Simultaneous Equations, Matrix Inversion, and Cramer’s Rule

Many circuit problems (nodal and mesh analysis especially) produce a set of simultaneous linear equations. This appendix reviews how to solve them.

A.1 Matrix form

A system of \(n\) linear equations in \(n\) unknowns can be written

\[ \mathbf{A}\mathbf{x} = \mathbf{b}, \]

where \(\mathbf{A}\) is the \(n\times n\) coefficient matrix, \(\mathbf{x}\) the unknown vector, and \(\mathbf{b}\) the source (right-hand-side) vector.

A.2 Cramer’s rule

For the system \(\mathbf{A}\mathbf{x} = \mathbf{b}\), if \(\Delta = \det(\mathbf{A}) \ne 0\), each unknown is

\[ \boxed{x_k = \frac{\Delta_k}{\Delta}}, \]

where \(\Delta_k\) is the determinant of the matrix formed by replacing the \(k\)-th column of \(\mathbf{A}\) with \(\mathbf{b}\).

2×2 case

For

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} e \\ f \end{bmatrix}, \]

\[ \Delta = ad - bc, \]

\[ x_1 = \frac{ed - bf}{\Delta}, \qquad x_2 = \frac{af - ec}{\Delta}. \]

3×3 case

For a \(3\times3\) matrix \(\mathbf{A}\) with entries \(a_{ij}\) and right-hand side \(\mathbf{b} = [e, f, g]^T\),

\[ \Delta = a_{11}(a_{22}a_{33}-a_{23}a_{32}) - a_{12}(a_{21}a_{33}-a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31}). \]

Each \(\Delta_k\) is computed the same way with the \(k\)-th column replaced by \([e,f,g]^T\), and \(x_k = \Delta_k/\Delta\).

A.3 Matrix inversion

If \(\mathbf{A}\) is invertible (\(\Delta \ne 0\)), the solution is

\[ \mathbf{x} = \mathbf{A}^{-1}\mathbf{b}, \]

where \(\mathbf{A}^{-1} = \frac{1}{\Delta}\,\text{adj}(\mathbf{A})\) and \(\text{adj}(\mathbf{A})\) is the adjugate (transpose of the cofactor matrix). For the \(2\times2\) case,

\[ \mathbf{A}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}. \]

A.4 Software

For larger systems, use computational tools: MATLAB (x = A\b), Mathcad, Maple, Python (numpy.linalg.solve), or a calculator with matrix functions.

Appendix B — Quick-Reference Formula Sheet

Basic quantities (Chapter 1)

• Charge: \(q\) (C); \(e = 1.602\times10^{-19}\) C
• Current: \(i = \dfrac{dq}{dt}\) (A); \(q = \displaystyle\int i\,dt\)
• Voltage: \(v = \dfrac{dw}{dq}\) (V = J/C); \(v_{ab} = -v_{ba}\)
• Power: \(p = vi\) (W); passive sign convention
• Energy: \(w = \displaystyle\int p\,dt\) (J); \(1\ \text{Wh} = 3600\) J
• Conservation: \(\sum p = 0\)

Ohm’s law and resistors (Chapter 2)

• \(v = iR\); \(R = \rho \ell/A\); \(G = 1/R\) (S)
• Power in resistor: \(p = i^2 R = v^2/R = v^2 G\)
• Series: \(R_{\text{eq}} = \sum R_k\); voltage divider \(v_n = v\,R_n/\sum R_k\)
• Parallel: \(1/R_{\text{eq}} = \sum 1/R_k\) (two: \(R_1R_2/(R_1+R_2)\)); \(G_{\text{eq}} = \sum G_k\)
• Current divider: \(i_1 = i\,R_2/(R_1+R_2)\)
• Wye–Delta: \(R_Y = R_{\text{adj1}}R_{\text{adj2}}/\sum R_\Delta\); \(R_\Delta = (R_1R_2+R_2R_3+R_3R_1)/R_{\text{opp}}\); balanced \(R_\Delta = 3R_Y\)

Kirchhoff’s laws (Chapter 2)

• KCL: \(\sum i_{\text{entering}} = 0\) (sum in = sum out)
• KVL: \(\sum v_{\text{around loop}} = 0\) (sum of drops = sum of rises)
• Topology: \(\ell = b - n + 1\)

Nodal and mesh analysis (Chapter 3)

• Nodal: \(i_{a\to b} = (V_a - V_b)/R\); KCL at each non-reference node
• Supernode: voltage source between two non-reference nodes → KCL on boundary + constraint \(v_2 - v_1 = V_s\)
• Mesh: clockwise currents; KVL in each mesh; diagonal = sum of mesh resistances, off-diagonal = −shared resistance
• Supermesh: current source on shared branch → KVL around supermesh + constraint \(i_2 - i_1 = i_s\)
• Cramer’s rule: \(x_k = \Delta_k/\Delta\)

Circuit theorems (Chapter 4)

• Linearity: homogeneity + additivity (does not apply to power)
• Superposition: turn off all but one independent source (V→short, I→open; keep dependent); sum contributions
• Source transformation: \(v_s = i_s R\) (series V/R ↔︎ parallel I/R); arrow toward + terminal
• Thévenin: \(V_{\text{Th}} = v_{oc}\); \(R_{\text{Th}}\) = input resistance with sources off (test source if dependent)
• Norton: \(I_N = i_{sc}\); \(R_N = R_{\text{Th}}\); \(V_{\text{Th}} = I_N R_{\text{Th}}\)
• Any two of \(\{v_{oc}, i_{sc}, R_{\text{Th}}\}\) determine the equivalent
• Max power transfer: \(R_L = R_{\text{Th}}\); \(P_{\max} = V_{\text{Th}}^2/(4R_{\text{Th}})\)

Capacitors and inductors (Chapter 5)

• Capacitor: \(q = Cv\); \(i = C\,dv/dt\); \(v(t) = v(0) + \frac{1}{C}\int i\,dt\); \(w = \tfrac12 Cv^2\); DC steady state → open; \(v_C\) continuous
• Inductor: \(\lambda = Li\); \(v = L\,di/dt\); \(i(t) = i(0) + \frac{1}{L}\int v\,dt\); \(w = \tfrac12 Li^2\); DC steady state → short; \(i_L\) continuous
• Capacitors: parallel \(C_{\text{eq}} = \sum C_k\); series \(1/C_{\text{eq}} = \sum 1/C_k\); series voltage divider \(v_1 = v\,C_2/(C_1+C_2)\)
• Inductors: series \(L_{\text{eq}} = \sum L_k\); parallel \(1/L_{\text{eq}} = \sum 1/L_k\); series voltage divider \(v_1 = v\,L_1/(L_1+L_2)\); parallel current divider \(i_1 = i\,L_2/(L_1+L_2)\)

First-order transients (Chapter 5)

• Complete response: \(x(t) = x(\infty) + [x(0)-x(\infty)]\,e^{-t/\tau}\)
• RC: \(\tau = RC\); charging \(v_C = V_s(1-e^{-t/\tau})\); discharging \(v_C = V_0 e^{-t/\tau}\)
• RL: \(\tau = L/R\); rising \(i_L = I_\infty(1-e^{-t/\tau})\); decaying \(i_L = I_0 e^{-t/\tau}\)

Second-order RLC (Chapter 5)

• \(s^2 + 2\alpha s + \omega_0^2 = 0\); series \(\alpha = R/(2L)\), parallel \(\alpha = 1/(2RC)\), \(\omega_0 = 1/\sqrt{LC}\)
• Overdamped (\(\alpha>\omega_0\)): \(x = A_1 e^{s_1 t} + A_2 e^{s_2 t}\)
• Critically damped (\(\alpha=\omega_0\)): \(x = (A_1 + A_2 t)e^{-\alpha t}\)
• Underdamped (\(\alpha<\omega_0\)): \(x = e^{-\alpha t}(A_1\cos\omega_d t + A_2\sin\omega_d t)\), \(\omega_d = \sqrt{\omega_0^2-\alpha^2}\)

Index

• Active element, Ch.1
• Additivity property, Ch.4
• Ammeter, Ch.2
• Ampere, Ch.1
• Branch, Ch.2
• Capacitor, Ch.5
  • energy, Ch.5
  • series and parallel, Ch.5
  • voltage–current relation, Ch.5
• Charge, Ch.1
  • conservation of, Ch.1
• Chassis ground, Ch.3
• Conductance, Ch.2
• Conservation of charge, Ch.1
• Conservation of power, Ch.1
• Cramer’s rule, Ch.3, Appendix A
• Current, Ch.1
  • conventional, Ch.1
  • DC, Ch.1
  • divider, Ch.2
  • mesh, Ch.3
• Damping factor, Ch.5
• Damping ratio, Ch.5
• Delta–wye conversion, Ch.2
• Dependent source, Ch.1
  • CCVS, VCVS, CCCS, VCCS, Ch.1
• Earth ground, Ch.3
• Energy, Ch.1
• Equivalent resistance, Ch.2
• First-order circuit, Ch.5
• Ground, Ch.3
• Homogeneity property, Ch.4
• Inductor, Ch.5
  • energy, Ch.5
  • series and parallel, Ch.5
  • voltage–current relation, Ch.5
• Kirchhoff’s current law (KCL), Ch.2
• Kirchhoff’s voltage law (KVL), Ch.2
• Linearity, Ch.4
• Loop, Ch.2
  • independent, Ch.2
• Maximum power transfer, Ch.4
• Mesh, Ch.3
  • analysis, Ch.3
  • supermesh, Ch.3
• Natural frequency, Ch.5
• Network topology, Ch.2
• Node, Ch.2, Ch.3
  • reference, Ch.3
  • supernode, Ch.3
• Nodal analysis, Ch.3
• Norton’s theorem, Ch.4
• Ohm’s law, Ch.2
• Open circuit, Ch.2
• Passive element, Ch.1
• Passive sign convention, Ch.1
• Potentiometer, Ch.2
• Power, Ch.1
  • absorbed/supplied, Ch.1
• Resistance, Ch.2
  • resistivity, Ch.2
• Resistor
  • linear, Ch.2
  • nonlinear, Ch.2
  • types, Ch.2
• Short circuit, Ch.2
• SI units, Ch.1
• Siemens, Ch.2
• Source transformation, Ch.4
• Superposition, Ch.4
• Thévenin’s theorem, Ch.4
• Time constant
  • capacitive (\(\tau = RC\)), Ch.5
  • inductive (\(\tau = L/R\)), Ch.5
• Transient analysis, Ch.5
• Voltage, Ch.1
  • division, Ch.2, Ch.5
  • drop/rise, Ch.1
  • source (independent, dependent), Ch.1
• Watt, Ch.1
• Wye–delta conversion, Ch.2