ECE 3336 Introduction to Circuits & Electronics Note Set #5 The Mesh-Current Method Spring 2015, TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

Mesh vs. Closed Loop Closed Loop – a closed contour in a circuit (may but does not have to follow components) Mesh – a closed path that does not enclose any other closed paths (empty) Planar Circuit – a circuit that can be drawn in just one plane. If some connecting wires are on another plane then the circuit is not planar and MSM will not apply.

Closed Contour (loop #1) A LOOP - not a mesh

Closed Contour (mesh #1) This is a MESH = an empty loop

Closed Contour (mesh #2) This is a MESH = an empty loop

Closed Contour (mesh #3) This is a MESH = an empty loop

Closed Contour (loop #2) A LOOP - not a mesh

Closed Path (loop #3) A LOOP - not a mesh Total # of closed contours: three meshes three loops

The Mesh-Current Method (MCM) The Mesh-Current Method (MCM) is a systematic way to write only necessary but complete set of equations required to solve a circuit. In complicated circuits MCM gives all the equations that we need, and no extras. It simplifies solutions. Any other current or voltage can be found from these mesh-currents. MCM works in planar circuits.

Mesh-Current Method (MCM) Details The Mesh-Current Method steps are: Ensure that the circuit is planar (redraw if necessary) Define the mesh currents, by labeling them. This includes showing the polarity of each mesh current. Apply KVL for each mesh. Write an equation for each current or voltage upon which dependent sources depend.

Kirchhoff’s Voltage Law (KVL) The algebraic (or signed i.e. directions defined for voltages) summation of voltages around a closed contour (loop or mesh) must equal zero. It results from energy conservation -v1+v2=0 We will always go around mesh (loop) clockwise. positive sign assigned for a voltage drop negative sign assigned to a voltage rise.

KVL an Example (from Set #2) Entering Negative See mnemonics positive sign assigned for a voltage drop negative sign assigned to a voltage rise. KVL, when starting at the bottom, will give the following equation: Mesh must be empty Entering Positive

Number of equations in MCM Determine that first i.e. before beginning a problem. For nm meshes, we need to write nm equations i.e. one KVL equation for each mesh. Dependent sources increase the number of equations i.e. one eq. for each source (v) so we need nm+v equations.

Solving Circuits by MCM Ensure that the circuit is planar (redraw if necessary) Identify all meshes. Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Write an equation for each current or voltage dependent source (if any). This circuit is already drawn in planar form. Most circuits (here) will be like that.

MCM – 1st Example iA iB iC Identify all meshes. Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Write an equation for each current or voltage dependent source (if any). Mesh current notation. Reference polarity is clockwise (choice is arbitrary) Choose your convention and do not change it. iA iB iC

MCM – 1st Example A mesh current is defined as a current that flows only around that mesh. In places where the meshes share their branches, both mesh currents flow simultaneously. In resistor R1, two mesh currents, iA and iB, flow. In resistor R3, two mesh currents, iB and iC, flow. The mesh currents are not real (cannot be measured). They give the net value and direction of the current (this can be measured). {Note: it would be good if the mesh current arrows (the three red circular arrows) were animated in this slide.} iC iA iB

MCM - mesh A i2 i1 iA iB iC Apply KVL for each mesh. Here, we have labeled the branch currents and voltages for each term of the equation. A branch current is the current in the component, which is the summation of the mesh currents that go through that branch (signs). Note that in this circuit, i2 = iA, i1 = (iA – iB) Apply KVL for each mesh. i2 i1 iA iB iC

Solving Circuits by MCM (mesh A) Ensure that the circuit is planar (redraw if necessary) Define the mesh currents: labels and directions i.e. polarity for each mesh current. Apply KVL for each mesh. (3 equations) Write an equation for each current or voltage for dependent sources. Ohms Law & KVL In branches net currents ONLY A iA iB iC

Solving Circuits by MCM (mesh B and C) Apply KVL for mesh B. Apply KVL for mesh C. B C iA iB iC

Solving Circuits by MCM (all meshes: A, B, C) We have the same number of equations (3) as unknowns (3). solve iA iB iC

MCM – 2nd Example Find the current ix Circuit is planar

MCM – 2nd Example Defined the mesh currents for the three meshes. Use clockwise directions for all meshes. Circuit is planar iB iA iC

Solving circuit - 2nd Example KVL equations for meshes A, B, and C. iB iA 3 eqs. 5 unknowns? iC Dependent sources

Complete MCM eqs. iB iA iC Dependent sources Now, we have: 5 equations for 5 unknowns. iA iC

Circuits with Current Sources A current source has a voltage across it determined by what it is connected to (not from Ohm’s law). Current source can be a part of one mesh only or shared by two meshes.

Sequence of Steps in MCM with Current Sources Identify all meshes. Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Problem: The voltage across a current source can be anything; the voltage depends on the rest of the circuit. Write an equation for each current or voltage dependent source (if any). Solution depends on where in the circuit the current source will appear: as a part of one mesh as a part of two meshes.

MCM with Current Source in One Mesh w/o Sharing Identify all meshes. Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Problem: The voltage across current sources can be anything. Vis1=? Vis2=? Solution: KVL in meshes A and D are not needed. The goal was to find mesh currents and we already know them. iD iA iB iC

Solution iD iA iB iC Equations: Mesh current iA is equal to the current source iS1, Mesh current iD is equal to but opposite in sign of the current source iS2.. Equations: iD iA iB iC

MCM with Current Source Shared by Two Meshes Define meshes in the circuit Write KVL equations for the three meshes, A, B, and C. Difficulties writing the equations for meshes B and C, because we do not know voltage across the current source and we know that the currents iB and iC are not equal to iS. iA iB iC

MCM with Current Source Use a Voltage on is Define the voltage across the current source to be vX. Write KVL equations for meshes B and C, using vX. + - vx iA iB iC

MCM with Current Source Eliminate the Voltage vx Eliminate the new variable vX by using KVL in mesh B and C. Add the B equation to the C equation to get: Supermesh Equation + - vx iA iB iC Supermesh

MCM with Current Source Find is as functions of Mesh Currents iB and iC The current source determines the difference between iB and iC. From KCL in Node 1 we have iC-iS-iB=0 Constraint Equation Supermesh Equation 1 • iA iB iC

MCM with Current Source Complete Equations One dependent source That gives us four equations for four unknowns. iA iB iC

Number of Equations for Mesh Current Method With or Without Current Sources Number of equations will be the same w/ or w/o current sources. However, dependent sources will always add one equation (per source) since the source has to be defined by its parameters used in the circuit. Location of the current source matters in what steps will be used. Current sources in a mesh not shared with the others will be your mesh currents. Current sources in a mesh, which is shared with the other meshes require supermesh. Here KVL will include all voltages from adjacent meshes and the constrain equation will be used to calculate mesh currents related to the source.