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4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
PRC Registered Electrical Engineer REE
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1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
Key Takeaways Explain This Topic AI
Vector differential operators—gradient ( ∇ ϕ \nabla \phi ∇ ϕ ), divergence ( ∇ ⋅ A \nabla \cdot \mathbf{A} ∇ ⋅ A ), curl ( ∇ × A \nabla \times \mathbf{A} ∇ × A ), and Laplacian ( ∇ 2 ϕ \nabla^2 \phi ∇ 2 ϕ )—form the mathematical foundation of Maxwell's electromagnetic field equations.
Gauss's Divergence Theorem connects volume integrals of field divergence to closed surface flux integrals ( ∭ V ∇ ⋅ A d V = ∬ S A ⋅ d S \iiint_V \nabla\cdot\mathbf{A}\,dV = \iint_S \mathbf{A}\cdot d\mathbf{S} ∭ V ∇ ⋅ A d V = ∬ S A ⋅ d S ), while Stokes' Theorem relates surface curl integrals to closed line integrals ( ∬ S ( ∇ × A ) ⋅ d S = ∮ C A ⋅ d r \iint_S (\nabla\times\mathbf{A})\cdot d\mathbf{S} = \oint_C \mathbf{A}\cdot d\mathbf{r} ∬ S ( ∇ × A ) ⋅ d S = ∮ C A ⋅ d r ).
Square matrices and determinants are used to analyze multi-mesh electrical networks, where matrix inversion A − 1 = adj ( A ) det ( A ) A^{-1} = \frac{\text{adj}(A)}{\det(A)} A − 1 = d e t ( A ) adj ( A ) and Cramer's rule solve simultaneous linear nodal and mesh equations.
System characteristic equations det ( A − λ I ) = 0 \det(\mathbf{A} - \lambda \mathbf{I}) = 0 det ( A − λ I ) = 0 yield eigenvalues λ \lambda λ and eigenvectors that determine modal stability, natural frequencies, and state-space dynamic responses.
Complex analytic functions satisfy the Cauchy-Riemann equations ( ∂ u ∂ x = ∂ v ∂ y \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} ∂ x ∂ u = ∂ y ∂ v , ∂ u ∂ y = − ∂ v ∂ x \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} ∂ y ∂ u = − ∂ x ∂ v ), enabling contour integration via Cauchy's Residue Theorem ( ∮ C f ( z ) d z = 2 π j ∑ Res \oint_C f(z)\,dz = 2\pi j \sum \text{Res} ∮ C f ( z ) d z = 2 πj ∑ Res ).
Last updated: August 2026
4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
Advanced engineering mathematics unifies spatial electromagnetic field theory, multi-variable circuit network equations, system dynamics, and complex boundary integrals. For electrical engineers preparing for the PRC Board Exam, proficiency in vector calculus differential and integral operators, matrix transformations (Cramer's Rule and eigenvalue analysis), and complex analytical residue integration provides essential tools for electromagnetic fields, power system stability, and advanced circuit synthesis.
- 1. Vector Operations & Differential Calculus
- Vector Multiplication
Dot (Scalar) Product: A ⋅ B = ∣ A ∣ ∣ B ∣ cos θ = A x B x + A y B y + A z B z \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}| \cos\theta = A_x B_x + A_y B_y + A_z B_z A ⋅ B = ∣ A ∣∣ B ∣ cos θ = A x B x + A y B y + A z B z
Cross (Vector) Product: A × B = ∣ A ∣ ∣ B ∣ sin θ n ^ = det [ i ^ j ^ k ^ A x A y A z B x B y B z ] \mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}| \sin\theta \, \hat{\mathbf{n}} = \det \begin{bmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{bmatrix} A × B = ∣ A ∣∣ B ∣ sin θ n ^ = det i ^ A x B x j ^ A y B y k ^ A z B z
Scalar Triple Product: A ⋅ ( B × C ) = det [ A x A y A z B x B y B z C x C y C z ] \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \det \begin{bmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{bmatrix} A ⋅ ( B × C ) = det A x B x C x A y B y C y A z B z C z (Represents parallelepiped volume)
Vector Triple Product: A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B ) C \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C}) \mathbf{B} - (\mathbf{A} \cdot \mathbf{B}) \mathbf{C} A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B ) C (BAC-CAB rule)
Vector Differential Operators (Cartesian Coordinates)
Using Del operator ∇ = i ^ ∂ ∂ x + j ^ ∂ ∂ y + k ^ ∂ ∂ z \nabla = \hat{\mathbf{i}}\frac{\partial}{\partial x} + \hat{\mathbf{j}}\frac{\partial}{\partial y} + \hat{\mathbf{k}}\frac{\partial}{\partial z} ∇ = i ^ ∂ x ∂ + j ^ ∂ y ∂ + k ^ ∂ z ∂ :
Gradient of a Scalar Field ϕ \phi ϕ : ∇ ϕ = ∂ ϕ ∂ x i ^ + ∂ ϕ ∂ y j ^ + ∂ ϕ ∂ z k ^ \nabla \phi = \frac{\partial \phi}{\partial x} \hat{\mathbf{i}} + \frac{\partial \phi}{\partial y} \hat{\mathbf{j}} + \frac{\partial \phi}{\partial z} \hat{\mathbf{k}} ∇ ϕ = ∂ x ∂ ϕ i ^ + ∂ y ∂ ϕ j ^ + ∂ z ∂ ϕ k ^
Physical Meaning: Points in the direction of maximum rate of increase of scalar potential ϕ \phi ϕ (e.g., Electric Field E = − ∇ V \mathbf{E} = -\nabla V E = − ∇ V ).
Divergence of a Vector Field A \mathbf{A} A : ∇ ⋅ A = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z \nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} ∇ ⋅ A = ∂ x ∂ A x + ∂ y ∂ A y + ∂ z ∂ A z
Physical Meaning: Measures net outward flux per unit volume from an infinitesimal region. If ∇ ⋅ A = 0 \nabla \cdot \mathbf{A} = 0 ∇ ⋅ A = 0 , field A \mathbf{A} A is solenoidal (e.g., Magnetic Field ∇ ⋅ B = 0 \nabla \cdot \mathbf{B} = 0 ∇ ⋅ B = 0 ).
Curl of a Vector Field A \mathbf{A} A : ∇ × A = ( ∂ A z ∂ y − ∂ A y ∂ z ) i ^ + ( ∂ A x ∂ z − ∂ A z ∂ x ) j ^ + ( ∂ A y ∂ x − ∂ A x ∂ y ) k ^ \nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \hat{\mathbf{i}} + \left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) \hat{\mathbf{j}} + \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \hat{\mathbf{k}} ∇ × A = ( ∂ y ∂ A z − ∂ z ∂ A y ) i ^ + ( ∂ z ∂ A x − ∂ x ∂ A z ) j ^ + ( ∂ x ∂ A y − ∂ y ∂ A x ) k ^
Physical Meaning: Measures local circulation or rotational intensity per unit area. If ∇ × A = 0 \nabla \times \mathbf{A} = \mathbf{0} ∇ × A = 0 , field A \mathbf{A} A is irrotational or conservative .
Laplacian Operator ∇ 2 \nabla^2 ∇ 2 : ∇ 2 ϕ = ∇ ⋅ ( ∇ ϕ ) = ∂ 2 ϕ ∂ x 2 + ∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ z 2 \nabla^2 \phi = \nabla \cdot (\nabla \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} ∇ 2 ϕ = ∇ ⋅ ( ∇ ϕ ) = ∂ x 2 ∂ 2 ϕ + ∂ y 2 ∂ 2 ϕ + ∂ z 2 ∂ 2 ϕ
- 2. Integral Theorems of Vector Calculus
- Gauss's Divergence Theorem
Relates a volume integral of field divergence over region V V V to a surface flux integral over bounding closed surface S S S : ∭ V ( ∇ ⋅ A ) d V = ∬ S A ⋅ d S \iiint_V (\nabla \cdot \mathbf{A}) \, dV = \iint_S \mathbf{A} \cdot d\mathbf{S} ∭ V ( ∇ ⋅ A ) d V = ∬ S A ⋅ d S
Stokes' Theorem
Relates a surface integral of field curl over open surface S S S to a closed line integral along bounding perimeter curve C C C : ∬ S ( ∇ × A ) ⋅ d S = ∮ C A ⋅ d r \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_C \mathbf{A} \cdot d\mathbf{r} ∬ S ( ∇ × A ) ⋅ d S = ∮ C A ⋅ d r
Green's Theorem in the Plane
Special 2D case of Stokes' Theorem for planar region R R R bounded by simple closed curve C C C : ∮ C ( M d x + N d y ) = ∬ R ( ∂ N ∂ x − ∂ M ∂ y ) d A \oint_C (M \, dx + N \, dy) = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA ∮ C ( M d x + N d y ) = ∬ R ( ∂ x ∂ N − ∂ y ∂ M ) d A
- 3. Linear Algebra, Matrices & Cramer's Rule
- Matrix Inversion
For an non-singular n × n n \times n n × n square matrix A \mathbf{A} A with det ( A ) ≠ 0 \det(\mathbf{A}) \neq 0 det ( A ) = 0 : A − 1 = 1 det ( A ) adj ( A ) \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \text{adj}(\mathbf{A}) A − 1 = det ( A ) 1 adj ( A )
where adj ( A ) = C T \text{adj}(\mathbf{A}) = \mathbf{C}^T adj ( A ) = C T is the transpose of the cofactor matrix C \mathbf{C} C .
Cramer's Rule for Solving Linear Systems
For a system of n n n linear equations A x = b \mathbf{A} \mathbf{x} = \mathbf{b} Ax = b : x i = det ( A i ) det ( A ) ( i = 1 , 2 , … , n ) x_i = \frac{\det(\mathbf{A}_i)}{\det(\mathbf{A})} \quad (i = 1, 2, \dots, n) x i = det ( A ) det ( A i ) ( i = 1 , 2 , … , n )
where matrix A i \mathbf{A}_i A i is formed by replacing the i i i -th column of coefficient matrix A \mathbf{A} A with column vector b \mathbf{b} b .
4. Eigenvalues & Eigenvectors
For a square matrix A \mathbf{A} A , non-zero eigenvector v \mathbf{v} v and scalar eigenvalue λ \lambda λ satisfy: A v = λ v ⟹ ( A − λ I ) v = 0 \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \implies (\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = \mathbf{0} Av = λ v ⟹ ( A − λ I ) v = 0
Characteristic Equation
To obtain non-trivial solutions v ≠ 0 \mathbf{v} \neq \mathbf{0} v = 0 : det ( A − λ I ) = 0 \det(\mathbf{A} - \lambda \mathbf{I}) = 0 det ( A − λ I ) = 0
Key Eigenvalue Properties
Trace Identity: Sum of eigenvalues equals the matrix trace: ∑ i = 1 n λ i = Tr ( A ) = ∑ i = 1 n a i i \sum_{i=1}^n \lambda_i = \text{Tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii} ∑ i = 1 n λ i = Tr ( A ) = ∑ i = 1 n a ii .
Determinant Identity: Product of eigenvalues equals matrix determinant: ∏ i = 1 n λ i = det ( A ) \prod_{i=1}^n \lambda_i = \det(\mathbf{A}) ∏ i = 1 n λ i = det ( A ) .
System Stability: State space matrix x ˙ = A x \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} x ˙ = Ax is asymptotically stable if and only if all eigenvalues λ i \lambda_i λ i have strictly negative real parts ( Re { λ i } < 0 \text{Re}\{\lambda_i\} < 0 Re { λ i } < 0 ).
- 5. Complex Variable Theory & Residue Calculus
- Analytic Functions & Cauchy-Riemann Equations
A complex function f ( z ) = u ( x , y ) + j v ( x , y ) f(z) = u(x, y) + j v(x, y) f ( z ) = u ( x , y ) + j v ( x , y ) is analytic (differentiable) in a region if continuous first partial derivatives satisfy the Cauchy-Riemann Equations : ∂ u ∂ x = ∂ v ∂ y and ∂ u ∂ y = − ∂ v ∂ x \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} ∂ x ∂ u = ∂ y ∂ v and ∂ y ∂ u = − ∂ x ∂ v
Both u ( x , y ) u(x,y) u ( x , y ) and v ( x , y ) v(x,y) v ( x , y ) satisfy Laplace's equation ( ∇ 2 u = 0 , ∇ 2 v = 0 \nabla^2 u = 0, \nabla^2 v = 0 ∇ 2 u = 0 , ∇ 2 v = 0 ) and are called harmonic conjugates .
Cauchy's Integral Theorem & Formula
Cauchy's Theorem: If f ( z ) f(z) f ( z ) is analytic inside and on simple closed curve C C C : ∮ C f ( z ) d z = 0 \oint_C f(z) \, dz = 0 ∮ C f ( z ) d z = 0
Cauchy's Integral Formula: If f ( z ) f(z) f ( z ) is analytic inside C C C containing point z 0 z_0 z 0 : ∮ C f ( z ) z − z 0 d z = 2 π j ⋅ f ( z 0 ) \oint_C \frac{f(z)}{z - z_0} \, dz = 2\pi j \cdot f(z_0) ∮ C z − z 0 f ( z ) d z = 2 πj ⋅ f ( z 0 )
Cauchy's Residue Theorem
For a function f ( z ) f(z) f ( z ) with isolated singular points (poles) z 1 , z 2 , … , z k z_1, z_2, \dots, z_k z 1 , z 2 , … , z k enclosed by contour C C C : ∮ C f ( z ) d z = 2 π j ∑ i = 1 k Res ( f , z i ) \oint_C f(z) \, dz = 2\pi j \sum_{i=1}^k \text{Res}(f, z_i) ∮ C f ( z ) d z = 2 πj i = 1 ∑ k Res ( f , z i )
For a simple pole at z = z 0 z = z_0 z = z 0 , the residue is calculated as: Res ( f , z 0 ) = lim z → z 0 ( z − z 0 ) f ( z ) \text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z) Res ( f , z 0 ) = z → z 0 lim ( z − z 0 ) f ( z )
- Step-by-Step Worked Numerical Examples
- Example 4.3.1: Vector Divergence & Flux Calculation via Gauss's Theorem
Problem: Evaluate the outward flux of vector field A = 2 x i ^ + 3 y j ^ + 4 z k ^ \mathbf{A} = 2x \hat{\mathbf{i}} + 3y \hat{\mathbf{j}} + 4z \hat{\mathbf{k}} A = 2 x i ^ + 3 y j ^ + 4 z k ^ through the closed surface of a sphere x 2 + y 2 + z 2 = 9 x^2 + y^2 + z^2 = 9 x 2 + y 2 + z 2 = 9 (radius R = 3 R = 3 R = 3 ).
Solution:
Calculate the divergence of vector field A \mathbf{A} A : ∇ ⋅ A = ∂ ∂ x ( 2 x ) + ∂ ∂ y ( 3 y ) + ∂ ∂ z ( 4 z ) = 2 + 3 + 4 = 9 \nabla \cdot \mathbf{A} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(4z) = 2 + 3 + 4 = 9 ∇ ⋅ A = ∂ x ∂ ( 2 x ) + ∂ y ∂ ( 3 y ) + ∂ z ∂ ( 4 z ) = 2 + 3 + 4 = 9
Apply Gauss's Divergence Theorem: Flux = ∬ S A ⋅ d S = ∭ V ( ∇ ⋅ A ) d V = ∭ V 9 d V = 9 ∭ V d V \text{Flux} = \iint_S \mathbf{A} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{A}) \, dV = \iiint_V 9 \, dV = 9 \iiint_V dV Flux = ∬ S A ⋅ d S = ∭ V ( ∇ ⋅ A ) d V = ∭ V 9 d V = 9 ∭ V d V
Compute sphere volume V V V : V = 4 3 π R 3 = 4 3 π ( 3 ) 3 = 4 3 π ( 27 ) = 36 π V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi V = 3 4 π R 3 = 3 4 π ( 3 ) 3 = 3 4 π ( 27 ) = 36 π
Calculate total flux: Flux = 9 ⋅ ( 36 π ) = 324 π ≈ 1017.876 units \text{Flux} = 9 \cdot (36\pi) = 324\pi \approx 1017.876 \text{ units} Flux = 9 ⋅ ( 36 π ) = 324 π ≈ 1017.876 units
Example 4.3.2: Multi-Mesh Circuit Solution via Cramer's Rule
Problem: Solve for mesh currents I 1 I_1 I 1 and I 2 I_2 I 2 in the 2-mesh network system: [ 5 − 2 − 2 4 ] [ I 1 I 2 ] = [ 12 0 ] \begin{bmatrix} 5 & -2 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} 12 \\ 0 \end{bmatrix} [ 5 − 2 − 2 4 ] [ I 1 I 2 ] = [ 12 0 ]
Solution:
Calculate system determinant D = det ( A ) D = \det(\mathbf{A}) D = det ( A ) : D = det [ 5 − 2 − 2 4 ] = ( 5 ) ( 4 ) − ( − 2 ) ( − 2 ) = 20 − 4 = 16 D = \det \begin{bmatrix} 5 & -2 \\ -2 & 4 \end{bmatrix} = (5)(4) - (-2)(-2) = 20 - 4 = 16 D = det [ 5 − 2 − 2 4 ] = ( 5 ) ( 4 ) − ( − 2 ) ( − 2 ) = 20 − 4 = 16
Formulate modified matrix A 1 \mathbf{A}_1 A 1 for I 1 I_1 I 1 : A 1 = [ 12 − 2 0 4 ] ⟹ D 1 = det ( A 1 ) = ( 12 ) ( 4 ) − ( − 2 ) ( 0 ) = 48 \mathbf{A}_1 = \begin{bmatrix} 12 & -2 \\ 0 & 4 \end{bmatrix} \implies D_1 = \det(\mathbf{A}_1) = (12)(4) - (-2)(0) = 48 A 1 = [ 12 0 − 2 4 ] ⟹ D 1 = det ( A 1 ) = ( 12 ) ( 4 ) − ( − 2 ) ( 0 ) = 48
Formulate modified matrix A 2 \mathbf{A}_2 A 2 for I 2 I_2 I 2 : A 2 = [ 5 12 − 2 0 ] ⟹ D 2 = det ( A 2 ) = ( 5 ) ( 0 ) − ( 12 ) ( − 2 ) = 24 \mathbf{A}_2 = \begin{bmatrix} 5 & 12 \\ -2 & 0 \end{bmatrix} \implies D_2 = \det(\mathbf{A}_2) = (5)(0) - (12)(-2) = 24 A 2 = [ 5 − 2 12 0 ] ⟹ D 2 = det ( A 2 ) = ( 5 ) ( 0 ) − ( 12 ) ( − 2 ) = 24
Apply Cramer's Rule: I 1 = D 1 D = 48 16 = 3.0 A I_1 = \frac{D_1}{D} = \frac{48}{16} = 3.0 \text{ A} I 1 = D D 1 = 16 48 = 3.0 A I 2 = D 2 D = 24 16 = 1.5 A I_2 = \frac{D_2}{D} = \frac{24}{16} = 1.5 \text{ A} I 2 = D D 2 = 16 24 = 1.5 A
Example 4.3.3: System Matrix Eigenvalue Analysis
Problem: Determine the eigenvalues λ 1 \lambda_1 λ 1 and λ 2 \lambda_2 λ 2 for state dynamic matrix A = [ − 4 2 1 − 3 ] \mathbf{A} = \begin{bmatrix} -4 & 2 \\ 1 & -3 \end{bmatrix} A = [ − 4 1 2 − 3 ] and assess system stability.
Solution:
Formulate characteristic equation det ( A − λ I ) = 0 \det(\mathbf{A} - \lambda \mathbf{I}) = 0 det ( A − λ I ) = 0 : det [ − 4 − λ 2 1 − 3 − λ ] = 0 \det \begin{bmatrix} -4 - \lambda & 2 \\ 1 & -3 - \lambda \end{bmatrix} = 0 det [ − 4 − λ 1 2 − 3 − λ ] = 0
Expand determinant quadratic: ( − 4 − λ ) ( − 3 − λ ) − ( 2 ) ( 1 ) = 0 (-4 - \lambda)(-3 - \lambda) - (2)(1) = 0 ( − 4 − λ ) ( − 3 − λ ) − ( 2 ) ( 1 ) = 0 λ 2 + 7 λ + 12 − 2 = 0 ⟹ λ 2 + 7 λ + 10 = 0 \lambda^2 + 7 \lambda + 12 - 2 = 0 \implies \lambda^2 + 7 \lambda + 10 = 0 λ 2 + 7 λ + 12 − 2 = 0 ⟹ λ 2 + 7 λ + 10 = 0
Factor quadratic to find roots: ( λ + 2 ) ( λ + 5 ) = 0 ⟹ λ 1 = − 2 , λ 2 = − 5 (\lambda + 2)(\lambda + 5) = 0 \implies \lambda_1 = -2, \quad \lambda_2 = -5 ( λ + 2 ) ( λ + 5 ) = 0 ⟹ λ 1 = − 2 , λ 2 = − 5
Verify properties:
Trace check: Tr ( A ) = − 4 + ( − 3 ) = − 7 \text{Tr}(\mathbf{A}) = -4 + (-3) = -7 Tr ( A ) = − 4 + ( − 3 ) = − 7 . Sum λ 1 + λ 2 = − 2 + ( − 5 ) = − 7 \lambda_1 + \lambda_2 = -2 + (-5) = -7 λ 1 + λ 2 = − 2 + ( − 5 ) = − 7 (Matches).
Determinant check: det ( A ) = ( − 4 ) ( − 3 ) − ( 2 ) ( 1 ) = 10 \det(\mathbf{A}) = (-4)(-3) - (2)(1) = 10 det ( A ) = ( − 4 ) ( − 3 ) − ( 2 ) ( 1 ) = 10 . Product λ 1 λ 2 = ( − 2 ) ( − 5 ) = 10 \lambda_1 \lambda_2 = (-2)(-5) = 10 λ 1 λ 2 = ( − 2 ) ( − 5 ) = 10 (Matches).
Assess Stability: Because both eigenvalues are real and strictly negative ( λ 1 , λ 2 < 0 \lambda_1, \lambda_2 < 0 λ 1 , λ 2 < 0 ), the dynamic system is asymptotically stable .