Network Analysis (EC-305) - Unit 2 Short Notes

UNIT 2: NETWORK ANALYSIS - SHORT NOTES

1. FUNDAMENTAL LAWS AND NETWORK TOPOLOGY

Kirchhoff's Laws

Kirchhoff's Current Law (KCL):

At any node, the algebraic sum of currents leaving the node is zero.

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

Example: For a node with currents $$\displaystyle i_1=2A $$, $$\displaystyle i_2=-3A $$ (entering), $$\displaystyle i_3=1A $$ leaving: $$\displaystyle i_1 + (-i_2) + i_3 = 0 \Rightarrow 2 + 3 + 1 = 6A $$? Wait—signs matter. If all currents are defined as leaving, then $$\displaystyle i_1 + i_2 + i_3 = 0 $$. If $$\displaystyle i_2 $$ is actually entering, its value in the sum is $$\displaystyle -i_2 $$. Common pitfall: inconsistent sign convention.

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 sources $$\displaystyle V_1=10V $$, $$\displaystyle V_2=5V $$, and resistor drops $$\displaystyle V_R=3V $$, $$\displaystyle V_R'=2V $$ (polarity consistent with loop direction): $$\displaystyle V_1 - V_2 - V_R - V_R' = 0 \Rightarrow 10 - 5 - 3 - 2 = 0 $$.

Graph Theory Fundamentals

• Graph: Set of vertices (nodes) connected by edges (branches). No isolated nodes.

• Oriented Graph: Graph with assigned direction to each branch.

• Tree: Connected subgraph containing all nodes with no loops. For a graph with $n$ nodes, tree has $n-1$ branches.

• Co-Tree: Complement of a tree; branches not in the tree are links (or chords).

• Twigs: Branches belonging to the tree.

• Links: Branches belonging to the co-tree.

Exam Tip: Number of links $$\displaystyle l = b - (n-1) $$, where $b$ = total branches, $n$ = nodes.

Incidence Matrix

• Complete Incidence Matrix $A$: $n \times b$ matrix. Row for each node, column for each branch. Entry $$\displaystyle a_{ij} = 1 $$ if branch $j$ leaves node $i$, $-1$ if enters, $0$ otherwise.

• Reduced Incidence Matrix $$\displaystyle A_r $$: Remove any one row (usually reference node) from complete matrix. Rank $$\displaystyle = n-1 $$.

Example: For a 3-node graph with 4 branches, $A$ is $3\times4$. Removing row 1 gives $$\displaystyle A_r $$ ($2\times4$).

Cut Set Matrix

• Cut Set: Minimal set of branches whose removal disconnects the graph into exactly two parts, with exactly one branch from the tree in each cut set.

• Basic Cut Sets: For a tree with $n-1$ twigs, there are $n-1$ fundamental cut sets, each containing one twig and some links.

• Cut Set Matrix $Q$: Rows correspond to basic cut sets, columns to branches. Entry $$\displaystyle q_{ij}=1 $$ if branch $j$ is in cut set $i$ with orientation leaving the partitioned part, $-1$ if entering, $0$ otherwise.

For planar networks: Cut sets correspond to loops in the dual graph.

Tie Set Matrix

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

• Basic Tie Sets: For a tree, each link forms one fundamental loop with tree branches. Number $$\displaystyle = l = b - n + 1 $$.

• Tie Set Matrix $B$: Rows = basic tie sets, columns = branches. $$\displaystyle b_{ij}=1 $$ if branch $j$ is in loop $i$ with orientation same as link, $-1$ if opposite, $0$ otherwise.

Property: $$\displaystyle A_r \cdot B^T = 0 $$ (orthogonality).

2. CIRCUIT THEOREMS AND ANALYSIS TECHNIQUES

Superposition Theorem

In a linear bilateral network with multiple sources, the response (voltage/current) is the algebraic sum of responses due to each source acting alone, with all other independent sources replaced by their internal impedances (voltage sources shorted, current sources opened).

DC/AC Example: For a circuit with DC voltage source and AC current source, compute DC response (capacitor open, inductor short), AC response (DC source shorted), then sum. Pitfall: Does not apply to power calculations directly (power is nonlinear).

Thevenin's Theorem

Any linear bilateral network can be replaced by an equivalent circuit: a voltage source $$\displaystyle V_{th} $$ in series with impedance $$\displaystyle Z_{th} $$.

• $$\displaystyle V_{th} $$ = open-circuit voltage at terminals.

• $$\displaystyle Z_{th} $$ = impedance seen at terminals with all independent sources killed (voltage sources shorted, current sources opened). For AC, $$\displaystyle Z_{th} $$ is complex.

AC Extension: $$\displaystyle V_{th} $$ is phasor open-circuit voltage; $$\displaystyle Z_{th} $$ is complex impedance.

Norton's Theorem

Equivalent to a current source $$\displaystyle I_N $$ in parallel with admittance $$\displaystyle Y_N $$ (or impedance $$\displaystyle Z_N $$).

• $$\displaystyle I_N $$ = short-circuit current at terminals.

• $$\displaystyle Y_N = 1/Z_{th} $$ (same as Thevenin impedance).

AC Extension: $$\displaystyle I_N $$ is phasor short-circuit current; $$\displaystyle Y_N $$ is complex admittance.

Maximum Power Transfer Theorem

DC Networks:

Maximum power delivered to load $$\displaystyle R_L $$ occurs when $$\displaystyle R_L = R_{th} $$ (Thevenin resistance).

Maximum power: $$\displaystyle P_{max} = \frac{V_{th}^2}{4R_{th}} $$.
Efficiency: $$\displaystyle \eta = \frac{P_{max}}{P_{source}} = \frac{R_{th}}{R_{th}+R_{th}} = 0.5 $$ (50%).

Proof: $$\displaystyle P_L = \frac{V_{th}^2 R_L}{(R_{th}+R_L)^2} $$. Differentiate w.r.t $$\displaystyle R_L $$, set to zero → $$\displaystyle R_L = R_{th} $$. Then $$\displaystyle P_{max} = V_{th}^2/(4R_{th}) $$. Source power $$\displaystyle = V_{th}^2/(2R_{th}) $$, so $$\displaystyle \eta = 0.5 $$.

AC Networks:

Maximum power when $$\displaystyle Z_L = Z_{th}^* $$ (complex conjugate). Then $$\displaystyle P_{max} = \frac{|V_{th}|^2}{4R_{th}} $$, where $$\displaystyle R_{th} = \text{Re}(Z_{th}) $$. Efficiency still 50% if source impedance is purely resistive, but generally lower if reactive.

Millman's Theorem

For multiple voltage sources $$\displaystyle V_1, V_2, ..., V_n $$ with series impedances $$\displaystyle Z_1, Z_2, ..., Z_n $$ connected in parallel, the common voltage $V$ at the junction is:

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

For current sources in parallel, use admittances.

Application: Simplifies parallel-source networks.

Tellegen's Theorem

For any two networks (not necessarily linear or passive) with the same topology (graph), if branch voltages $$\displaystyle v_{1k} $$ and branch currents $$\displaystyle i_{2k} $$ satisfy KVL and KCL respectively in their networks, then:

$$\sum_{k=1}^{b} v_{1k} i_{2k} = 0$$

Verification: Often used to check network analysis solutions. For same network, $$\displaystyle \sum v_k i_k = 0 $$ (power sum zero for lossless? Actually for any network, sum of instantaneous powers is zero if KVL/KCL hold—Tellegen's is more general).

Compensation Theorem

If a branch with impedance $Z$ has current $I$, and its impedance is changed to $Z + \Delta Z$, the resulting changes in all other branch currents/voltages are equivalent to injecting a voltage source $-\Delta Z \cdot I$ in series with that branch (with original current $I$ maintained).

Application: Sensitivity analysis, effect of parameter changes.

Substitution Theorem

If the voltage across and current through any branch of a network are known, that branch can be replaced by any combination of elements that maintains the same voltage-current relationship (e.g., voltage source, current source, impedance). The rest of the network remains unaffected.

Example: Replace a resistor with a voltage source equal to the voltage across it.

3. RESONANCE AND COUPLED CIRCUITS

Series Resonant Circuit

Circuit: $R$, $L$, $C$ in series.
Impedance: $$\displaystyle Z = R + j(\omega L - 1/(\omega C)) $$.
Resonant Frequency $$\displaystyle \omega_0 $$: $$\displaystyle \omega_0 L = 1/(\omega_0 C) \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}} $$, $$\displaystyle f_0 = \frac{1}{2\pi\sqrt{LC}} $$.

At resonance: $$\displaystyle Z = R $$ (minimum), current $$\displaystyle I = V/R $$ (maximum). Phase angle $$\displaystyle \phi = 0 $$.
Voltage Magnification (Q-factor):
$$\displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} $$.

Voltage across $L$ or $C$: $$\displaystyle V_L = V_C = Q \cdot V $$ (magnitude).

Derivation: $$\displaystyle |V_L| = I \cdot \omega_0 L = (V/R) \cdot \omega_0 L = Q V $$.

Parallel Resonant Circuit

Circuit: $R$, $L$, $C$ in parallel (often $R$ represents inductor resistance).
Admittance: $$\displaystyle Y = 1/R + j(\omega C - 1/(\omega L)) $$.
Resonant Frequency $$\displaystyle \omega_0 $$: $$\displaystyle \omega_0 C = 1/(\omega_0 L) \Rightarrow \omega_0 = 1/\sqrt{LC} $$ (if $R$ large, approximately). Exact: $$\displaystyle \omega_0 = \sqrt{\frac{1}{LC} - \frac{R^2}{L^2}} $$ for $R$ in series with $L$? Actually for parallel RLC with loss in L: $$\displaystyle \omega_0 \approx 1/\sqrt{LC} $$ if $Q$ high.

At resonance: $$\displaystyle Y = 1/R $$ (minimum), impedance $$\displaystyle Z = R $$ (maximum). Phase angle $$\displaystyle \phi = 0 $$.
Current Magnification (Q-factor):
$$\displaystyle Q = R \sqrt{\frac{C}{L}} = \frac{R}{\omega_0 L} = \omega_0 R C $$.

Current through $L$ or $C$: $$\displaystyle I_L = I_C = Q \cdot I_{source} $$ (magnitude).

Mutual Inductance and Coupling

Mutual Inductance $M$: Voltage induced in one coil due to current change in coupled coil: $$\displaystyle v_2 = M \frac{di_1}{dt} $$.
Coefficient of Coupling $k$:

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

Dot Convention: Determines polarity of induced voltage. If currents enter dotted terminals, mutual voltage is positive (aiding).
Coupled Circuit Equations:

For two coils:

$$v_1 = L_1 \frac{di_1}{dt} \pm M \frac{di_2}{dt}$$

$$v_2 = \pm M \frac{di_1}{dt} + L_2 \frac{di_2}{dt}$$

Sign depends on dot positions.

Example: Two coils in series aiding: $$\displaystyle L_{eq} = L_1 + L_2 + 2M $$; opposing: $$\displaystyle L_1 + L_2 - 2M $$.

4. LAPLACE TRANSFORM IN NETWORK ANALYSIS

Laplace Transform of Standard Waveforms

Piecewise: Use time-shifting and unit step functions.

Initial Value Theorem (IVT) and Final Value Theorem (FVT)

IVT: If $sF(s)$ has no poles in $$\displaystyle \text{Re}(s) > 0 $$, then

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

FVT: If $sF(s)$ has no poles in $\text{Re}(s) \ge 0$ except possibly at $$\displaystyle s=0 $$ (simple pole), then

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

Conditions: Crucial—FVT fails if poles on imaginary axis (except origin) or right half-plane.

Verification: Check poles of $sF(s)$.

Solving Circuits Using Laplace Transform

1. Replace circuit elements with impedances: $R \to R$, $L \to sL$, $C \to 1/(sC)$. Include initial conditions as sources: inductor current $$\displaystyle i_L(0^-) $$ becomes voltage source $$\displaystyle L i_L(0^-) $$ in series; capacitor voltage $$\displaystyle v_C(0^-) $$ becomes current source $$\displaystyle C v_C(0^-) $$ in parallel.

2. Apply KVL/KCL in s-domain.

3. Solve for desired variable $X(s)$.

4. Inverse Laplace (partial fractions, tables) to get $x(t)$.

Switch Operations: At $$\displaystyle t=0 $$, assume steady-state before switch for initial conditions. Use step functions for switching.

Pole-Zero Plot

• Poles: Roots of denominator of $H(s)$ (where $H(s) \to \infty$).

• Zeros: Roots of numerator (where $$\displaystyle H(s)=0 $$).

• Plot: s-plane (Re vs Im).
  Interpretation:

• Poles in LHP → stable, decaying response.

• Poles on Im-axis → sustained oscillations (marginally stable).

• Poles in RHP → unstable, growing response.

• Zero locations affect shape but not stability.

Transfer Functions and Driving Point Impedances

• Driving Point Impedance $Z(s)$: Ratio of Laplace voltage to current at same port with all sources killed: $$\displaystyle Z(s) = V(s)/I(s) $$.

• Transfer Function $G(s)$ (or $H(s)$): Ratio of output to input (voltage/current) in s-domain, e.g., voltage gain $$\displaystyle G_{21}(s) = V_2(s)/V_1(s) $$ with $$\displaystyle I_2=0 $$ (open-circuit).

5. FOURIER SERIES ANALYSIS

Trigonometric Fourier Series

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

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

Coefficients:

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

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

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

Symmetry Properties:

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

• Odd function: $$\displaystyle a_0 = a_n = 0 $$, only sine terms.

• Half-wave symmetry: $$\displaystyle f(t+T/2) = -f(t) $$ → only odd harmonics ($$\displaystyle a_0=0 $$, even $n$ coefficients zero).

• Quarter-wave symmetry: Combination of half-wave and even/odd → specific simplifications.

Exponential Fourier Series

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

Coefficients:

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

Relation: $$\displaystyle C_n = \frac{1}{2}(a_n - jb_n) $$ for $$\displaystyle n>0 $$; $$\displaystyle C_0 = a_0 $$; $$\displaystyle C_{-n} = C_n^* $$.

Fourier Series of Standard Waveforms

Tip: Use symmetry to reduce integration.

6. TWO-PORT NETWORK PARAMETERS

Impedance Parameters (Z-Parameters)

Definition (Open-Circuit Impedance 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)

Circuit Model: Two voltage sources in series with impedances: $$\displaystyle V_1 = Z_{11}I_1 + Z_{12}I_2 $$, etc.

Admittance Parameters (Y-Parameters)

Definition (Short-Circuit Admittance Parameters):

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

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

• $$\displaystyle Y_{11} = \left. \frac{I_1}{V_1} \right|_{V_2=0} $$, etc. Circuit Model: Two current sources in parallel with admittances.

Hybrid Parameters (h-Parameters)

Definition:

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

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

• $$\displaystyle h_{11} = \left. \frac{V_1}{I_1} \right|_{V_2=0} $$ (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) Units: $$\displaystyle h_{11} $$: $\Omega$, $$\displaystyle h_{12} $$: dimensionless, $$\displaystyle h_{21} $$: dimensionless, $$\displaystyle h_{22} $$: S.

Transmission Parameters (ABCD Parameters)

Definition:

$$V_1 = A V_2 + B I_2$$

$$-I_1 = C V_2 + D I_2$$

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

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

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

• $$\displaystyle D = \left. \frac{-I_1}{I_2} \right|_{V_2=0} $$ (current ratio, short-circuit) Property: For reciprocal network, $$\displaystyle AD - BC = 1 $$. For symmetric network, $$\displaystyle A = D $$.

Circuit Model: Cascade-friendly.

Interconversion of Parameters

Y in terms of ABCD:

$$Y_{11} = \frac{D}{B}, \quad Y_{12} = \frac{AD - BC}{B} = \frac{1}{B} \text{ (if reciprocal)}$$

$$Y_{21} = -\frac{1}{B}, \quad Y_{22} = \frac{A}{B}$$

Derivation: From ABCD equations, solve for $$\displaystyle I_1, I_2 $$ in terms of $$\displaystyle V_1, V_2 $$.

Z in terms of ABCD:

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

$$Z_{21} = -\frac{1}{C}, \quad Z_{22} = \frac{D}{C}$$

h in terms of Z:

$$h_{11} = \frac{Z_{11}}{Z_{21}}, \quad h_{12} = \frac{Z_{12}Z_{21} - Z_{11}Z_{22}}{Z_{21}}$$

$$h_{21} = \frac{1}{Z_{21}}, \quad h_{22} = -\frac{Z_{22}}{Z_{21}}$$

h in terms of ABCD:

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

$$h_{21} = -\frac{1}{C}, \quad h_{22} = \frac{D}{C}$$

General: Use matrix relations. Parameter matrix $P$ relates $$\displaystyle [V_1, I_1]^T = P [V_2, I_2]^T $$ or similar. Conversion involves inverting and rearranging.

Cascade Connection

For two two-ports in cascade (output of first to input of second), overall transmission matrix is product:

$$\begin{bmatrix} A_{total} & B_{total} \\ C_{total} & D_{total} \end{bmatrix} = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix} \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}$$

Proof: Apply first network equations to get $$\displaystyle V_1, I_1 $$ in terms of $$\displaystyle V_2, I_2 $$ (at first output). Then these become input to second network. Substitute.

Terminated Two-Port Network

Two-port with load impedance $$\displaystyle Z_L $$ at output (port 2). Input impedance:

$$Z_{in} = \frac{V_1}{I_1} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22} + Z_L}$$

Gains:

• Voltage gain: $$\displaystyle G_v = V_2/V_1 = \frac{Z_{21}}{Z_{11} + Z_{in}} $$? Actually: $$\displaystyle V_2 = \frac{Z_L}{Z_{22}+Z_L} \cdot (-Z_{21} I_1) $$? Better:

From Z-params: $$\displaystyle V_2 = Z_{21}I_1 + Z_{22}I_2 $$, $$\displaystyle I_2 = V_2/Z_L $$. Solve: $$\displaystyle V_2 = \frac{Z_{21}}{1 - Z_{22}/Z_L} I_1 $$?

Standard: $$\displaystyle V_2 = \frac{Z_{21} Z_L}{Z_{22} + Z_L} I_1 $$, so $$\displaystyle G_v = V_2/V_1 = \frac{Z_{21} Z_L}{Z_{22} + Z_L} \cdot \frac{1}{Z_{in}} $$.

• Current gain: $$\displaystyle G_i = I_2/I_1 = \frac{Z_{21}}{Z_{22} + Z_L} $$.

• Power gain: $$\displaystyle G_p = \frac{|I_2|^2 R_{eff,L}}{|I_1|^2 R_{eff,in}} $$ or using $$\displaystyle G_v $$ and $$\displaystyle G_i $$.

With ABCD parameters: Easier: $$\displaystyle V_1 = A V_2 + B I_2 $$, $$\displaystyle -I_1 = C V_2 + D I_2 $$. With $$\displaystyle I_2 = V_2/Z_L $$, solve for $$\displaystyle V_2/V_1 $$.

7. SPECIAL TOPICS FOR SHORT NOTES

Controlled Sources

Example: Op-amp models use VCVS.

Dual Networks and Duality Principle

• Dual elements: Series ↔ parallel, voltage ↔ current, resistance ↔ conductance, impedance ↔ admittance, KVL ↔ KCL.

• Dual circuit: For a planar graph, replace each branch with its dual in the dual graph.

• Duality in theorems: Superposition ↔ ? Actually, many theorems have duals: Thevenin (voltage source + Z) ↔ Norton (current source + Y).

Application: Solve a difficult network by solving its dual (often simpler).

Network Topology (Overview)

Graph theory applied to circuits: nodes, branches, loops, cut sets.
Tree: $n-1$ twigs. Co-tree: $l$ links.
Fundamental circuits: Each link + tree branches → $l$ tie sets.
Fundamental cut sets: Each twig + links → $n-1$ cut sets.
Matrices: Incidence $A$, Tie Set $B$, Cut Set $Q$. Relationships: $$\displaystyle A_r B^T = 0 $$, $$\displaystyle Q B^T = I $$ (if properly ordered).

Compensation Theorem (Detailed)

If in a branch with impedance $Z$, current $I$ flows, and $Z$ is changed to $$\displaystyle Z' = Z + \Delta Z $$, then the new currents/voltages are same as original plus the effect of a voltage source $$\displaystyle v_c = -I \Delta Z $$ inserted in series with the original branch (with $I$ unchanged).

Proof: From original: $$\displaystyle v = Z I $$. New: $$\displaystyle v' = Z' I' = (Z+\Delta Z) I' $$. But $I'$ unknown. Instead, consider original branch replaced by $$\displaystyle v_c $$ and $Z$: KVL gives $$\displaystyle v' = v + v_c = Z I + v_c $$. But $$\displaystyle v' = Z I' + \Delta Z I' $$. Equate: $$\displaystyle Z I' + \Delta Z I' = Z I + v_c $$. If we set $$\displaystyle v_c = -\Delta Z I $$, then $$\displaystyle Z I' = Z I \Rightarrow I'=I $$ for that branch? Actually careful: The theorem states that the change in any other branch is same as if we inserted $$\displaystyle v_c = -I \Delta Z $$ with original network (all sources as before). So we solve modified network with $$\displaystyle v_c $$ added.

Open Circuit Impedance Parameters (Z-Parameters)

Already covered in Section 6.1. Emphasize: all measurements with output open ($$\displaystyle I_2=0 $$) for $$\displaystyle Z_{11}, Z_{21} $$; input open ($$\displaystyle I_1=0 $$) for $$\displaystyle Z_{22}, Z_{12} $$.

Short Circuit Admittance Parameters (Y-Parameters)

Already covered in Section 6.2. Measurements with output shorted ($$\displaystyle V_2=0 $$) for $$\displaystyle Y_{11}, Y_{21} $$; input shorted ($$\displaystyle V_1=0 $$) for $$\displaystyle Y_{22}, Y_{12} $$.

Final Exam Tips:

• For graph theory, always verify $$\displaystyle b = n-1 + l $$.

• In two-port conversions, remember reciprocal condition $$\displaystyle AD-BC=1 $$ for Z/Y parameters? Actually for Z/Y, reciprocal means $$\displaystyle Z_{12}=Z_{21} $$, $$\displaystyle Y_{12}=Y_{21} $$. For ABCD, reciprocal means $$\displaystyle AD-BC=1 $$.

• Laplace: Always check FVT conditions before applying.

• Fourier: Identify symmetry first to avoid full integration.

• Resonance: Series → current max, voltage across L/C magnified; Parallel → voltage max, current through L/C magnified.

• Max power in AC: Conjugate matching $$\displaystyle Z_L = Z_{th}^* $$.