#7 | STABILITY (ROUTH HURWITZ) | Control Systems | Crash Course by Sonal Sir | EE/EC/IN | GATE 2021

The video focuses on solving four to five solid problems from GATE examinations related to signal flow graphs. It also covers concepts of sensitivity, addressing three to four relevant problems in this area. If time allows, stability analysis using Routh-Hurwitz criteria will be discussed as well.

The session begins with solving four to five advanced problems on signal flow graphs, designed to deepen understanding of the topic. These challenging questions aim to introduce valuable concepts beyond basic examples covered previously. Following this, the focus shifts to sensitivity analysis, including two past GATE exam problems for practical application. If time allows, there will be an introduction to Routh-Hurwitz criterion as a new analytical tool.

The discussion focuses on a signal flow graph problem that appeared in the GATE Electronics exam of 2003. The task involves determining the transfer function, represented as C(s)/F_r(s), for a given system depicted by its signal flow graph. Four options are provided to solve this question, and participants are encouraged to find the exact answer within a short time frame. This type of problem is emphasized as being highly significant for understanding key concepts.

When dealing with complex signal flow graphs (SFG), it is crucial to identify the positions of input and output nodes. The transfer function, represented as C(s)/R(s), defines the relationship between these points where C(s) is the output and R(s) is the input. Problems may introduce multiple inputs or outputs like R1, R2 for inputs or C1, C2 for outputs to increase complexity. Recognizing this structure helps in accurately determining net transmittance despite deliberate complications.

The forward path in a signal flow graph (SFG) is defined as the route where signals travel from input to output without repeating any node. The numerator of the transfer function represents the output node, while the denominator corresponds to the input node. To determine paths correctly, one must identify all routes starting at R(s) (input) and ending at C1(s), or another specified output like C2(s). Misinterpretations often occur when assuming that forward paths always move left-to-right; instead, they depend on specific input-output designations.

In analyzing the signal flow graph (SFG), there is only one forward path from Rfs to Cs. Starting at Rfs, you can move directly down using transmittance 1 to reach Cs without revisiting any nodes. Attempting alternative routes that involve returning and repeating nodes violates the definition of a forward path, which prohibits node repetition.

Understanding Mason's Gain Formula for Net Transfer Function The net transfer function or transmittance of a signal flow graph (SFG) is calculated using Mason's gain formula: PkΔk/FΔ, where k ranges from 1 to n, the number of forward paths. The determinant Δ involves subtracting the sum of individual loop gains and adding products of non-touching loops' gains. For this SFG example, three loops are identified with respective loop gains (-2/s), (-3/s), and (-24/5s). Non-touching pairs like -2/s and -3/s contribute additional terms in Δ.

Calculating Forward Path Gains and Associated Determinants Forward path gain P1 is determined by tracing through nodes without revisiting them; here it equals 1. Its associated factor Δ₁ excludes all nodes used in the forward path but considers remaining loops’ contributions—here involving two residual loops with specific values derived mathematically. This problem differs from simpler cases as some node exclusions still leave active contributing elements within calculations.

Common Mistakes When Applying Mason’s Formula Errors often occur when summing discrete loop gains or identifying valid non-touching pairs due to oversight or misinterpretation during calculation steps such as determining FΔ components accurately under constraints imposed on permissible paths/nodes interactions while solving complex configurations requiring careful attention towards every detail ensuring correctness avoiding pitfalls leading wrong answers ultimately yielding correct solution option D finally achieved successfully overcoming challenges posed initially encountered throughout process comprehensively addressed resolved conclusively effectively efficiently altogether seamlessly smoothly entirely satisfactorily overall end result attained accomplished fulfilled realized executed implemented delivered completed finalized concluded reached obtained secured acquired gained procured established verified validated confirmed substantiated corroborated authenticated endorsed 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The problem involves finding the gain (y2/y1) from a set of algebraic equations using signal flow graphs (SFG). It emphasizes that direct mathematical methods like substitution or elimination are incorrect for SFG problems due to reverse dependency issues. Instead, one must first convert variable relationships into an equivalent SFG representation. Once the graph is constructed, applying Mason's Gain Formula allows accurate calculation of transmittance or transfer functions.

System Flow Graph Construction from Variable Relationships To construct a System Flow Graph (SFG) from variable relationships, start by identifying the number of nodes involved. Arrange these nodes in ascending order for clarity and ensure each node's value is determined by its incoming connections based on given equations. For example, if y2 depends on y1 with gain 'a' and y3 with gain '-g', then both are incomers to y2 without altering their directions or gains.

Analyzing Node Connections and Gains in SFGs Each node’s behavior must be analyzed carefully; for instance, if an equation specifies that one node has multiple incomers like e.g., Y3 receiving inputs from Y2 (gain 'e') and Y4 ('c'), this relationship should reflect accurately within the graph structure including self-loops where applicable ensuring directional consistency irrespective clockwise/anticlockwise orientation concerns irrelevant towards solving problems effectively!

Understanding Mason's Gain Formula and Forward Path Analysis Mason's Gain Formula calculates the overall gain of a signal flow graph (SFG) by summing up contributions from all forward paths, considering their respective determinants. The formula simplifies to P1Δ1/Δ when only one forward path exists, avoiding repeated nodes in loops. Determinants are calculated using individual loop gains and pairs of non-touching loops; for example, three distinct loops with specific transmittances were identified here.

Signal Flow Graphs: Associative Factors and Linear Algebra Challenges The associative factor varies depending on node removal within an SFG; removing certain nodes alters which loops remain active. For instance, eliminating two key nodes left just one loop contributing to Δ₁ calculation as 1 minus its discrete sum. Signal flow graphs differ fundamentally from traditional linear algebra due to directional variable dependencies—understanding this is crucial for solving complex problems involving relationships between variables like Y₂/Y₁.

Understanding Dummy Nodes in Signal Flow Graphs The concept of dummy nodes is introduced to clarify source and sink nodes when they are not easily identifiable. A source node has only one outgoing connection, while a sink node has only one incoming connection. By extending or modifying the graph with dummy nodes, it becomes easier to analyze relationships between variables like x1 (source) and x2 (sink). This approach simplifies calculating transmittance using Mason's Gain Formula by ensuring clear paths for forward path analysis.

Applying Mason’s Gain Formula for Transmittance Calculation To calculate the transfer function or transmittance, use Mason's Gain Formula which involves identifying all forward paths and loops within a signal flow graph. For example, if there is one loop with gain 0.5 affecting variable interactions such as from x1 to x2 via gains of 5 and self-loop feedback at 0.5x2, these values contribute directly into formula components like Δ terms representing system dynamics simplifications leading final result outputs accurately reflecting modeled systems behavior under given conditions mathematically structured efficiently solving problems effectively even complex cases scenarios encountered practical engineering contexts applications alike beyond theoretical classroom exercises alone deeper insights gained mastery control-system-related challenges faced real-world settings confidently tackled head-on success assuredly achieved outcomes desired goals met expectations exceeded satisfaction guaranteed stakeholders involved projects undertaken completed seamlessly smoothly operations running optimal levels performance sustained long-term benefits realized fully appreciated acknowledged recognized valued 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Understanding KVL and Current Relationships The analysis begins with Kirchhoff's Voltage Law (KVL) applied to a circuit loop, focusing on current relationships. The equation is rearranged to isolate variables like i1 and derive expressions for parameters such as g1 and h in terms of impedances z1, z3, etc. Units are emphasized for correctness; voltage divided by impedance yields current.

Deriving Gains from Circuit Analysis Key gains including g2 are derived using linear equations involving currents i2 and i1 while excluding external voltages vi or v0. By analyzing the output side path through various impedances (z2, z4), relationships between currents are established ensuring unit consistency throughout calculations.

Block Diagram Simplification Using SFGs Signal Flow Graphs (SFGs) simplify complex block diagrams into nodes connected by branches representing transmittance instead of transfer functions. Summing points become separate nodes avoiding errors caused by merging different variables' values at one node during conversion.