Formula Sheet: Network Theorems

Superposition Theorem

Fundamental Principle

• In a linear network with multiple independent sources, the total response (voltage or current) at any element is the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources deactivated.
• Voltage sources are replaced by short circuits (0 V)
• Current sources are replaced by open circuits (0 A)
• Dependent sources remain active during all calculations

Application Procedure

1. Identify all independent sources in the network
2. Deactivate all but one independent source
3. Calculate the desired response due to the active source
4. Repeat for each independent source
5. Sum all individual responses algebraically (considering polarity/direction)

Mathematical Expression

\[V_{total} = V_1 + V_2 + V_3 + ... + V_n\] \[I_{total} = I_1 + I_2 + I_3 + ... + I_n\]

• V_total, I_total = total voltage or current response
• V_n, I_n = response due to the n^th source alone

Limitations

• Only applicable to linear circuits
• Cannot be used to calculate power directly (power is not a linear function)
• Does not apply to circuits with nonlinear elements

Thévenin's Theorem

Fundamental Principle

• Any linear two-terminal network containing voltage sources, current sources, and resistances can be replaced by an equivalent circuit consisting of a voltage source (V_TH) in series with a resistance (R_TH)

Thévenin Voltage (V_TH)

• V_TH = open-circuit voltage across the terminals of interest
• Calculate with load disconnected
• Also called V_OC (open-circuit voltage)

Thévenin Resistance (R_TH)

Method 1: Source Deactivation (No Dependent Sources)

• Deactivate all independent sources:
  • Replace voltage sources with short circuits
  • Replace current sources with open circuits
• Calculate equivalent resistance looking into the terminals

\[R_{TH} = R_{eq}\]

Method 2: Short-Circuit Current Method

\[R_{TH} = \frac{V_{OC}}{I_{SC}}\]

• V_OC = open-circuit voltage (volts, V)
• I_SC = short-circuit current when terminals are shorted (amperes, A)
• This method works for circuits with or without dependent sources

Method 3: External Source Method (For Dependent Sources)

• Deactivate all independent sources
• Apply a test voltage source V_test or test current source I_test at the terminals
• Calculate the resulting current or voltage

\[R_{TH} = \frac{V_{test}}{I_{test}}\]

Load Current and Voltage

\[I_L = \frac{V_{TH}}{R_{TH} + R_L}\] \[V_L = V_{TH} \cdot \frac{R_L}{R_{TH} + R_L}\]

• I_L = load current (A)
• V_L = load voltage (V)
• R_L = load resistance (Ω)

Maximum Power Transfer

\[P_{max} = \frac{V_{TH}^2}{4R_{TH}}\]

• Maximum power occurs when R_L = R_TH
• P_max = maximum power delivered to load (watts, W)
• Efficiency at maximum power transfer = 50%

Norton's Theorem

Fundamental Principle

• Any linear two-terminal network can be replaced by an equivalent circuit consisting of a current source (I_N) in parallel with a resistance (R_N)

Norton Current (I_N)

• I_N = short-circuit current through the shorted terminals
• Also called I_SC (short-circuit current)
• Calculate with terminals short-circuited

Norton Resistance (R_N)

• R_N = R_TH (same as Thévenin resistance)
• Use same methods as for calculating R_TH

\[R_N = R_{TH} = \frac{V_{OC}}{I_{SC}}\]

Load Current and Voltage

\[I_L = I_N \cdot \frac{R_N}{R_N + R_L}\] \[V_L = I_N \cdot \frac{R_N \cdot R_L}{R_N + R_L}\]

Thévenin-Norton Conversion

\[V_{TH} = I_N \cdot R_N\] \[I_N = \frac{V_{TH}}{R_{TH}}\] \[R_{TH} = R_N\]

• Thévenin and Norton equivalents are dual representations
• Either can be converted to the other

Maximum Power Transfer Theorem

Resistive Load

\[R_L = R_{TH}\] \[P_{max} = \frac{V_{TH}^2}{4R_{TH}} = \frac{I_N^2 \cdot R_N}{4}\]

• Maximum power is transferred when load resistance equals source resistance
• R_L = load resistance for maximum power (Ω)
• Efficiency = 50% at maximum power transfer

Power Delivered to Load

\[P_L = I_L^2 \cdot R_L = \frac{V_{TH}^2 \cdot R_L}{(R_{TH} + R_L)^2}\] \[P_L = V_L \cdot I_L\]

Complex Load (AC Circuits)

• For maximum power transfer with complex impedances:

\[Z_L = Z_{TH}^*\]

• Z_L = load impedance
• Z_TH* = complex conjugate of Thévenin impedance
• If Z_TH = R_TH + jX_TH, then Z_L = R_TH - jX_TH

\[P_{max} = \frac{|V_{TH}|^2}{8R_{TH}}\]

• |V_TH| = magnitude of Thévenin voltage (V)
• R_TH = real part of Thévenin impedance (Ω)

Purely Resistive Load on Complex Source

• If load must be purely resistive but source has complex impedance:

\[R_L = |Z_{TH}| = \sqrt{R_{TH}^2 + X_{TH}^2}\] \[P_{max} = \frac{|V_{TH}|^2}{2(R_{TH} + |Z_{TH}|)}\]

Source Transformation

Voltage Source to Current Source

• A voltage source V_S in series with resistance R_S can be transformed to:

\[I_S = \frac{V_S}{R_S}\]

• Current source I_S in parallel with resistance R_S
• The resistance value remains the same
• Current source direction: from negative to positive terminal of original voltage source (through the source)

Current Source to Voltage Source

• A current source I_S in parallel with resistance R_S can be transformed to:

\[V_S = I_S \cdot R_S\]

• Voltage source V_S in series with resistance R_S
• The resistance value remains the same
• Voltage polarity: positive terminal in direction of current source arrow

Conditions and Limitations

• Transformations apply only to independent sources
• The internal resistance must be finite and nonzero
• Ideal voltage sources (R_S = 0) cannot be transformed
• Ideal current sources (R_S = ∞) cannot be transformed
• Transformations are valid for external circuit analysis only

Reciprocity Theorem

Fundamental Principle

• In a linear, bilateral network with a single source, if a voltage source in branch A produces current in branch B, then moving the same voltage source to branch B will produce the same magnitude of current in branch A
• Similarly applies to current sources and resulting voltages

Mathematical Statement

• For voltage source excitation:

\[\frac{I_2}{V_1} = \frac{I_1}{V_2}\]

• V₁ = voltage source in position 1
• I₂ = resulting current at position 2
• V₂ = voltage source moved to position 2
• I₁ = resulting current at position 1

• For current source excitation:

\[\frac{V_2}{I_1} = \frac{V_1}{I_2}\]

Conditions for Applicability

• Network must be linear
• Network must be bilateral (same properties in both directions)
• Network must be time-invariant
• Only one independent source at a time
• Initial conditions must be zero

Millman's Theorem

Fundamental Principle

• Simplifies analysis of parallel voltage sources with series resistances
• Converts multiple voltage sources in parallel branches to a single equivalent source

Equivalent Voltage

\[V_{eq} = \frac{\frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3} + ... + \frac{V_n}{R_n}}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}}\] \[V_{eq} = \frac{\sum_{i=1}^{n}\frac{V_i}{R_i}}{\sum_{i=1}^{n}\frac{1}{R_i}}\]

Equivalent Resistance

\[R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}}\] \[R_{eq} = \frac{1}{\sum_{i=1}^{n}\frac{1}{R_i}} = \left(R_1^{-1} + R_2^{-1} + ... + R_n^{-1}\right)^{-1}\]

Alternative Form with Conductances

\[V_{eq} = \frac{V_1 G_1 + V_2 G_2 + V_3 G_3 + ... + V_n G_n}{G_1 + G_2 + G_3 + ... + G_n}\]

• G_i = 1/R_i = conductance of branch i (siemens, S)

Including Current Sources

• If parallel branches include current sources, convert them to voltage sources first using source transformation
• Or directly incorporate as:

\[V_{eq} = \frac{\sum\frac{V_i}{R_i} + \sum I_k}{\sum\frac{1}{R_i}}\]

• I_k = current sources (with appropriate sign based on direction)

Substitution Theorem

Fundamental Principle

• Any branch in a network with known voltage V and current I can be replaced by:
  • A voltage source of value V, or
  • A current source of value I, or
  • An impedance Z = V/I
• The replacement does not affect voltages and currents in the rest of the network

Application

• Useful for simplifying complex network analysis
• Branch to be substituted must have known voltage and current
• Applicable to both DC and AC circuits

Compensation Theorem

Fundamental Principle

• When the resistance of a branch changes from R to R + ΔR, the change in current distribution can be found by inserting a compensating voltage source in that branch

Compensating Voltage

\[V_c = -I \cdot \Delta R\]

• V_c = compensating voltage source (V)
• I = original current in the branch before change (A)
• ΔR = change in resistance (Ω)
• Negative sign indicates voltage opposes original current direction

Change in Current

• The change in any branch current due to ΔR can be found using superposition:
• Calculate current due to compensating voltage source alone (with all other sources deactivated)
• Add to original current distribution

Applications

• Analyzing effect of component tolerance variations
• Studying sensitivity of circuits to parameter changes
• Incremental analysis in nonlinear circuits

Tellegen's Theorem

Fundamental Principle

• In any lumped network satisfying Kirchhoff's laws, the sum of instantaneous powers in all branches is zero
• Applicable to linear and nonlinear networks
• Applicable to time-varying networks

Power Form

\[\sum_{k=1}^{b} v_k(t) \cdot i_k(t) = 0\]

• v_k(t) = voltage across branch k at time t (V)
• i_k(t) = current through branch k at time t (A)
• b = total number of branches
• Sign convention: follow associated reference directions

General Form

\[\sum_{k=1}^{b} v_k \cdot i_k = 0\]

• Voltages and currents need not be from the same network, but must satisfy Kirchhoff's laws independently

Implications

• Conservation of energy in electrical networks
• Total power supplied equals total power absorbed
• Valid for any network topology
• Does not depend on element characteristics (linear or nonlinear)

Nodal Analysis

Fundamental Principle

• Application of Kirchhoff's Current Law (KCL) at each node
• Solves for node voltages with respect to a reference node (ground)

Node Voltage Equation

\[\sum_{k} \frac{V_n - V_k}{R_{nk}} + I_{sources} = 0\]

• V_n = voltage at node n (V)
• V_k = voltage at adjacent node k (V)
• R_nk = resistance between nodes n and k (Ω)
• I_sources = algebraic sum of current sources connected to node n (A)

Matrix Form

\[[G][V] = [I]\]

• [G] = conductance matrix (S)
• [V] = node voltage vector (V)
• [I] = current source vector (A)

Conductance Matrix Elements

• Diagonal element G_ii:

\[G_{ii} = \sum \text{(all conductances connected to node i)}\]

• Off-diagonal element G_ij (i ≠ j):

\[G_{ij} = -\sum \text{(conductances between nodes i and j)}\]

Supernode

• Used when a voltage source exists between two non-reference nodes
• Enclose the voltage source and connected nodes in a supernode
• Apply KCL to the entire supernode
• Add constraint equation from the voltage source

\[V_i - V_j = V_s\]

• V_s = voltage of source between nodes i and j

Mesh Analysis (Loop Analysis)

Fundamental Principle

• Application of Kirchhoff's Voltage Law (KVL) around each mesh
• Solves for mesh currents
• Applicable to planar circuits only

Mesh Current Equation

\[\sum_{k} R_k \cdot I_m - \sum_{adj} R_{shared} \cdot I_{adj} = V_{sources}\]

• I_m = mesh current m (A)
• R_k = resistances in mesh m (Ω)
• I_adj = mesh currents in adjacent meshes (A)
• R_shared = resistances shared with adjacent meshes (Ω)
• V_sources = algebraic sum of voltage sources in mesh m (V)

Matrix Form

\[[R][I] = [V]\]

• [R] = resistance matrix (Ω)
• [I] = mesh current vector (A)
• [V] = voltage source vector (V)

Resistance Matrix Elements

• Diagonal element R_ii:

\[R_{ii} = \sum \text{(all resistances in mesh i)}\]

• Off-diagonal element R_ij (i ≠ j):

\[R_{ij} = -\sum \text{(resistances shared between meshes i and j)}\]

• Sign depends on whether mesh currents flow in same or opposite directions through shared resistance

Supermesh

• Used when a current source exists in a branch shared by two meshes
• Create a supermesh by excluding the current source branch
• Apply KVL to the supermesh
• Add constraint equation from the current source

\[I_i - I_j = I_s\]

• I_s = current of source between meshes i and j

Network Equations Summary

Number of Equations Required

• For nodal analysis:

\[N = n - 1\]

• N = number of independent node equations
• n = total number of nodes
• Reference node (ground) reduces equation count by 1

• For mesh analysis:

\[M = b - n + 1\]

• M = number of independent mesh equations
• b = number of branches
• n = number of nodes
• This equals the number of meshes in a planar circuit

Choice of Analysis Method

• Use nodal analysis when:
  • Circuit has more meshes than nodes
  • Circuit contains many current sources
• Use mesh analysis when:
  • Circuit has more nodes than meshes
  • Circuit contains many voltage sources
  • Circuit is planar

Delta-Wye (Δ-Y) and Wye-Delta (Y-Δ) Transformations

Delta to Wye Transformation (Δ → Y)

\[R_1 = \frac{R_b \cdot R_c}{R_a + R_b + R_c}\] \[R_2 = \frac{R_c \cdot R_a}{R_a + R_b + R_c}\] \[R_3 = \frac{R_a \cdot R_b}{R_a + R_b + R_c}\]

• R_a, R_b, R_c = delta-connected resistances (Ω)
• R₁, R₂, R₃ = wye-connected resistances (Ω)
• Each wye resistance = product of two adjacent delta resistances / sum of all delta resistances

Wye to Delta Transformation (Y → Δ)

\[R_a = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1}\] \[R_b = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_2}\] \[R_c = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3}\]

• Each delta resistance = sum of all products of wye resistances taken two at a time / opposite wye resistance

Balanced Networks (Equal Resistances)

• For balanced delta (R_a = R_b = R_c = R_Δ):

\[R_Y = \frac{R_\Delta}{3}\]

• For balanced wye (R₁ = R₂ = R₃ = R_Y):

\[R_\Delta = 3R_Y\]

Application

• Convert non-series-parallel resistor networks to equivalent series-parallel forms
• Simplify bridge circuits
• Three-phase power system analysis
• Transformations preserve terminal characteristics but not internal power dissipation

AC Circuit Theorems

Thévenin's Theorem for AC Circuits

• Replace linear AC network with:
• Thévenin voltage V_TH (phasor, V)
• Thévenin impedance Z_TH (complex, Ω)

\[Z_{TH} = R_{TH} + jX_{TH}\] \[Z_{TH} = \frac{V_{OC}}{I_{SC}}\]

• V_OC = open-circuit phasor voltage
• I_SC = short-circuit phasor current

Norton's Theorem for AC Circuits

• Replace linear AC network with:
• Norton current I_N (phasor, A)
• Norton impedance Z_N = Z_TH (complex, Ω)

\[I_N = \frac{V_{TH}}{Z_{TH}}\]

Load Current for AC Circuits

\[I_L = \frac{V_{TH}}{Z_{TH} + Z_L}\] \[V_L = V_{TH} \cdot \frac{Z_L}{Z_{TH} + Z_L}\]

• Z_L = load impedance (Ω)
• All quantities are phasors

Maximum Power Transfer for AC (Resistive Load)

\[R_L = |Z_{TH}| = \sqrt{R_{TH}^2 + X_{TH}^2}\] \[P_{max} = \frac{|V_{TH}|^2}{2(R_{TH} + |Z_{TH}|)}\]

Maximum Power Transfer for AC (Complex Load)

\[Z_L = Z_{TH}^* = R_{TH} - jX_{TH}\] \[P_{max} = \frac{|V_{TH}|^2}{8R_{TH}}\]

• Load impedance must be complex conjugate of source impedance
• Reactances must be opposite in sign

Two-Port Networks

Impedance Parameters (Z-parameters)

\[V_1 = Z_{11} I_1 + Z_{12} I_2\] \[V_2 = Z_{21} I_1 + Z_{22} I_2\]

• Z₁₁ = V₁/I₁ with I₂ = 0 (output open) = open-circuit input impedance (Ω)
• Z₁₂ = V₁/I₂ with I₁ = 0 = reverse transfer impedance (Ω)
• Z₂₁ = V₂/I₁ with I₂ = 0 = forward transfer impedance (Ω)
• Z₂₂ = V₂/I₂ with I₁ = 0 (input open) = open-circuit output impedance (Ω)

Admittance Parameters (Y-parameters)

\[I_1 = Y_{11} V_1 + Y_{12} V_2\] \[I_2 = Y_{21} V_1 + Y_{22} V_2\]

• Y₁₁ = I₁/V₁ with V₂ = 0 (output shorted) = short-circuit input admittance (S)
• Y₁₂ = I₁/V₂ with V₁ = 0 = reverse transfer admittance (S)
• Y₂₁ = I₂/V₁ with V₂ = 0 = forward transfer admittance (S)
• Y₂₂ = I₂/V₂ with V₁ = 0 (input shorted) = short-circuit output admittance (S)

Hybrid Parameters (h-parameters)

\[V_1 = h_{11} I_1 + h_{12} V_2\] \[I_2 = h_{21} I_1 + h_{22} V_2\]

• h₁₁ = V₁/I₁ with V₂ = 0 = short-circuit input impedance (Ω)
• h₁₂ = V₁/V₂ with I₁ = 0 = open-circuit reverse voltage gain (dimensionless)
• h₂₁ = I₂/I₁ with V₂ = 0 = short-circuit forward current gain (dimensionless)
• h₂₂ = I₂/V₂ with I₁ = 0 = open-circuit output admittance (S)

Transmission Parameters (ABCD parameters)

\[V_1 = A V_2 - B I_2\] \[I_1 = C V_2 - D I_2\]

• A = V₁/V₂ with I₂ = 0 = open-circuit voltage ratio (dimensionless)
• B = -V₁/I₂ with V₂ = 0 = negative short-circuit transfer impedance (Ω)
• C = I₁/V₂ with I₂ = 0 = open-circuit transfer admittance (S)
• D = -I₁/I₂ with V₂ = 0 = negative short-circuit current ratio (dimensionless)

Reciprocal Networks

• For reciprocal (bilateral) networks:

\[Z_{12} = Z_{21}\] \[Y_{12} = Y_{21}\] \[AD - BC = 1\]

Symmetrical Networks

• For symmetrical networks (identical when viewed from either port):

\[Z_{11} = Z_{22}\] \[Y_{11} = Y_{22}\] \[A = D\]

Dependent Sources in Circuit Analysis

Types of Dependent Sources

• Voltage-Controlled Voltage Source (VCVS): V_out = μV_control
• Current-Controlled Voltage Source (CCVS): V_out = rI_control
• Voltage-Controlled Current Source (VCCS): I_out = gV_control
• Current-Controlled Current Source (CCCS): I_out = βI_control

Parameters

• μ = voltage gain (dimensionless)
• r = transresistance (Ω)
• g = transconductance (S)
• β = current gain (dimensionless)

Treatment in Network Theorems

• Dependent sources always remain active when finding Thévenin/Norton equivalents
• Only independent sources are deactivated
• Use test source method or open-circuit/short-circuit method for R_TH calculation
• Superposition theorem: dependent sources remain active for all source combinations

R_TH Calculation with Dependent Sources

• Method 1: Apply test voltage V_test, measure resulting I_test

\[R_{TH} = \frac{V_{test}}{I_{test}}\]

• Method 2: Apply test current I_test, measure resulting V_test

\[R_{TH} = \frac{V_{test}}{I_{test}}\]

• Method 3: Use V_OC and I_SC

\[R_{TH} = \frac{V_{OC}}{I_{SC}}\]

Special Cases and Conditions

Ideal Sources

• Ideal voltage source: internal resistance = 0 Ω
• Ideal current source: internal resistance = ∞ Ω
• Ideal voltage sources cannot be placed in parallel (unless same voltage)
• Ideal current sources cannot be placed in series (unless same current)

Source Combinations

• Voltage sources in series: add algebraically

\[V_{total} = V_1 + V_2 + ... + V_n\]

• Current sources in parallel: add algebraically

\[I_{total} = I_1 + I_2 + ... + I_n\]

Short Circuit and Open Circuit Conditions

• Short circuit: V = 0, I may be any value
• Open circuit: I = 0, V may be any value
• To deactivate voltage source: replace with short circuit (wire)
• To deactivate current source: replace with open circuit (remove)

Power Sign Conventions

• Passive sign convention: current enters positive terminal → power absorbed (positive)
• Current leaves positive terminal → power supplied (negative)
• For sources: typically power supplied is considered positive
• For loads: power absorbed is positive

\[P = V \times I\]

• P > 0: element absorbs power (load)
• P < 0:="" element="" supplies="" power="">