Network Theorems & Topology

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### Network Theorems with Dependent Sources Dependent sources (voltage or current controlled) are common in active circuits (e.g., op-amps, transistors). Their presence modifies the application of some theorems. #### Superposition Theorem - **Rule:** Not directly applicable to dependent sources. Only independent sources are turned off. - **Method:** 1. Turn off all independent sources (voltage sources become short circuits, current sources become open circuits). 2. Calculate the contribution of each independent source to the desired response. 3. The dependent sources remain active throughout the analysis. 4. Sum the individual contributions. #### Thevenin's Theorem - **Procedure with Dependent Sources:** 1. **Find $R_{Th}$ (Thevenin Resistance):** - Turn off all independent sources. - Apply an external voltage source $V_x$ (or current source $I_x$) at the terminals where $R_{Th}$ is to be found. - Calculate the current $I_x$ (or voltage $V_x$) flowing from the applied source. - $R_{Th} = V_x / I_x$. 2. **Find $V_{Th}$ (Thevenin Voltage):** - This is the open-circuit voltage at the terminals. - Use standard circuit analysis techniques (KVL, KCL, mesh, nodal) to find $V_{OC}$ with all sources (independent and dependent) active. - **Equivalent Circuit:** A voltage source $V_{Th}$ in series with a resistor $R_{Th}$. #### Norton's Theorem - **Procedure with Dependent Sources:** 1. **Find $R_N$ (Norton Resistance):** - Same as $R_{Th}$, so $R_N = R_{Th}$. (Turn off independent sources, apply $V_x$ or $I_x$, calculate ratio). 2. **Find $I_N$ (Norton Current):** - This is the short-circuit current flowing through the terminals. - Short-circuit the terminals and use standard circuit analysis to find $I_{SC}$ with all sources active. - **Equivalent Circuit:** A current source $I_N$ in parallel with a resistor $R_N$. #### Maximum Power Transfer Theorem - **Rule:** For a circuit with dependent sources, maximum power is transferred to a load $R_L$ when $R_L = R_{Th}$ (or $R_L = R_N$). - **Maximum Power:** $P_{max} = V_{Th}^2 / (4 R_{Th})$. - **Note:** $R_{Th}$ must be calculated considering dependent sources as described above. #### Reciprocity Theorem - **Rule:** Generally NOT applicable to circuits containing dependent sources, as they are not linear bilateral elements in the same way. - **Applies to:** Linear, bilateral networks with only independent sources. #### Tellegen's Theorem - **Rule:** Applicable to ANY lumped network (linear or non-linear, time-invariant or time-varying, passive or active, with or without dependent sources). - **Statement:** The sum of the products of voltage and current for each branch in a network is zero at any instant of time. $$\sum_{k=1}^b v_k(t) i_k(t) = 0$$ where $v_k$ and $i_k$ are the voltage and current of the $k$-th branch, adhering to the passive sign convention. - **Significance:** A powerful general theorem used to derive other network properties and conservation laws. #### Millman's Theorem - **Rule:** Applicable to parallel branches with voltage sources in series with resistors. - **Statement:** The voltage across a set of parallel branches, each containing a voltage source $V_k$ in series with a resistance $R_k$, is given by: $$V_{eq} = \frac{\sum_{k=1}^N (V_k / R_k)}{\sum_{k=1}^N (1 / R_k)}$$ - **Equivalent Resistance:** $R_{eq} = 1 / \sum_{k=1}^N (1 / R_k)$. - **Note:** Can be used with dependent sources if they are part of a branch that can be converted to a Thevenin equivalent (voltage source in series with resistance). ### Network Topology and Graph Concepts Network topology is the study of the geometric properties of a circuit without regard to the actual components or their values. It simplifies complex circuits into graphs. #### Basic Definitions - **Graph (G):** A collection of nodes (vertices) and branches (edges) connecting them. - **Node (Vertex):** A junction point where two or more circuit elements connect. - **Branch (Edge):** A circuit element (resistor, source, etc.) connecting two nodes. - **Oriented Graph (Directed Graph):** A graph where each branch has a defined direction, usually indicating assumed current flow. - **Path:** A sequence of branches that can be traversed from one node to another without repeating any node. - **Loop:** A closed path (starts and ends at the same node). - **Tree:** A subgraph of a connected graph that includes all nodes of the original graph but contains no loops. A tree with $N$ nodes has $N-1$ branches. - **Co-tree:** The set of branches in the original graph that are not part of the chosen tree. - **Twig (Tree Branch):** A branch belonging to a tree. - **Link (Co-tree Branch):** A branch belonging to the co-tree. #### Fundamental Matrices - **Incidence Matrix (A):** - Describes the connection between branches and nodes. - Size: $N \times B$ (Nodes $\times$ Branches). - $A_{ij} = 1$ if branch $j$ leaves node $i$. - $A_{ij} = -1$ if branch $j$ enters node $i$. - $A_{ij} = 0$ if branch $j$ is not incident to node $i$. - **Reduced Incidence Matrix ($A_r$):** Formed by removing one row (corresponding to the reference node). Used for nodal analysis. - $A_r \mathbf{i}_b = \mathbf{0}$ (KCL in terms of branch currents). - **Fundamental Cut-Set Matrix (Bf):** - A **cut-set** is a minimal set of branches whose removal divides the graph into two separate parts. - A **fundamental cut-set** is formed by replacing one twig of a tree with a link, which creates a unique loop. - Size: $(N-1) \times B$ (number of twigs $\times$ Branches). - Each row corresponds to a fundamental cut-set defined by a twig. - $B_f \mathbf{v}_b = \mathbf{0}$ (KCL in terms of branch voltages, dual of KVL). - Used in nodal analysis formulations. - **Fundamental Loop Matrix (Bf):** - A **fundamental loop (f-loop)** is a unique loop formed by adding one link to a tree. - Size: $L \times B$ (number of links $\times$ Branches), where $L = B - (N-1)$. - Each row corresponds to a fundamental loop defined by a link. - $B_f \mathbf{v}_b = \mathbf{0}$ (KVL in terms of branch voltages). - Used in mesh analysis formulations. #### Circuit Analysis using Graph Theory Graph theory provides systematic methods for forming KCL and KVL equations. #### Nodal Analysis with Graph Theory 1. **Choose a reference node.** 2. **Form the reduced incidence matrix ($A_r$)**. 3. **Apply KCL at each non-reference node:** $A_r \mathbf{i}_b = \mathbf{0}$. 4. **Relate branch currents to node voltages using Ohm's Law:** $\mathbf{i}_b = \mathbf{Y}_b (\mathbf{v}_b - \mathbf{v}_{sources})$, where $\mathbf{Y}_b$ is the branch admittance matrix. 5. **Express branch voltages in terms of node voltages:** $\mathbf{v}_b = A_r^T \mathbf{v}_n$. 6. **Substitute to get:** $A_r \mathbf{Y}_b A_r^T \mathbf{v}_n = A_r \mathbf{i}_{sources}$. - The matrix $A_r \mathbf{Y}_b A_r^T$ is the **nodal admittance matrix ($Y_{node}$)**. #### Mesh Analysis with Graph Theory 1. **Choose a tree and identify all fundamental loops.** 2. **Form the fundamental loop matrix ($B_f$)**. 3. **Apply KVL around each fundamental loop:** $B_f \mathbf{v}_b = \mathbf{0}$. 4. **Relate branch voltages to mesh currents using Ohm's Law:** $\mathbf{v}_b = \mathbf{Z}_b (\mathbf{i}_b - \mathbf{i}_{sources})$, where $\mathbf{Z}_b$ is the branch impedance matrix. 5. **Express branch currents in terms of mesh currents:** $\mathbf{i}_b = B_f^T \mathbf{i}_m$. 6. **Substitute to get:** $B_f \mathbf{Z}_b B_f^T \mathbf{i}_m = B_f \mathbf{v}_{sources}$. - The matrix $B_f \mathbf{Z}_b B_f^T$ is the **mesh impedance matrix ($Z_{mesh}$)**. #### Advantages of Graph Theory in Circuit Analysis - Provides a systematic and algorithmic approach for formulating circuit equations. - Especially useful for large, complex networks where manual equation formulation is prone to errors. - Forms the basis for computer-aided circuit analysis tools (SPICE, etc.).