UNIT 1: Network Analysis – Short Notes

1. Fundamental Laws and Circuit Elements

Kirchhoff’s Current Law (KCL)

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

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

Key Point: Based on conservation of charge. Currents entering are positive, leaving are negative (or vice-versa, but be consistent).

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 $$\displaystyle i_1 = i_2 + i_3 $$.

Application: Foundation of nodal analysis.

Kirchhoff’s Voltage Law (KVL)

Statement: The algebraic sum of voltages around any closed loop is zero.

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

Key Point: Based on conservation of energy. Traverse the loop in a consistent direction (clockwise/anticlockwise); voltage rises are positive, drops are negative (or follow the passive sign convention).

Example: Loop with sources and resistors: $$\displaystyle +V_s - V_{R1} - V_{R2} = 0 $$.

Application: Foundation of mesh analysis.

[!TIP] Common Pitfall: KCL applies to nodes (junctions of 2+ elements). KVL applies to closed loops. Do not apply KCL to a simple two-element connection.

Circuit Elements

Ideal vs. Practical Models:

• Ideal: Pure R, L, C as above.

• Practical: Include series/parallel parasitic elements.

  • Practical Inductor: $R$ in series with $L$ (winding resistance).

  • Practical Capacitor: $R$ in parallel with $C$ (leakage resistance).

Dependent (Controlled) Sources

Definition: Source value depends on another voltage or current elsewhere in the circuit.

Four Types:

1. VCVS (Voltage-Controlled Voltage Source): $$\displaystyle v_o = \mu v_i $$

2. VCCS (Voltage-Controlled Current Source): $$\displaystyle i_o = g v_i $$

3. CCVS (Current-Controlled Voltage Source): $$\displaystyle v_o = r i_i $$

4. CCCS (Current-Controlled Current Source): $$\displaystyle i_o = \alpha i_i $$

Symbol: Diamond shape with control variable indicated.

Example: Transistor small-signal models (e.g., $$\displaystyle i_c = \beta i_b $$ is a CCCS).

Mutual Inductance and Coupling Coefficient

• Mutual Inductance (M): Voltage induced in one coil due to current change in a coupled coil.

$$v_1 = L_1\frac{di_1}{dt} + M\frac{di_2}{dt}, \quad v_2 = M\frac{di_1}{dt} + L_2\frac{di_2}{dt}$$

**Dot Convention** determines polarity of induced voltage.

• Coefficient of Coupling (k): Measures how tightly coupled two inductors are.

$$k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \le k \le 1$$

$$\displaystyle k=1 $$ is perfect coupling (ideal transformer). $$\displaystyle M_{max} = \sqrt{L_1 L_2} $$.

2. Graph Theory and Network Topology

Graph and Oriented Graph

• Graph: A representation of a network consisting of nodes (vertices, where elements meet) and branches (edges, representing circuit elements). A graph has no isolated loops.

• Oriented Graph: A graph where each branch is assigned a direction (arrow). Essential for applying KCL/KVL via matrix methods.

• Key Terms: $b$ = number of branches, $n$ = number of nodes.

Tree and Co-Tree

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

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

• Twigs: The branches that form the tree.

[!TIP] Finding a Tree: Start at any node, add branches one by one without forming a loop until all nodes are included.

Incidence Matrix

• Complete Incidence Matrix ($A$): An $n \times b$ matrix describing graph topology.

  • Rows = Nodes.

  • Columns = Branches.

  • Entries: $$\displaystyle a_{ij} = +1 $$ if branch $j$ leaves node $i$, $-1$ if it enters, $0$ if not incident.

  • Property: Each column has exactly one $+1$ and one $-1$ (for a connected graph). Sum of all rows = zero row (dependent).

  • Reduced Incidence Matrix ($$\displaystyle A_r $$): Remove any one row (usually reference node) to get a full-rank $(n-1) \times b$ matrix.

Cut Set Matrix

• Cut Set: A minimal set of branches whose removal disconnects the graph into exactly two parts.

• Fundamental Cut Set: Associated with a tree twig. Removing that twig creates a cut set containing that twig and some links. There are $(n-1)$ fundamental cut sets.

• Cut Set Matrix ($Q$): Rows = fundamental cut sets, Columns = branches. $$\displaystyle q_{ij} = +1 $$ if branch $j$ is in cut set $i$ and oriented from the side containing the reference node, $-1$ if opposite, $0$ otherwise.

$$Q = A_r \cdot B^{-1}$$

where $B$ is the Tie Set Matrix.

Tie Set (Basic Tie Set) Matrix

• Tie Set (Loop): A minimal set of branches forming a closed loop.

• Fundamental Tie Set: Associated with a tree link. Adding that link to the tree creates exactly one loop (the fundamental tie set). There are $b-(n-1)$ fundamental tie sets.

• Tie Set Matrix ($B$): Rows = fundamental tie sets, Columns = branches. $$\displaystyle b_{ij} = +1 $$ if branch $j$ is in loop $i$ and oriented in the same direction as the loop, $-1$ if opposite, $0$ otherwise.

  Key Relation: $$\displaystyle A_r \cdot B^T = 0 $$ (Orthogonality).

Dual Networks

• Duality Principle: For any network theorem/property in a primal network, there exists a dual network where:

  • Mesh ↔ Node

  • Branch ↔ Branch

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

  • $$\displaystyle L \leftrightarrow C $$

  • Series ↔ Parallel

  • KVL ↔ KCL

• Dual Circuit: Obtained by drawing the dual graph (nodes in faces of primal, branches perpendicular). Element values are transformed accordingly.

• Example: Series RLC circuit ↔ Parallel RLC circuit.

3. Network Theorems

[!TIP] General Application Steps: 1. Identify linear, bilateral network. 2. Remove the load (for Thevenin/Norton). 3. Zero all independent sources (voltage → short, current → open) to find $$\displaystyle R_{th} $$. 4. Re-activate sources one at a time (Superposition). 5. Combine results algebraically.

Superposition Theorem

Statement: In a linear network with multiple independent sources, the response (voltage/current) in any branch is the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources zeroed (but dependent sources remain active).

Limitation: Only for linear circuits. Power calculation requires $$\displaystyle P_{total} \neq \sum P_{individual} $$ (use total $V$ or $I$).

Thevenin’s Theorem

Statement: Any linear two-terminal network can be replaced by an equivalent circuit consisting of a voltage source $$\displaystyle V_{th} $$ in series with an impedance $$\displaystyle Z_{th} $$.

• $$\displaystyle V_{th} $$ = Open-circuit voltage across terminals.

• $$\displaystyle Z_{th} $$ = Impedance seen across terminals with all independent sources zeroed. (Can also be $$\displaystyle V_{oc}/I_{sc} $$).

AC Version: $$\displaystyle Z_{th} $$ is complex (includes $j\omega L$, $1/j\omega C$).

Norton’s Theorem

Statement: Any linear two-terminal network can be replaced by an equivalent circuit consisting of a current source $$\displaystyle I_{N} $$ in parallel with an impedance $$\displaystyle Z_{N} $$.

• $$\displaystyle I_{N} $$ = Short-circuit current across terminals.

• $$\displaystyle Z_{N} = Z_{th} $$ (same as Thevenin).

Relation: $$\displaystyle V_{th} = I_N Z_{th} $$, $$\displaystyle I_N = V_{th}/Z_{th} $$.

Maximum Power Transfer Theorem

DC Condition: Maximum power is delivered to the load when $$\displaystyle R_L = R_{th} $$ (load resistance equals Thevenin resistance). For AC, $$\displaystyle Z_L = Z_{th}^* $$ (complex conjugate).

Efficiency at Max Power:

$$\eta = \frac{P_{max}}{P_{source}} = \frac{R_L}{R_L + R_{th}} = \frac{1}{2} = 50\% \quad \text{(when } R_L = R_{th}\text{)}$$

Note: 50% efficiency is often undesirable in power systems.

Millman’s Theorem

Statement: For a circuit with several parallel branches between two nodes, each containing a voltage source in series with an impedance (or a current source in parallel with an impedance), the voltage between the nodes is:

$$V = \frac{\sum_{k=1}^{n} \frac{V_k}{Z_k}}{\sum_{k=1}^{n} \frac{1}{Z_k}}$$

For pure resistive parallel voltage sources: $$\displaystyle V = \frac{\sum (V_k/R_k)}{\sum (1/R_k)} $$.

Use: Simplifies analysis of parallel source networks.

Tellegen’s Theorem

Statement: For any two networks (not necessarily the same) with the same topology (graph), if their branch voltages and currents satisfy KVL and KCL respectively, then:

$$\sum_{b=1}^{B} v_i \cdot i_j' = 0$$

where $$\displaystyle v_i $$ are voltages in Network 1, $$\displaystyle i_j' $$ are currents in Network 2.

Interpretation: A statement of power balance or orthogonality of the incidence matrix. Often verified by calculating $$\displaystyle \sum vk \cdot ik' $$ for both networks.

Use: Proving other theorems, network synthesis.

Compensation Theorem

Statement: If the impedance $Z$ of a branch in a network is changed by $\Delta Z$, the resulting change in current/voltage anywhere is the same as that produced by injecting a compensating source $-\Delta Z \cdot i$ (where $i$ is the original current through that branch) in series with the modified branch, with all other sources killed.

Application: Analyzing effect of small changes in component values.

Substitution Theorem

Statement: If the voltage across and current through any branch of a network are known (from analysis or measurement), that branch can be replaced by any combination of elements that maintains the same $v$ and $i$ at its terminals, without affecting the rest of the network.

Condition: The replacement must be behaviorally equivalent at the terminals (same port equation).

Example: Replace a resistor with a voltage source $V$ and a series resistance $R$ if $$\displaystyle V = IR $$.

4. Resonance in AC Circuits

Series Resonant Circuit

• Circuit: $R$, $L$, $C$ in series with AC source $$\displaystyle v_s = V_m \sin \omega t $$.

• Impedance: $$\displaystyle Z = R + j(\omega L - 1/\omega C) $$

• Resonant Frequency ($$\displaystyle \omega_0 $$): When $$\displaystyle X_L = X_C $$.

$$\omega_0 = \frac{1}{\sqrt{LC}}, \quad f_0 = \frac{1}{2\pi\sqrt{LC}}$$

• At Resonance:

  • $$\displaystyle Z = R $$ (minimum, purely resistive).

  • $$\displaystyle I = V_s/R $$ (maximum).

  • $$\displaystyle V_L = V_C = Q \cdot V_s $$ (can be much larger than source voltage).

• Quality Factor (Q):

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

Measures sharpness of resonance and voltage magnification.

• Bandwidth (BW): $$\displaystyle \text{BW} = \frac{\omega_0}{Q} $$ (in rad/s) or $$\displaystyle \frac{f_0}{Q} $$ (in Hz). Frequencies where power is half of max ($$\displaystyle |Z| = \sqrt{2}R $$).

Parallel Resonant Circuit

• Circuit: $R$, $L$, $C$ in parallel (often $R$ represents inductor loss in series with $L$).

• Admittance: $$\displaystyle Y = 1/R + j(\omega C - 1/\omega L) $$ (for ideal parallel RLC).

• Resonant Frequency ($$\displaystyle \omega_0 $$): When $$\displaystyle B = 0 $$ ( susceptance zero).

$$\omega_0 = \frac{1}{\sqrt{LC}} \quad \text{(same as series, for ideal case)}$$

For practical parallel RLC (R in series with L), $$\displaystyle \omega_0 \approx \omega_0(1 - \frac{R^2 C}{2L}) $$ but often approximated as $1/\sqrt{LC}$.

• At Resonance:

  • $$\displaystyle Y = 1/R $$ (minimum admittance).

  • $$\displaystyle Z = R $$ (maximum impedance, purely resistive).

  • Branch currents $$\displaystyle I_L $$ and $$\displaystyle I_C $$ are large and opposite, summing to small input current.

• Q-Factor (Parallel): $$\displaystyle Q = R \sqrt{\frac{C}{L}} = \frac{R}{\omega_0 L} = \omega_0 R C $$.

• Bandwidth: Same relation $$\displaystyle \text{BW} = \omega_0 / Q $$.

5. Laplace Transform Analysis

Laplace Transform of Standard Waveforms

s-Domain Circuit Analysis

• Equivalent Circuits:

  • Resistor: $R$ (same).

  • Inductor: $sL$ with initial current $$\displaystyle i(0^-) $$ represented by a current source $$\displaystyle i(0^-) $$ in parallel.

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

• Procedure: 1. Draw s-domain equivalent circuit. 2. Apply KCL/KVL or network theorems. 3. Solve for desired $I(s)$ or $V(s)$. 4. Inverse Laplace to get $i(t)$ or $v(t)$.

Initial Value Theorem (IVT)

Statement: If $f(t)$ and $f'(t)$ are Laplace transformable and $sF(s)$ has no poles on the $j\omega$ axis or in the right-half plane (RHP), then:

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

Use: Finds initial value directly from $F(s)$ without inverse transform.

Example: $$\displaystyle I(s) = \frac{2s+3}{(s+1)(s+3)} $$, $$\displaystyle i(0^+) = \lim_{s\to\infty} s \cdot I(s) = 2 $$.

Final Value Theorem (FVT)

Statement: If $f(t)$ and $f'(t)$ are Laplace transformable and $sF(s)$ has no poles in the RHP or on the $j\omega$ axis (except possibly a simple pole at $$\displaystyle s=0 $$), then:

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

CRITICAL: Check pole locations first! System must be stable.

Example: $$\displaystyle I(s) = \frac{0.42}{s(s^2+0.35s+0.816)} $$. Poles at $$\displaystyle s=0 $$ and roots of $$\displaystyle s^2+0.35s+0.816=0 $$ (both LHP since discriminant $$\displaystyle 0.1225-3.264<0 $$). So $$\displaystyle i(\infty) = \lim_{s\to0} s \cdot I(s) = 0.42 / 0.816 \approx 0.515 $$.

Pole-Zero Plot & System Response

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

  • Real negative pole → decaying exponential.

  • Complex LHP pole → damped sinusoid.

  • RHP pole → unstable (growing).

  • Repeated pole → $$\displaystyle t e^{-at} $$ terms.

• Zeros: Roots of numerator. Affect forced response amplitude and can cause cancellation.

• Plot: s-plane with $\sigma$ (real) and $j\omega$ (imaginary). Location dictates time-domain behavior.

6. Fourier Series

Trigonometric Fourier Series (TFS)

For a periodic signal $f(t)$ with period $T$, fundamental frequency $$\displaystyle \omega_0 = 2\pi/T $$:

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

Coefficients:

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

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

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

Even/Odd Functions:

• Even $f(t)$: $$\displaystyle b_n = 0 $$, only cosine terms (and $$\displaystyle a_0 $$).

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

Exponential Fourier Series (EFS)

Complex Form:

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

Coefficient:

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

Relation to TFS:

$$c_0 = a_0$$

$$c_n = \frac{1}{2}(a_n - jb_n) \quad \text{for } n>0$$

$$c_{-n} = \frac{1}{2}(a_n + jb_n) = c_n^*$$

Magnitude & Phase: $$\displaystyle |c_n| $$ gives amplitude of $$\displaystyle n^{th} $$ harmonic, $$\displaystyle \angle c_n $$ gives phase.

Fourier Series of Standard Waveforms

[!TIP] For waveforms with half-wave symmetry ($$\displaystyle f(t+T/2) = -f(t) $$), DC component $$\displaystyle a_0=0 $$ and even harmonics ($$\displaystyle a_{2n}, b_{2n} $$) are zero.

7. Two-Port Network Parameters

Z-Parameters (Open-Circuit Impedance)

Definition: $$\displaystyle V_1 $$ and $$\displaystyle V_2 $$ expressed in terms of $$\displaystyle I_1 $$ and $$\displaystyle I_2 $$ with output port open ($$\displaystyle I_2=0 $$).

$$\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$$

Determination:

• $$\displaystyle Z_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} $$ (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)

Reciprocity: For passive, linear, bilateral networks, $$\displaystyle Z_{12} = Z_{21} $$.

Y-Parameters (Short-Circuit Admittance)

Definition: $$\displaystyle I_1 $$ and $$\displaystyle I_2 $$ expressed in terms of $$\displaystyle V_1 $$ and $$\displaystyle V_2 $$ with output port shorted ($$\displaystyle V_2=0 $$).

$$\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$$

Determination:

• $$\displaystyle Y_{11} = \left. \frac{I_1}{V_1} \right|_{V_2=0} $$

• $$\displaystyle Y_{12} = \left. \frac{I_1}{V_2} \right|_{V_1=0} $$

• $$\displaystyle Y_{21} = \left. \frac{I_2}{V_1} \right|_{V_2=0} $$

• $$\displaystyle Y_{22} = \left. \frac{I_2}{V_2} \right|_{V_1=0} $$

Reciprocity: $$\displaystyle Y_{12} = Y_{21} $$.

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

h-Parameters (Hybrid)

Definition: Mix of voltage and current equations. Common for transistor modeling.

$$\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$$

Determination:

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

• $$\displaystyle h_{12} = \left. \frac{V_1}{V_2} \right|_{I_1=0} $$ (Reverse voltage gain with input open) → $$\displaystyle h_{re} $$

• $$\displaystyle h_{21} = \left. \frac{I_2}{I_1} \right|_{V_2=0} $$ (Forward current gain with output shorted) → $$\displaystyle h_{fe} $$

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

Reciprocity: Generally NOT reciprocal ($$\displaystyle h_{12} \ne h_{21} $$).

ABCD-Parameters (Transmission)

Definition: Relates input voltage/current to output voltage/current. Ideal for cascading.

$$\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$$

Note: Sign convention: $$\displaystyle I_2 $$ is entering the output port in the defining equation, but often defined as leaving. The matrix form above uses $$\displaystyle -I_2 $$ for consistency with cascade property.

Determination:

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

• $$\displaystyle B = \left. -\frac{V_1}{I_2} \right|_{V_2=0} $$ (Open-circuit transfer impedance)

• $$\displaystyle C = \left. \frac{I_1}{V_2} \right|_{I_2=0} $$ (Short-circuit transfer admittance)

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

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

Reciprocity: For reciprocal networks, $$\displaystyle AD - BC = 1 $$.

Parameter Conversions (Key Relationships)

• From Z to ABCD:

$$A = \frac{Z_{11}}{Z_{21}}, \quad B = \frac{Z_{11}Z_{22} - Z_{12}Z_{21}}{Z_{21}}, \quad C = \frac{1}{Z_{21}}, \quad D = \frac{Z_{22}}{Z_{21}}$$

• From Y to ABCD:

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

• From h to ABCD:

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

• From ABCD to Z (if $A \ne 0$):

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

Terminated Two-Port Network

• Scenario: Two-port with load $$\displaystyle Z_L $$ at output port 2.

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

$$Z_{in} = \frac{V_1}{I_1} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22} + Z_L} \quad \text{(using Z-params)}$$

Or using ABCD: $$\displaystyle Z_{in} = \frac{AV_2 + B(-I_2)}{CV_2 + D(-I_2)} = \frac{AZ_L + B}{CZ_L + D} $$.

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

$$G_v = \frac{V_2}{V_1} = \frac{Z_L}{Z_{22} + Z_L} \cdot \frac{1}{1 + \frac{Z_{12}Z_{21}}{Z_{11}(Z_{22}+Z_L)}} \quad \text{(simplified)}$$

Using ABCD: $$\displaystyle G_v = \frac{V_2}{V_1} = \frac{1}{A + B/Z_L} $$ (if $$\displaystyle I_1 $$ is output? Careful: from $$\displaystyle V_1 = AV_2 + B(-I_2) $$, and $$\displaystyle -I_2 = V_2/Z_L $$, so $$\displaystyle V_1 = (A + B/Z_L)V_2 $$ → $$\displaystyle V_2/V_1 = 1/(A + B/Z_L) $$).

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

8. Additional Short Note Topics (Concise Summaries)

Network Topology (Overview)

Graph-theoretic analysis of circuits. Uses incidence matrix ($A$), fundamental cut set matrix ($Q$), and fundamental tie set matrix ($B$). Key relations: $$\displaystyle A_r B^T = 0 $$, $$\displaystyle Q = A_r B^{-1} $$. Provides systematic method for writing KCL ($$\displaystyle Q \cdot i = 0 $$) and KVL ($$\displaystyle B \cdot v = 0 $$) equations.

s-Domain Theorems (Laplace Applications)

All circuit theorems (Superposition, Thevenin, Norton) apply in s-domain with impedances ($Z(s)$) and sources (including initial conditions as sources). Initial/Final Value Theorems bridge s-domain and time-domain at $$\displaystyle t=0^+ $$ and $$\displaystyle t=\infty $$.

Compensation Theorem

Change in branch impedance $\Delta Z$ is equivalent to injecting a compensating source $$\displaystyle -\Delta Z \cdot i_{old} $$ in series with that branch, with all other independent sources killed. Useful for sensitivity analysis.

Millman’s Theorem

For $n$ parallel branches between two nodes, each with voltage source $$\displaystyle V_k $$ in series with impedance $$\displaystyle Z_k $$:

$$V = \frac{\sum_{k=1}^{n} \frac{V_k}{Z_k}}{\sum_{k=1}^{n} \frac{1}{Z_k}}$$

Directly gives common node voltage.

Tellegen’s Theorem

$$\displaystyle \sum_{b} v_k \cdot i_k' = 0 $$ for any two networks with identical topology. A powerful, general statement of energy conservation in networks. Used to prove other theorems and establish relationships.

Dual Networks

Every network has a dual obtained by: replacing nodes with meshes, series with parallel, $$\displaystyle R \leftrightarrow G $$, $$\displaystyle L \leftrightarrow C $$, KVL with KCL. Theorems in primal have corresponding duals.

Controlled Sources

Four types: VCVS, VCCS, CCVS, CCCS. Represent active elements (transistors, op-amps). Their parameters ($\mu, g, r, \alpha$) are dimensionless or have dimensions (transconductance $g$, transresistance $r$). Essential for modeling amplifiers.

Open Circuit Impedance Parameters (Z-parameters)

$[Z]$ matrix from $$\displaystyle V_1 = Z_{11}I_1 + Z_{12}I_2 $$, $$\displaystyle V_2 = Z_{21}I_1 + Z_{22}I_2 $$. Found by open-circuiting output ($$\displaystyle I_2=0 $$) and input ($$\displaystyle I_1=0 $$). Reciprocal if $$\displaystyle Z_{12}=Z_{21} $$. Symmetric if also $$\displaystyle Z_{11}=Z_{22} $$.

Short Circuit Admittance Parameters (Y-parameters)

$[Y]$ matrix from $$\displaystyle I_1 = Y_{11}V_1 + Y_{12}V_2 $$, $$\displaystyle I_2 = Y_{21}V_1 + Y_{22}V_2 $$. Found by short-circuiting output ($$\displaystyle V_2=0 $$) and input ($$\displaystyle V_1=0 $$). $$\displaystyle [Y] = [Z]^{-1} $$. Reciprocal if $$\displaystyle Y_{12}=Y_{21} $$.

Terminated Two-Port Network

Analysis when load $$\displaystyle Z_L $$ is connected to output port. Key results:

• Input Impedance: $$\displaystyle Z_{in} = \frac{AZ_L + B}{CZ_L + D} $$ (ABCD params).

• Voltage Gain: $$\displaystyle G_v = V_2/V_1 = 1/(A + B/Z_L) $$.

• Current Gain: $$\displaystyle G_i = I_2/I_1 = -1/(C Z_L + D) $$.

• Power Gain: Can be derived from these.