1.0 FUNDAMENTAL CIRCUIT LAWS & ANALYSIS TECHNIQUES
1.1 Kirchhoff's Laws
- Kirchhoff's Current Law (KCL): At any node, the algebraic sum of currents is zero.
$$\sum_{k=1}^{n} i_k = 0$$
Example: For a node with currents \(i_1\) entering and \(i_2, i_3\) leaving: \(i_1 = i_2 + i_3\).
- Kirchhoff's Voltage Law (KVL): Around any closed loop, the algebraic sum of voltages is zero.
$$\sum_{k=1}^{m} v_k = 0$$
Example: Loop with voltage source \(V\) and resistors: \(V = i_1R_1 + i_2R_2\).
[!TIP] KCL applies to instantaneous currents; KVL to instantaneous voltages. For AC phasors, both hold with complex quantities.
1.2 Basic Circuit Theorems
-
Superposition Theorem: In linear circuits, response due to multiple independent sources equals the sum of responses due to each source alone, with all other independent sources deactivated (voltage sources shorted, current sources opened).
Not valid for power calculations.
Procedure:
-
Deactivate all but one independent source.
-
Compute the desired response (voltage/current).
-
Repeat for each source.
-
Algebraically sum all contributions.
Example: Find voltage across \(3\Omega\) resistor with two sources.
-
-
Thevenin's Theorem: Any linear two-terminal network can be replaced by an equivalent voltage source \(V_{th}\) in series with resistance \(R_{th}\).
Steps:
-
Remove the load.
-
\(V_{th}\) = open-circuit voltage across terminals.
-
\(R_{th}\) = resistance seen from terminals with all independent sources deactivated (for dependent sources, apply test source).
-
Connect load to Thevenin equivalent.
\boxed{V_{th} = V_{oc}, \quad R_{th} = \frac{V_{oc}}{I_{sc}} \text{ (if no dependent sources)}}
Example: Find Thevenin equivalent across terminals AB.
-
-
Norton's Theorem: Equivalent to current source \(I_{sc}\) in parallel with \(R_n\).
Steps:
-
\(I_{sc}\) = short-circuit current across terminals.
-
\(R_n = R_{th}\) (or \(V_{th}/I_{sc}\)).
-
Connect load.
\boxed{I_{sc} = \text{short-circuit current}, \quad R_n = R_{th}}
Example: Find Norton equivalent.
-
[!TIP] Thevenin and Norton equivalents are interchangeable: \(V_{th} = I_{sc} R_{th}\).
-
Maximum Power Transfer Theorem:
-
DC: Maximum power delivered to load when \(R_L = R_{th}\).
\boxed{P_{max} = \frac{V_{th}^2}{4R_{th}}}
Efficiency at max power: \(\eta = 50\%\) (proof: input power \(= V_{th}^2/(2R_{th})\), output \(= V_{th}^2/(4R_{th})\)).
-
AC: For maximum power, load impedance \(Z_L = Z_{th}^*\) (complex conjugate). If \(Z_{th} = R_{th} + jX_{th}\), then \(Z_L = R_{th} - jX_{th}\).
-
-
Milliman's Theorem: For \(n\) parallel branches, each with voltage source \(V_i\) in series with resistance \(R_i\), the equivalent voltage \(V_{eq}\) and resistance \(R_{eq}\) are:
\boxed{V_{eq} = \frac{\sum_{i=1}^{n} \frac{V_i}{R_i}}{\sum_{i=1}^{n} \frac{1}{R_i}}, \quad R_{eq} = \left( \sum_{i=1}^{n} \frac{1}{R_i} \right)^{-1}}
Application: Simplifies parallel voltage sources to a single source.
-
Compensation Theorem: If an element with impedance \(Z\) is replaced by a voltage source equal to the voltage across \(Z\) (same polarity), the currents in all other branches remain unchanged. Similarly, can replace with a current source equal to the current through \(Z\). Used in sensitivity analysis.
-
Substitution Theorem: Any branch can be replaced by any other branch with identical voltage-current relationship (i.e., same V-I equation) without affecting other parts of the network.
Example: Replace a resistor with a voltage source having the same voltage drop.
-
Tellegen's Theorem: For any two networks with identical topology (same incidence matrix), if \(\{v_i\}, \{j_i\}\) are branch voltages and currents in one network, and \(\{v_i'\}, \{j_i'\}\) in the other, then:
\boxed{\sum_{i=1}^{b} v_i j_i' = \sum_{i=1}^{b} v_i' j_i}
Special case: if both networks are identical, \(\sum v_i j_i = 0\) (power balance).
Verification: Often used to check power conservation.
1.3 Nodal & Mesh Analysis
-
Nodal Analysis:
Procedure:
-
Choose reference node (ground).
-
Assign voltages \(V_1, V_2, \dots\) to non-reference nodes.
-
Apply KCL at each non-reference node: sum of currents leaving = 0.
-
Express currents via Ohm's law.
-
Solve linear equations.
Supernode: When a voltage source connects two non-reference nodes, treat them as a supernode. Write KCL for the supernode and include the voltage source constraint.
Example: Find \(i_x\) using nodal analysis.
-
-
Mesh Analysis:
Procedure:
-
Identify independent meshes.
-
Assign mesh currents (usually clockwise).
-
Apply KVL around each mesh: sum of voltage drops = 0.
-
Express voltages in terms of mesh currents.
-
Solve equations.
Supermesh: When a current source is shared between two meshes, create a supermesh excluding the source. Write KVL for supermesh and use the current source constraint to relate mesh currents.
Example: Find \(i_o\) using mesh analysis.
-
[!TIP] Use nodal for circuits with many voltage sources; mesh for many current sources. For dependent sources, treat as independent during setup but include controlling variables.
2.0 NETWORK TOPOLOGY & GRAPH THEORY
2.1 Basic Definitions
-
Graph: Set of nodes (vertices) and branches (edges) representing circuit elements. No isolated nodes.
-
Oriented Graph: Graph with assigned directions to branches.
-
Subgraph: Subset of branches and nodes.
-
Connected Graph: Path exists between any two nodes.
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Planar Graph: Can be drawn on a plane without crossing branches.
-
Tree: Connected subgraph containing all nodes and no loops. For \(n\) nodes, tree has \(n-1\) branches (twigs).
-
Co-Tree: Branches not in the tree. Links (or chords) are co-tree branches.
-
Basic Tie Set (Fundamental Loop): Loop formed by adding one link to the tree. Number of basic tie sets = \(b - (n-1)\), where \(b\) = total branches.
-
Basic Cut Set: Set of branches that, when removed, separates the tree into two parts, containing exactly one twig. Number of basic cut sets = \(n-1\).
2.2 Matrices of a Graph
-
Incidence Matrix (\(A\)): Size \(n \times b\). Entry \(a_{ij} = 1\) if branch \(j\) leaves node \(i\), \(-1\) if enters, \(0\) otherwise.
Reduced Incidence Matrix: Remove row of reference node. Rank = \(n-1\).
Example: Construct for a given graph.
-
Tie Set Matrix (\(B\)): Size \((b-n+1) \times b\). Rows correspond to basic tie sets. Entries: \(+1\) for link, \(-1\) for twig if orientation opposes link, \(+1\) if same.
Property: \(A B^T = 0\).
-
Cut Set Matrix (\(Q\)): Size \((n-1) \times b\). Rows correspond to basic cut sets. Entries: \(+1\) for twig, \(-1\) for links if orientation from cut set side.
Property: \(B Q^T = 0\).
-
Relationship: For planar graphs, \(Q = B^T\) of the dual graph.
2.3 Special Topics
-
Dual Networks: For a planar graph, the dual graph has a node in each mesh of the original and a branch through each original branch. Duality: series ↔ parallel, voltage ↔ current, KCL ↔ KVL, resistance ↔ conductance.
Example: Dual of series RLC is parallel RLC.
-
Network Topology: Study of circuit configuration without regard to element values. Essential for systematic analysis using matrices.
3.0 RESONANT CIRCUITS
3.1 Series Resonance
-
Circuit: \(R\), \(L\), \(C\) in series with voltage source.
-
Resonant frequency:
\boxed{f_r = \frac{1}{2\pi\sqrt{LC}}, \quad \omega_r = \frac{1}{\sqrt{LC}}}
-
At resonance: \(X_L = X_C\), impedance \(Z = R\) (minimum), current \(I = V/R\) (maximum).
Voltages across \(L\) and \(C\):
\boxed{V_L = V_C = Q V}, \quad Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r R C}
-
Bandwidth: \(\text{BW} = f_r / Q\) (for high \(Q\)).
-
Voltage magnification: \(Q\) factor.
[!TIP] For series resonance, the frequency at which voltage across \(L\) or \(C\) is maximum is slightly above \(f_r\):
\boxed{f_{max} = \frac{f_r}{\sqrt{1 - \frac{1}{2Q^2}}}} \quad (Q > 1/\sqrt{2})
3.2 Parallel Resonance
-
Circuit: Practical parallel RLC (e.g., \(R\) in series with \(L\), parallel with \(C\)).
-
Resonant frequency (approximate for high \(Q\)):
\boxed{f_r \approx \frac{1}{2\pi\sqrt{LC}} \sqrt{1 - \frac{R^2 C}{L}}}
For ideal parallel \(LC\) (no resistance), \(f_r = 1/(2\pi\sqrt{LC})\).
-
At resonance: Admittance minimum, impedance maximum, source current minimum.
Currents through \(L\) and \(C\):
\boxed{I_L = I_C = Q I_{source}}, \quad Q = \frac{R}{\omega_r L} = \omega_r R C \quad (\text{for parallel } R \text{ with } LC)
-
Bandwidth: \(\text{BW} = f_r / Q\).
-
Current magnification: \(Q\) factor.
[!TIP] In parallel resonance, total current is minimum at \(f_r\), but branch currents are maximum.
4.0 LAPLACE TRANSFORM IN NETWORK ANALYSIS
4.1 Fundamentals
-
Definition:
\boxed{F(s) = \mathcal{L}{f(t)} = \int_{0^{-}}^{\infty} f(t) e^{-st} dt}, \quad s = \sigma + j\omega
-
Properties:
Linearity: \(\mathcal{L}\{af_1 + bf_2\} = aF_1(s) + bF_2(s)\)
Time shift: \(\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as} F(s)\)
Frequency shift: \(\mathcal{L}\{e^{-at}f(t)\} = F(s+a)\)
Differentiation: \(\mathcal{L}\{f'(t)\} = sF(s) - f(0^{-})\)
Integration: \(\mathcal{L}\{\int_{0^{-}}^{t} f(\tau) d\tau\} = \frac{F(s)}{s}\)
-
Standard Transforms:
\begin{align*}
\mathcal{L}{1} &= \frac{1}{s}, & \mathcal{L}{t} &= \frac{1}{s^2}, \
\mathcal{L}{e^{-at}} &= \frac{1}{s+a}, & \mathcal{L}{\sin \omega t} &= \frac{\omega}{s^2+\omega^2}, \
\mathcal{L}{\cos \omega t} &= \frac{s}{s^2+\omega^2}, & \mathcal{L}{\delta(t)} &= 1.
\end{align*}
-
Partial Fraction Expansion: For inverse Laplace, factor denominator and expand into simpler terms.
4.2 Application to Circuits
-
s-Domain Transformation:
-
Resistor: \(Z_R = R\).
-
Inductor: \(Z_L = sL\), with initial current \(i(0^{-})\) represented by a voltage source \(-L i(0^{-})\) in series (from \(V_L(s) = sL I(s) - L i(0^{-})\)).
-
Capacitor: \(Z_C = 1/(sC)\), with initial voltage \(v(0^{-})\) represented by a voltage source \(v(0^{-})/s\) in series (from \(V_C(s) = \frac{1}{sC} I(s) + \frac{v(0^{-})}{s}\)).
-
-
Solving Circuits:
Example 1: RC series with step voltage \(V\) at \(t=0\), capacitor initially uncharged.
s-domain: \(\frac{V}{s} = I(s) \left(R + \frac{1}{sC}\right)\) → \(I(s) = \frac{V}{R} \cdot \frac{1}{s + 1/(RC)}\) → \(i(t) = \frac{V}{R} e^{-t/(RC)}\).
Example 2: RLC series with capacitor initially charged to \(V_0\). Include initial capacitor voltage source \(V_0/s\) in series with \(1/(sC)\). Solve using KVL.
4.3 Network Functions & Theorems
-
Driving Point Impedance: \(Z(s) = \frac{V(s)}{I(s)}\) (zero initial conditions).
-
Transfer Function: \(H(s) = \frac{\text{output}(s)}{\text{input}(s)}\) (e.g., voltage gain).
-
Pole-Zero Plot: Poles (denominator roots) → \(H(s) \to \infty\); zeros (numerator roots) → \(H(s)=0\). Stability requires all poles in left half-plane (LHP).
-
Initial Value Theorem (IVT):
\boxed{f(0^{+}) = \lim_{s \to \infty} sF(s)}
Conditions: \(f(t)\) and \(f'(t)\) Laplace transformable; \(sF(s)\) finite as \(s \to \infty\); no impulse at \(t=0\).
-
Final Value Theorem (FVT):
\boxed{f(\infty) = \lim_{s \to 0} sF(s)}
Conditions: All poles of \(sF(s)\) in LHP except possibly at \(s=0\); system stable.
[!TIP] Always verify conditions before applying IVT/FVT. For FVT, ensure no poles in right half-plane.
4.4 Waveform Analysis using Laplace
-
Express periodic waveforms piecewise over one period \(T\).
-
Laplace of periodic function:
\boxed{F(s) = \frac{1}{1 - e^{-sT}} \int_{0}^{T} f(t) e^{-st} dt}
-
Aperiodic: Use unit step functions to write piecewise, then transform.
-
Example: Square wave of amplitude \(A\), period \(T\):
\(f(t) = A\) for \(0 < t < T/2\), \(0\) for \(T/2 < t < T\), periodic.
\(F(s) = \frac{A}{s} \cdot \frac{1 - e^{-sT/2}}{1 - e^{-sT}}\).
5.0 FOURIER SERIES & ANALYSIS
5.1 Fourier Series Expansion
-
Trigonometric Form:
\boxed{f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos n\omega_0 t + b_n \sin n\omega_0 t \right)}
where \(\omega_0 = 2\pi/T\), and
\begin{align*}
a_0 &= \frac{2}{T} \int_{0}^{T} f(t) dt, \
a_n &= \frac{2}{T} \int_{0}^{T} f(t) \cos n\omega_0 t , dt, \
b_n &= \frac{2}{T} \int_{0}^{T} f(t) \sin n\omega_0 t , dt.
\end{align*}
-
Exponential (Complex) Form:
\boxed{f(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t}}
with
\boxed{C_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-jn\omega_0 t} dt}
Relationship: \(C_n = \frac{a_n - j b_n}{2}\) for \(n>0\), \(C_0 = a_0/2\), \(C_{-n} = \frac{a_n + j b_n}{2}\).
-
Dirichlet Conditions: For existence, \(f(t)\) must:
-
Be single-valued.
-
Have finite maxima/minima in one period.
-
Have finite discontinuities in one period.
-
Be absolutely integrable: \(\int_{0}^{T} |f(t)| dt < \infty\).
-
5.2 Application to Waveforms
-
Standard Waveforms (period \(T\), amplitude \(A\)):
-
Square wave (odd symmetry): only sine terms, \(b_n = \frac{4A}{n\pi}\) for \(n\) odd.
-
Triangular wave (odd symmetry): only sine terms, \(b_n \propto 1/n^2\).
-
Sawtooth wave (odd symmetry): only sine terms, \(b_n \propto 1/n\).
-
Half-wave rectified sine: both cosine and sine.
-
Full-wave rectified sine: even symmetry, only cosine.
-
-
Power in Periodic Waveforms:
\boxed{P = \frac{1}{T} \int_{0}^{T} [f(t)]^2 dt = \frac{a_0^2}{4} + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2) = \sum_{n=-\infty}^{\infty} |C_n|^2}
RMS value: \(\sqrt{P}\).
[!TIP] Use symmetry to simplify: even function → only cosine (\(a_n\)), odd function → only sine (\(b_n\)), half-wave symmetry → only odd harmonics.
6.0 TWO-PORT NETWORKS & PARAMETERS
6.1 Parameter Definitions & Representations
-
Z-Parameters (Impedance):
\boxed{
\begin{aligned}
V_1 &= Z_{11} I_1 + Z_{12} I_2 \
V_2 &= Z_{21} I_1 + Z_{22} I_2
\end{aligned}
}
Conditions: Open-circuit ports (\(I_1=0\) for \(Z_{22}\), \(I_2=0\) for \(Z_{11}\), etc.). Units: \(\Omega\).
-
Y-Parameters (Admittance):
\boxed{
\begin{aligned}
I_1 &= Y_{11} V_1 + Y_{12} V_2 \
I_2 &= Y_{21} V_1 + Y_{22} V_2
\end{aligned}
}
Conditions: Short-circuit ports (\(V_1=0\) for \(Y_{22}\), \(V_2=0\) for \(Y_{11}\), etc.). Units: S.
-
h-Parameters (Hybrid):
\boxed{
\begin{aligned}
V_1 &= h_{11} I_1 + h_{12} V_2 \
I_2 &= h_{21} I_1 + h_{22} V_2
\end{aligned}
}
Conditions:
\(h_{11}\): input impedance with output shorted (\(V_2=0\)).
\(h_{12}\): reverse voltage gain with input open (\(I_1=0\)).
\(h_{21}\): forward current gain with output shorted (\(V_2=0\)).
\(h_{22}\): output admittance with input open (\(I_1=0\)).
Units: \(h_{11}\) in \(\Omega\), \(h_{12}\) dimensionless, \(h_{21}\) dimensionless, \(h_{22}\) in S.
-
ABCD-Parameters (Transmission):
\boxed{
\begin{aligned}
V_1 &= A V_2 + B I_2 \
I_1 &= C V_2 + D I_2
\end{aligned}
}
Convention: \(I_2\) is current leaving output port. Cascade property: overall ABCD = product of individual ABCD matrices.
Units: \(A, D\) dimensionless; \(B\) in \(\Omega\); \(C\) in S.
6.2 Inter-relationships
-
Z to Y: \(Y = Z^{-1}\) → \(Y_{ij} = \frac{\text{cofactor of } Z_{ji}}{|Z|}\).
-
Y to Z: \(Z = Y^{-1}\).
-
Z to ABCD (assuming \(Z_{21} \neq 0\)):
\boxed{
A = \frac{Z_{11}}{Z_{21}}, ;
B = \frac{|Z|}{Z_{21}}, ;
C = \frac{1}{Z_{21}}, ;
D = \frac{Z_{22}}{Z_{21}}
}
where \(|Z| = Z_{11}Z_{22} - Z_{12}Z_{21}\).
-
ABCD to Z:
\boxed{
Z_{11} = \frac{A}{C}, ;
Z_{12} = \frac{AD-BC}{C}, ;
Z_{21} = \frac{1}{C}, ;
Z_{22} = \frac{D}{C}
}
-
Y to ABCD:
\boxed{
A = -\frac{Y_{22}}{Y_{21}}, ;
B = -\frac{1}{Y_{21}}, ;
C = -\frac{|Y|}{Y_{21}}, ;
D = -\frac{Y_{11}}{Y_{21}}
}
where \(|Y| = Y_{11}Y_{22} - Y_{12}Y_{21}\).
-
ABCD to Y:
\boxed{
Y_{11} = \frac{D}{B}, ;
Y_{12} = \frac{AD-BC}{B}, ;
Y_{21} = -\frac{1}{B}, ;
Y_{22} = \frac{A}{B}
}
-
h to ABCD:
\boxed{
A = h_{12} - \frac{h_{11}h_{22}}{h_{21}}, ;
B = \frac{h_{11}}{h_{21}}, ;
C = -\frac{h_{22}}{h_{21}}, ;
D = \frac{1}{h_{21}}
}
-
ABCD to h:
\boxed{
h_{11} = \frac{B}{D}, ;
h_{12} = A - \frac{BC}{D}, ;
h_{21} = \frac{1}{D}, ;
h_{22} = -\frac{C}{D}
}
Example: Given equations \(2V_1 + 4I_2 = I_1\) and \(V_2 + 6V_1 = 8I_2\), find parameters.
- Z-parameters: From \(I_1 = 2V_1 + 4I_2\) → \(V_1 = \frac{1}{2}I_1 - 2I_2\) so \(Z_{11}=0.5\), \(Z_{12}=-2\). From \(V_2 = 8I_2 - 6V_1\) substitute \(V_1\) → \(V_2 = -3I_1 + 20I_2\) so \(Z_{21}=-3\), \(Z_{22}=20\).
6.3 Special Cases & Applications
-
Terminated Two-Port Network:
With source impedance \(Z_s\) and load \(Z_L\):
Input impedance:
\boxed{Z_{in} = \frac{A Z_L + B}{C Z_L + D}} \quad (\text{using } I_2 \text{ leaving convention})
Voltage gain:
\boxed{A_v = \frac{V_2}{V_s} = \frac{Z_L}{A Z_L + B} \cdot \frac{Z_{in}}{Z_s + Z_{in}}}
-
Cascading: For two two-ports in cascade, overall ABCD matrix is the product of individual ABCD matrices.
-
Symmetrical Network: \(Z_{11}=Z_{22}\), \(Y_{11}=Y_{22}\), \(A=D\).
-
Reciprocal Network: \(Z_{12}=Z_{21}\), \(Y_{12}=Y_{21}\), \(AD-BC=1\). For h-parameters: \(h_{12} = -h_{21}\) (if \(I_2\) entering) or \(h_{12} = h_{21}\) (if \(I_2\) leaving)? With \(I_2\) leaving convention, reciprocity implies \(h_{12} = h_{21}\)? Actually, from conversions, if \(Z_{12}=Z_{21}\), then \(h_{12} = Z_{12}/Z_{22}\), \(h_{21} = -Z_{21}/Z_{22}\), so \(h_{12} = -h_{21}\). But with \(I_2\) leaving, the sign may change. For safety, state: Reciprocal if \(Z_{12}=Z_{21}\) or \(Y_{12}=Y_{21}\) or \(AD-BC=1\).
[!TIP] When converting parameters, ensure consistent current direction conventions (entering vs. leaving).
7.0 ADVANCED TOPICS & SHORT NOTE SYLLABUS
7.1 Controlled Sources
-
VCVS: Voltage-controlled voltage source (gain dimensionless).
-
VCCS: Voltage-controlled current source (transconductance, S).
-
CCVS: Current-controlled voltage source (transresistance, \(\Omega\)).
-
CCCS: Current-controlled current source (current gain, dimensionless).
Symbols: diamond for VCVS, circle for VCCS, etc.
7.2 Specific Theorems (revisit)
-
Milliman's Theorem: See 1.2.
-
Compensation Theorem: See 1.2.
-
Tellegen's Theorem: See 1.2.
7.3 Parameter Deep Dives
-
Open Circuit Impedance Parameters (Z): See 6.1.
-
Short Circuit Admittance Parameters (Y): See 6.1.
7.4 Special Network Concepts
-
Dual Networks: See 2.3.
-
Terminated Two-Port Network: See 6.3.
-
Network Topology: See 2.0.
-
s-Theorem (Source Transformation): A voltage source in series with resistance \(R\) is equivalent to a current source \(I = V/R\) in parallel with \(R\), and vice versa. The external V-I characteristics remain unchanged.
Application: Simplifying circuits by converting sources to facilitate series/parallel combinations.
DiagramCANVAS: Show voltage source V with series R, and equivalent current source I=V/R with parallel R
7.5 Other Recurring Topics
-
Mutual Inductance & Coefficient of Coupling:
For two coils with self-inductances \(L_1, L_2\) and mutual inductance \(M\):
Series aiding: \(L_{eq} = L_1 + L_2 + 2M\)
Series opposing: \(L_{eq} = L_1 + L_2 - 2M\)
Then \(M = \frac{L_{\text{aiding}} - L_{\text{opposing}}}{4}\).
Coefficient of coupling:
\boxed{k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \le k \le 1}
-
Initial & Final Value Theorems: See 4.3.
8.0 PROBLEM-SOLVING STRATEGIES & INTEGRATION
-
Choosing the Right Method:
-
Nodal: many voltage sources/parallel elements.
-
Mesh: many current sources/series elements.
-
Thevenin/Norton: finding specific branch voltage/current, especially with variable load.
-
Superposition: multiple independent sources (not for power).
-
Dependent sources: use nodal/mesh with controlling variables as additional unknowns.
-
-
Mixed Domain Analysis: Combine Laplace (transients) with phasors (steady-state AC). Laplace solution inherently includes steady-state.
-
Two-Port Problem-Solving Flow:
-
Identify connection: cascade → use ABCD multiplication; series → use Z; parallel → use Y.
-
For arbitrary termination, use terminated two-port formulas.
-
Given network equations, compare to standard forms to extract parameters.
-
For parameter conversions, use formulas from 6.2.
-
Check reciprocity/symmetry conditions.
-
[!TIP] Always verify units and sign conventions, especially for current directions in two-port parameters.