EE484: Mathematical Circuit Theory + Analysis Node and Mesh Equations By: Jason Cho

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

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. 3

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. 4

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

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. 6

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. 7

Kirchhoff’s Circuit Laws 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. (cont’d) 8 This law can be derived from Faraday’s Law of Induction.

Kirchhoff’s Circuit Laws Define a closed loop path in a circuit. (cont’d) E = electric field B = magnetic field 9 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. Faraday’s Law of Induction.

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. 10

11 Node Equations (cont’d) Step 1: Identify and label the nodes. Step 2: Determine a reference node. Step 3: Apply KCL at each non-reference node. GND V 1 V 2 : V2V2

12 Node Equations (cont’d) Step 4: Solve the system of equations. GND V1V1 V2V2

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. 13

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

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. Branch current method: V1V1 V2V2 I2I2 I4I4 I3I3 I5I5 I1I1 5 different branch currents 2 non-reference nodes, 3 independent loops GND Mesh current method Or Node voltage method: 3 mesh loops, 2 non-reference nodes V1V1 V2V2 GND l3l3 NOO!!  3 KVL + 2 KCL = 5 equations with 5 variables!!  3 KVL or 2 KCL = 3 equations with 3 variables OR 2 equations with 2 variables.

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

Thank You! 18