Network Analysis (EX-304) - Unit 4 Short Notes

UNIT 4: Network Analysis

I. Fundamental Circuit Laws

Kirchhoff's Current Law (KCL)

Definition: The algebraic sum of currents entering any node (or closed boundary) is zero.

$$\sum_{k=1}^{n} i_k = 0$$

• Principle: Based on conservation of charge. Currents entering are positive, leaving are negative (or vice-versa).

• Example: For a node with three branches: $$\displaystyle i_1 $$ entering, $$\displaystyle i_2 $$ and $$\displaystyle i_3 $$ leaving → $$\displaystyle i_1 - i_2 - i_3 = 0 $$ or $$\displaystyle i_1 = i_2 + i_3 $$.

Kirchhoff's Voltage Law (KVL)

Definition: The algebraic sum of voltages around any closed loop (or mesh) is zero.

$$\sum_{k=1}^{n} v_k = 0$$

• Principle: Based on conservation of energy. Voltage rises are positive, drops are negative (or follow loop direction).

• Example: For a loop with a voltage source $$\displaystyle V_s $$ and two resistors $$\displaystyle V_1 $$, $$\displaystyle V_2 $$ → $$\displaystyle V_s - V_1 - V_2 = 0 $$.

[!TIP] Common Pitfall: KCL applies to instantaneous currents at a node. KVL applies to instantaneous voltages in a loop. Both are valid for AC/DC, transient, and steady-state.

II. Circuit Analysis Methods

A. Mesh Analysis

• Concept: Apply KVL to independent meshes (loops with no other loops inside). Unknowns are mesh currents.

• Steps:

  1. Identify all meshes. Assign mesh currents ($$\displaystyle I_1, I_2, ... $$) in a consistent direction (usually clockwise).

  2. Write KVL equation for each mesh. Express branch voltages in terms of mesh currents.

  3. Solve the simultaneous equations.

• Special Case: Supermesh – Used when a current source lies on the perimeter of two meshes. The supermesh excludes the current source and its branch. The current source equation provides an additional constraint.

• Key Formula (for $n$ meshes):

$$[R][I] = [V]$$

Where $[R]$ is the mesh resistance matrix (diagonal: self-resistance, off-diagonal: negative of shared resistance).

B. Nodal Analysis

• Concept: Apply KCL at essential nodes (excluding reference). Unknowns are node voltages ($$\displaystyle V_1, V_2, ... $$) w.r.t. reference.

• Steps:

  1. Select a reference node (usually ground). Assign voltages $$\displaystyle V_1, V_2, ... $$ to other nodes.

  2. Write KCL at each non-reference node. Express branch currents in terms of node voltages using Ohm's law.

  3. Solve the simultaneous equations.

• Special Case: Supernode – Used when a voltage source is connected between two non-reference nodes. The supernode encloses the voltage source. KCL is written for the supernode boundary. The voltage source equation provides an additional constraint.

• Key Formula (for $n$ nodes):

$$[G][V] = [I]$$

Where $[G]$ is the nodal conductance matrix (diagonal: sum of conductances connected to node, off-diagonal: negative of conductance between nodes).

[!TIP] Exam Focus: Nodal analysis is often more efficient for circuits with fewer nodes than meshes. Always check for supernodes/supermeshes when voltage/current sources are present.

III. Network Theorems

A. Superposition Theorem

Statement: In a linear, bilateral network with multiple sources, the response (voltage/current) in any element is the algebraic sum of the responses caused by each source acting alone, with all other independent sources deactivated (replaced by their internal resistances: voltage sources → short, current sources → open).

• Applicability: Only for linear circuits. Power cannot be found by superposition (since $$\displaystyle P \propto V^2 $$ or $$\displaystyle I^2 $$).

• Procedure: Turn off all but one source, find the contribution, repeat for all sources, sum contributions.

B. Thevenin's Theorem

Statement: Any linear, bilateral network can be replaced by an equivalent circuit consisting of a voltage source $$\displaystyle V_{Th} $$ in series with a resistance $$\displaystyle R_{Th} $$ as seen from the load terminals.

• $$\displaystyle V_{Th} $$: Open-circuit voltage across the load terminals.

• $$\displaystyle R_{Th} $$: Equivalent resistance seen from load terminals with all independent sources deactivated. (For circuits with dependent sources, apply a test source).

• Equivalent Circuit:

$$V_{load} = V_{Th} \frac{R_L}{R_{Th} + R_L}$$

\boxed{V_{Th} = \text{Open-circuit voltage across AB}}

C. Norton's Theorem

Statement: Any linear, bilateral network can be replaced by an equivalent circuit consisting of a current source $$\displaystyle I_N $$ in parallel with a resistance $$\displaystyle R_N $$ as seen from the load terminals.

• $$\displaystyle I_N $$: Short-circuit current through the load terminals.

• $$\displaystyle R_N $$: Same as $$\displaystyle R_{Th} $$ (Thevenin resistance). $$\displaystyle R_N = R_{Th} $$.

• Equivalent Circuit:

$$I_{load} = I_N \frac{R_N}{R_N + R_L}$$

\boxed{I_N = \text{Short-circuit current across AB}}

For AC Networks: Replace resistances with impedances $$\displaystyle Z_{Th} $$ and $$\displaystyle Z_N $$. $$\displaystyle V_{Th} $$, $$\displaystyle I_N $$ become phasors.

D. Maximum Power Transfer Theorem

For DC Networks:

Maximum power is delivered to the load $$\displaystyle R_L $$ when $$\displaystyle R_L = R_{Th} $$ (load resistance equals Thevenin resistance).

• Condition: $$\displaystyle R_L = R_{Th} $$ (for purely resistive load).

• Maximum Power:

$$P_{max} = \frac{V_{Th}^2}{4 R_{Th}}$$

• Efficiency: $$\displaystyle \eta = \frac{P_{load}}{P_{source}} = \frac{P_{max}}{P_{max} + P_{Th}} = \frac{1}{2} = 50\% $$ when $$\displaystyle R_L = R_{Th} $$.

\boxed{\eta = 50\% \text{ at max power}}

For AC Networks: Maximum power transfer occurs when load impedance $$\displaystyle Z_L $$ is the complex conjugate of Thevenin impedance $$\displaystyle Z_{Th} $$.

$$Z_L = Z_{Th}^* \quad \text{or} \quad R_L = R_{Th} \ \text{and} \ X_L = -X_{Th}$$

E. Millman's Theorem

Statement: For a circuit with several parallel branches between two nodes, the voltage between the nodes can be found directly.

\boxed{V = \frac{\sum_{k=1}^{n} \frac{V_k}{R_k}}{\sum_{k=1}^{n} \frac{1}{R_k}}}

Where $$\displaystyle V_k $$ is the voltage source in branch $k$ (with polarity consistent with $V$), and $$\displaystyle R_k $$ is the resistance in that branch. If a branch has only a current source, treat $$\displaystyle V_k=0 $$ and $$\displaystyle R_k = \infty $$ (or just add the current source to the numerator's sum of conductances).

F. Tellegen's Theorem

Statement: For any two networks (not necessarily the same) that have the same topology (graph) and satisfy KCL and KVL, the sum of the products of branch voltages and currents (from each network) is zero.

$$\sum_{b=1}^{b} v_b i_b = 0$$

Where $$\displaystyle v_b $$ and $$\displaystyle i_b $$ are voltage and current of branch $b$ in the two networks respectively (following a consistent orientation).

• Interpretation: It's a statement of power balance in a network. Often used to verify solutions or derive other theorems.

G. Compensation Theorem

Statement: If the impedance of a branch in a network is changed from $Z$ to $Z + \Delta Z$, the resulting changes in all branch currents and voltages are the same as those produced by injecting a compensating voltage source $$\displaystyle V_c = -I \Delta Z $$ in series with the modified branch, where $I$ is the original current through that branch.

• Use: Simplifies analysis of networks with small changes in impedance.

H. Substitution Theorem

Statement: If the voltage across a branch and the current through it are known (i.e., the branch is "told" what to do), then the branch can be replaced by any combination of elements that maintains the same voltage and current, without affecting the rest of the network.

• Use: Useful for simplifying parts of a circuit during analysis.

IV. Network Topology

A. Graph Theory

• Graph: A representation of a network consisting of nodes (vertices) and branches (edges) connecting them, where all elements are replaced by their admittance/impedance-free lines. A graph has no isolated nodes.

• Oriented Graph: A graph with directions (arrows) assigned to all branches. Used for matrix methods.

B. Trees and Co-Trees

• Tree: A connected subgraph of the original graph that contains all nodes and no loops. It has $(n-1)$ branches for $n$ nodes.

• Co-Tree: The set of branches not in the tree. These branches are called links or chords. Number of links = $b - (n-1)$, where $b$ = total branches.

• Twigs: The branches of the tree.

• Links: The branches of the co-tree.

C. Fundamental Circuits and Cuts

• Tie Set (Fundamental Loop): Adding one link to a tree forms a single closed loop called a fundamental tie set. Each link corresponds to one fundamental tie set. Number of tie sets = number of links.

• Basic Tie Set Matrix ($B$): A matrix ($b \times b$) where rows correspond to tie sets and columns to branches. Entry $$\displaystyle b_{ij} $$:

  • $+1$ if branch $j$ is in tie set $i$ and its orientation agrees with tie set loop.

  • $-1$ if branch $j$ is in tie set $i$ and its orientation opposes.

  • $0$ if branch $j$ is not in tie set $i$.

  • Property: $[B]$ is of rank $(b-n+1)$.

• Cut Set: A minimal set of branches whose removal disconnects the graph into exactly two parts. A cut set must contain at least one twig.

• Fundamental Cut Set: Formed by selecting one twig and all links that connect the two separated parts when that twig is removed. Each twig corresponds to one fundamental cut set. Number of fundamental cut sets = number of twigs = $(n-1)$.

• Cut Set Matrix ($Q$): Similar to $[B]$, but rows are fundamental cut sets. Property: $[Q]$ is of rank $(n-1)$.

D. Incidence Matrix

• Complete Incidence Matrix ($A$): A matrix ($n \times b$) describing the graph's topology. Rows = nodes, columns = branches. Entry $$\displaystyle a_{ij} $$:

  • $+1$ if branch $j$ is incident to node $i$ and leaves node $i$.

  • $-1$ if branch $j$ is incident to node $i$ and enters node $i$.

  • $0$ if branch $j$ is not incident to node $i$.

  • Property: Any row is linearly dependent on others. Rank = $(n-1)$. Sum of any column = 0 (KCL).

• Reduced Incidence Matrix ($$\displaystyle A_r $$): Formed by deleting the row corresponding to the reference node. It is of rank $(n-1)$ and non-singular.

[!TIP] Key Relations:

$$[A_r]^T [B] = 0 \quad \text{and} \quad [Q][A_r] = 0$$

These are fundamental in network topology.

V. Transient Analysis Using Laplace Transform

A. Laplace Transform: Definition & Properties

Definition: $$\displaystyle F(s) = \mathcal{L}\{f(t)\} = \int_{0^-}^{\infty} f(t) e^{-st} dt $$, where $$\displaystyle s = \sigma + j\omega $$.

• Key Properties:

  • Linearity: $$\displaystyle \mathcal{L}\{a f_1 + b f_2\} = a F_1(s) + b F_2(s) $$

  • Differentiation: $$\displaystyle \mathcal{L}\{f'(t)\} = sF(s) - f(0^-) $$

  • Integration: $$\displaystyle \mathcal{L}\{\int_0^t f(\tau) d\tau\} = \frac{F(s)}{s} $$

  • Time-shift: $$\displaystyle \mathcal{L}\{f(t-a)u(t-a)\} = e^{-as} F(s) $$

B. Laplace Transform of Standard Waveforms

• Step: $$\displaystyle u(t) \leftrightarrow \frac{1}{s} $$

• Impulse: $$\displaystyle \delta(t) \leftrightarrow 1 $$

• Ramp: $$\displaystyle t \cdot u(t) \leftrightarrow \frac{1}{s^2} $$

• Exponential: $$\displaystyle e^{-at} u(t) \leftrightarrow \frac{1}{s+a} $$

• Sinusoid: $$\displaystyle \sin \omega t \leftrightarrow \frac{\omega}{s^2 + \omega^2} $$, $$\displaystyle \cos \omega t \leftrightarrow \frac{s}{s^2 + \omega^2} $$

C. Initial Value Theorem

If $sF(s)$ has no poles in the right-half plane (RHP) and possibly a simple pole at $$\displaystyle s=0 $$, then:

$$\lim_{t \to 0^+} f(t) = \lim_{s \to \infty} sF(s)$$

\boxed{f(0^+) = \lim_{s \to \infty} sF(s)}

Use: Finds initial value directly from $F(s)$.

D. Final Value Theorem

If $sF(s)$ has no poles in the RHP and only simple poles on the imaginary axis (including $$\displaystyle s=0 $$), then:

$$\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s)$$

\boxed{f(\infty) = \lim_{s \to 0} sF(s)}

Use: Finds steady-state (final) value. Condition check is mandatory.

E. Analysis of Switching Circuits

• General Steps:

  1. Draw the circuit for $$\displaystyle t < 0 $$. Find initial conditions ($$\displaystyle i_C(0^-) $$, $$\displaystyle v_C(0^-) $$, $$\displaystyle i_L(0^-) $$, $$\displaystyle v_L(0^-) $$). Capacitor acts as open circuit ($$\displaystyle i_C $$ finite $\Rightarrow$ $$\displaystyle v_C $$ constant). Inductor acts as short circuit ($$\displaystyle v_L $$ finite $\Rightarrow$ $$\displaystyle i_L $$ constant).

  2. Draw the circuit for $$\displaystyle t > 0 $$. Replace sources with their Laplace impedances. Replace:

     • Capacitor: $$\displaystyle Z_C = \frac{1}{sC} $$ with initial voltage $$\displaystyle v_C(0^-) $$ represented by a voltage source $$\displaystyle \frac{v_C(0^-)}{s} $$ in series.

     • Inductor: $$\displaystyle Z_L = sL $$ with initial current $$\displaystyle i_L(0^-) $$ represented by a current source $$\displaystyle \frac{i_L(0^-)}{s} $$ in parallel.

  3. Apply KVL/KCL or use nodal/mesh analysis in s-domain to find the desired response $F(s)$.

  4. Perform inverse Laplace transform (using partial fractions) to get $f(t)$ for $$\displaystyle t > 0 $$.

• RC Circuit (Step Response): $$\displaystyle i(t) = \frac{V}{R} e^{-t/RC} $$ for charging from zero initial voltage.

• RL Circuit (Step Response): $$\displaystyle i(t) = \frac{V}{R} (1 - e^{-tL/R}) $$.

• RLC Circuit: Response depends on damping ($$\displaystyle \alpha = \frac{R}{2L} $$, $$\displaystyle \omega_0 = \frac{1}{\sqrt{LC}} $$). Can be overdamped, critically damped, or underdamped.

F. Pole-Zero Plot and Time Domain Response

• Poles: Roots of denominator of $F(s)$. Determine natural response (stability, oscillation).

• Zeros: Roots of numerator of $F(s)$. Affect the amplitude and shape of response.

• Interpretation:

  • Real negative pole $$\displaystyle \rightarrow $$ exponential decay.

  • Complex conjugate pole with $$\displaystyle \sigma < 0 $$ $$\displaystyle \rightarrow $$ damped sinusoid.

  • Pole on imaginary axis ($$\displaystyle \sigma=0 $$) $$\displaystyle \rightarrow $$ sustained oscillation (undamped).

  • Pole in RHP ($$\displaystyle \sigma>0 $$) $$\displaystyle \rightarrow $$ unstable (growing oscillation).

VI. Frequency Domain Analysis

A. Fourier Series

• Purpose: Represent a periodic waveform $f(t)$ with period $T$ as a sum of sinusoids.

• Trigonometric Form:

$$f(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos n\omega_0 t + b_n \sin n\omega_0 t)$$

Where $$\displaystyle \omega_0 = \frac{2\pi}{T} $$.

$$a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt$$

$$a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos n\omega_0 t \ dt$$

$$b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin n\omega_0 t \ dt$$

• Exponential (Complex) Form:

$$f(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t}$$

$$C_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-jn\omega_0 t} dt = \frac{1}{2}(a_n - jb_n)$$

*   $$\displaystyle C_0 = a_0 $$ (DC component).

*   $$\displaystyle C_n = C_{-n}^* $$ for real $f(t)$.

• Analysis of Periodic Waveforms: Use symmetry to simplify:

  • Even function ($$\displaystyle f(t)=f(-t) $$): $$\displaystyle b_n = 0 $$, only cosine terms.

  • Odd function ($$\displaystyle f(t)=-f(-t) $$): $$\displaystyle a_0 = a_n = 0 $$, only sine terms.

  • Half-wave symmetry ($$\displaystyle f(t) = -f(t \pm T/2) $$): Only odd harmonics ($$\displaystyle n=1,3,5... $$) exist.

B. Response to Sinusoidal Excitation

• For a linear circuit with sinusoidal input $$\displaystyle v_s(t) = V_m \sin(\omega t + \phi) $$, the steady-state response is also sinusoidal with same frequency $\omega$ but different amplitude and phase.

• Phasor Analysis: Replace $$\displaystyle v(t) \rightarrow \tilde{V} = V_m \angle \phi $$, $$\displaystyle i(t) \rightarrow \tilde{I} = I_m \angle \theta $$. Use complex impedance $$\displaystyle Z = R + jX $$ and Ohm's law $$\displaystyle \tilde{V} = \tilde{I} Z $$.

• Power: $$\displaystyle P = V_{rms} I_{rms} \cos \phi $$, $$\displaystyle Q = V_{rms} I_{rms} \sin \phi $$, $$\displaystyle S = V_{rms} I_{rms} $$.

VII. Resonance

A. Series Resonant Circuit

• Circuit: $R$, $L$, $C$ in series with voltage source $V$.

• Resonant Frequency ($$\displaystyle \omega_0 $$): Condition where imaginary part of impedance is zero ($$\displaystyle X_L = X_C $$).

$$\omega_0 L = \frac{1}{\omega_0 C} \quad \Rightarrow \quad \boxed{\omega_0 = \frac{1}{\sqrt{LC}}}$$

$$\displaystyle f_0 = \frac{1}{2\pi\sqrt{LC}} $$.

• Quality Factor ($Q$): Measure of sharpness of resonance.

$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R} \sqrt{\frac{L}{C}}$$

• Bandwidth (BW): $$\displaystyle \text{BW} = \frac{\omega_0}{Q} $$ (in rad/s) or $$\displaystyle \frac{f_0}{Q} $$ (in Hz).

• Voltage Magnification: At resonance, voltages across $L$ and $C$ are $Q$ times the input voltage (and equal in magnitude but opposite in phase).

$$|V_L| = |V_C| = Q \cdot V$$

B. Parallel Resonant Circuit

• Circuit: $R$, $L$, $C$ in parallel (often $R$ represents inductor's series resistance).

• Resonant Frequency ($$\displaystyle \omega_0 $$): Condition where imaginary part of admittance is zero ($$\displaystyle B_L + B_C = 0 $$).

  For practical parallel RLC (with $R$ in series with $L$), exact resonance frequency is:

$$\omega_0 = \sqrt{\frac{1}{LC} - \frac{R^2}{L^2}} \approx \frac{1}{\sqrt{LC}} \quad \text{if } R \text{ is small}.$$

\boxed{\omega_0 \approx \frac{1}{\sqrt{LC}} \ \text{(for high Q)}}.

• Quality Factor ($Q$): For parallel circuit with equivalent parallel resistance $$\displaystyle R_p $$:

$$Q = R_p \sqrt{\frac{C}{L}} = \frac{R_p}{\omega_0 L} = \omega_0 C R_p$$

• Characteristic: At resonance, impedance is maximum (purely resistive), current from source is minimum.

VIII. Two-Port Networks

A. Parameter Definitions

A two-port network has input port (1-1') and output port (2-2').

1. Impedance Parameters (Z-parameters):

$$V_1 = Z_{11} I_1 + Z_{12} I_2$$

$$V_2 = Z_{21} I_1 + Z_{22} I_2$$

*   $$\displaystyle Z_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} $$ (Input impedance with output open).

*   $$\displaystyle Z_{12} = \left. \frac{V_1}{I_2} \right|_{I_1=0} $$ (Reverse transfer impedance).

*   $$\displaystyle Z_{21} = \left. \frac{V_2}{I_1} \right|_{I_2=0} $$ (Forward transfer impedance).

*   $$\displaystyle Z_{22} = \left. \frac{V_2}{I_2} \right|_{I_1=0} $$ (Output impedance with input open).

\boxed{[Z] = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix}}

2. Admittance Parameters (Y-parameters):

$$I_1 = Y_{11} V_1 + Y_{12} V_2$$

$$I_2 = Y_{21} V_1 + Y_{22} V_2$$

*   $$\displaystyle Y_{ij} = \left. \frac{I_i}{V_j} \right|_{\text{other port voltage}=0} $$.

*   $$\displaystyle [Y] = [Z]^{-1} $$.

3. Hybrid Parameters (h-parameters):

$$V_1 = h_{11} I_1 + h_{12} V_2$$

$$I_2 = h_{21} I_1 + h_{22} V_2$$

*   $$\displaystyle h_{11} = \left. \frac{V_1}{I_1} \right|_{V_2=0} $$ (Input impedance with output shorted).

*   $$\displaystyle h_{12} = \left. \frac{V_1}{V_2} \right|_{I_1=0} $$ (Reverse voltage gain).

*   $$\displaystyle h_{21} = \left. \frac{I_2}{I_1} \right|_{V_2=0} $$ (Forward current gain).

*   $$\displaystyle h_{22} = \left. \frac{I_2}{V_2} \right|_{I_1=0} $$ (Output admittance with input open).

\boxed{[h] = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix}}

4. Transmission Parameters (ABCD-parameters):

$$V_1 = A V_2 + B I_2$$

$$I_1 = C V_2 + D I_2$$

*   $$\displaystyle A = \left. \frac{V_1}{V_2} \right|_{I_2=0} $$ (Open-circuit voltage ratio).

*   $$\displaystyle B = \left. \frac{V_1}{I_2} \right|_{V_2=0} $$ (Short-circuit input impedance).

*   $$\displaystyle C = \left. \frac{I_1}{V_2} \right|_{I_2=0} $$ (Open-circuit input admittance).

*   $$\displaystyle D = \left. \frac{I_1}{I_2} \right|_{V_2=0} $$ (Short-circuit current ratio).

*   **Cascade Property:** For two cascaded networks, overall $$\displaystyle [ABCD] = [ABCD]_1 \cdot [ABCD]_2 $$.

\boxed{[ABCD] = \begin{bmatrix} A & B \\ C & D \end{bmatrix}, \ \text{det}=AD-BC=1 \ \text{(for reciprocal, passive networks)}}

B. Parameter Conversions

• Z to Y: $$\displaystyle [Y] = [Z]^{-1} $$.

• Z to h: $$\displaystyle h_{11} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22}} $$, $$\displaystyle h_{12} = \frac{Z_{12}}{Z_{22}} $$, $$\displaystyle h_{21} = -\frac{Z_{21}}{Z_{22}} $$, $$\displaystyle h_{22} = \frac{1}{Z_{22}} $$.

• ABCD to Y:

$$Y_{11} = \frac{D}{B}, \quad Y_{12} = \frac{AD-BC}{B}, \quad Y_{21} = -\frac{1}{B}, \quad Y_{22} = \frac{A}{B}$$

• ABCD to h:

$$h_{11} = \frac{A}{C}, \quad h_{12} = \frac{AD-BC}{C}, \quad h_{21} = \frac{1}{C}, \quad h_{22} = \frac{D}{C}$$

C. Cascade Connection

• When two-ports are connected in cascade (output of first to input of second), the overall transmission matrix is the product of individual transmission matrices.

$$[ABCD]_{total} = [ABCD]_1 \cdot [ABCD]_2$$

• This is the primary advantage of ABCD parameters.

D. Terminated Two-Port Networks

• Input Impedance ($$\displaystyle Z_{in} $$) with load $$\displaystyle Z_L $$:

  Using Z-params: $$\displaystyle Z_{in} = \frac{V_1}{I_1} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22} + Z_L} $$

  Using ABCD: $$\displaystyle Z_{in} = \frac{AV_2 + BI_2}{CV_2 + DI_2} = \frac{AI_2Z_L + BI_2}{CI_2Z_L + DI_2} = \frac{AZ_L + B}{CZ_L + D} $$

• Voltage Gain ($$\displaystyle G_v = V_2/V_1 $$):

  Using ABCD: $$\displaystyle G_v = \frac{V_2}{V_1} = \frac{1}{A + B/Z_L} = \frac{Z_L}{AZ_L + B} $$ (if $$\displaystyle I_2 $$ is output current).

• Current Gain ($$\displaystyle G_i = I_2/I_1 $$):

  Using ABCD: $$\displaystyle G_i = \frac{I_2}{I_1} = \frac{1}{CZ_L + D} $$

E. Interconnection of Two-Ports

• Series-Series Connection: Two-ports connected in series at both ports. Z-parameters are additive.

• Parallel-Parallel Connection: Two-ports connected in parallel at both ports. Y-parameters are additive.

• Series-Parallel (Cascade): As above, ABCD-parameters multiply.

IX. Additional Topics

A. Controlled Sources (Dependent Sources)

• Key: They are linear and bilateral if controlling and controlled variables are in same network. Their presence requires special care in finding $$\displaystyle R_{Th} $$ (use test source).

B. Dual Networks

• Definition: Two networks are dual if the same system of equations describes them when the following pairs are interchanged:

  • Voltage $$\displaystyle \leftrightarrow $$ Current

  • Resistance $$\displaystyle \leftrightarrow $$ Conductance ($$\displaystyle R \leftrightarrow G $$)

  • Series $$\displaystyle \leftrightarrow $$ Parallel

  • Mesh $$\displaystyle \leftrightarrow $$ Node

  • Loop $$\displaystyle \leftrightarrow $$ Node-pair

  • KVL $$\displaystyle \leftrightarrow $$ KCL

• Rule: To find dual of a circuit:

  1. Place a node inside every mesh of original.

  2. Place a mesh around every node of original.

  3. Connect dual elements between corresponding dual nodes/meshes.

• Property: Dual networks have the same determinant for their impedance/admittance matrices.

C. Mutual Inductance and Coupled Circuits

• Mutual Inductance ($M$): Flux in one coil due to current in another. $$\displaystyle v_1 = L_1 \frac{di_1}{dt} \pm M \frac{di_2}{dt} $$.

• Dot Convention: Determines polarity of induced voltage. Currents entering dotted terminals produce positive mutual voltage.

• Coefficient of Coupling ($k$): $$\displaystyle k = \frac{M}{\sqrt{L_1 L_2}} $$, $0 \le k \le 1$.

• Equivalent Circuit: Can be represented by reflected impedance:

  • $$\displaystyle Z_{eq} = j\omega M $$ (transformer action).

  • Reflected impedance to primary: $$\displaystyle Z' = \frac{(j\omega M)^2}{Z_2 + j\omega L_2} $$.

D. Network Topology Overview

• Core Idea: Study of networks based on their connectivity (graph) rather than element values.

• Key Quantities:

  • $b$ = number of branches.

  • $n$ = number of nodes.

  • $l$ = number of independent loops = $b - n + 1$.

  • $t$ = number of twigs in a tree = $n-1$.

  • Number of links = $$\displaystyle b - n + 1 = l $$.

• Matrices: Incidence Matrix ($A$), Tie Set Matrix ($B$), Cut Set Matrix ($Q$). They are interrelated: $$\displaystyle [B] = [A_r]^T $$, $$\displaystyle [Q][A_r] = 0 $$.

• Advantage: Allows formulation of fundamental circuit equations (KVL in tie sets: $$\displaystyle [B][V] = 0 $$) and fundamental cut-set equations (KCL in cuts: $$\displaystyle [Q][I] = 0 $$) without writing individual equations.

[!TIP] Exam Focus: Be prepared to draw a graph from a circuit, select a tree, write tie set and cut set matrices, and relate them to the incidence matrix. Always verify matrix properties (row/column sums).