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EE484: Mathematical Circuit Theory + Analysis Node and Mesh Equations

EE484: Mathematical Circuit Theory + Analysis Node and Mesh Equations. By: Jason Cho 20076166. Overview. Review of Kirchhoff’s Circuit Laws Node Equations Mesh Equations Why these methods? Summary Questions. Definitions. Node : a point where two or more elements or branches connect.

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EE484: Mathematical Circuit Theory + Analysis Node and Mesh Equations

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  1. EE484: Mathematical Circuit Theory + AnalysisNode and Mesh Equations By: Jason Cho 20076166

  2. Overview • Review of Kirchhoff’s Circuit Laws • Node Equations • Mesh Equations • Why these methods? • Summary • Questions

  3. Definitions Node: a point where two or more elements or branches connect. a point where all the connecting branches have the same voltage. Branch: any path between two nodes. Mesh: a set of branches that make up a closed loop path in a circuit where the removal of one branch will result in an open loop.

  4. Kirchhoff’s Circuit Laws • Kirchhoff’s Current Law (KCL) .. which states that the algebraic sum of all currents entering or leaving a node is zero for all time instances. This law can be derived by using the Divergence Theorem, Gauss’ Law, and Ampere’s Law.

  5. Kirchhoff’s Circuit Laws (cont’d) • Enclose a node with a Gaussian surface, and apply Gauss’ Law, and the Divergence Theorem … (1) J = current density (vector) • Take the divergence of Ampere’s Law B = magnetic field (vector) D = electric displacement (vector) … (2) ρ = charge density (scalar)

  6. Kirchhoff’s Circuit Laws (cont’d) • Substitute Eq. 2 into Eq. 1 • Apply conservation of charge So the final equation states that the sum of all current densities entering and leaving the enclosed surface is always zero.

  7. Kirchhoff’s Circuit Laws (cont’d) Intuitively, the divergence of a vector field measures the magnitude of the vector fields source or sink. Integrating all these sinks and sources inside this closed surface yields the net flow. Since our answer was zero, this means the sum of all sinks and the sum of all sources are equal.

  8. Kirchhoff’s Circuit Laws (cont’d) • Kirchhoff’s Voltage Law (KVL) .. which states that the algebraic sum of all the voltage drops or rises in any closed loop path is zero for all time instances. This law can be derived from Faraday’s Law of Induction.

  9. Kirchhoff’s Circuit Laws (cont’d) • Faraday’s Law of Induction. • Define a closed loop path in a circuit. E = electric field B = magnetic field Since there is no fluctuating magnetic field linked to the loop, the equation becomes The LHS of the above equation is also known as the electric potential equation. So the above equation just states that the electric potential in the closed loop path is 0.

  10. Node Equations Node voltage analysis is one of many methods used in circuit analysis. This method involves a series of equations known as node equations. Each equation is expressed using Kirchhoff’s Current Law and Ohm’s Law. Therefore, this method can be thought of as a system of KCL equations, in terms of the node voltages. This method allows one to solve for the currents and voltages at any point in a circuit.

  11. Node Equations (cont’d) V1 V2 GND Step 1: Identify and label the nodes. Step 2: Determine a reference node. Step 3: Apply KCL at each non-reference node. @ V1: @ V2:

  12. Node Equations (cont’d) Step 4: Solve the system of equations. V1 V2 GND

  13. Mesh Equations Mesh current analysis is another method used to solve for the voltages and currents at any point in a circuit. Mesh current analysis involves a series of equations known as mesh equations. Each equation is expressed using Kirchhoff’s Voltage Law, and Ohm’s Law. Therefore, this method can be thought of as a system of KVL equations, in terms of the mesh currents. The equations are similar to KVL in the way that it is also written as the algebraic sum of voltage rises or drops around a mesh.

  14. Mesh Equations (cont’d) Step 1: Identify and label the mesh loops, and choose direction of current flow. Step 2: Apply KVL to each mesh loop. Loop 1: Loop 2:

  15. Mesh Equations (cont’d) Step 4: Solve the system of equations. Net current flow down the middle branch is (-1A) + 5A = 4A (upwards).

  16. Why? Consider a larger network. V1 V1 V2 V2 I1 I3 I5 l3 I2 I4 GND GND • Branch current method: 5 different branch currents 2 non-reference nodes, 3 independent loops  3 KVL + 2 KCL = 5 equations with 5 variables!! NOO!! • Mesh current method • Or Node voltage method: 3 mesh loops, 2 non-reference nodes • 3 KVL or 2 KCL = 3 equations with 3 variables • OR 2 equations with 2 variables.

  17. Summary • Revisted Kirchhoff’s Circuit Laws • Kirchhoff’s Current Law (KCL) • Kirchhoff’s Voltage Law (KVL) • Node Equations • Mesh Equations • Why these methods?

  18. Thank You!

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