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ECE 3336 Introduction to Circuits & Electronics Note Set #5 The Mesh-Current Method 1 Spring 2015, TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

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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. 2

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Closed Contour (loop #1) A LOOP - not a mesh 3

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Closed Contour (mesh #1) This is a MESH = an empty loop 4

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Closed Contour (mesh #2) This is a MESH = an empty loop 5

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Closed Contour (mesh #3) This is a MESH = an empty loop 6

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Closed Contour (loop #2) A LOOP - not a mesh 7

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Closed Path (loop #3) Total # of closed contours: three meshes three loops A LOOP - not a mesh 8

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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. 9

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Mesh-Current Method (MCM) Details The Mesh-Current Method steps are: 1.Ensure that the circuit is planar (redraw if necessary) 2.Define the mesh currents, by labeling them. This includes showing the polarity of each mesh current. 3.Apply KVL for each mesh. 4.Write an equation for each current or voltage upon which dependent sources depend. 10

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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. -v 1 +v 2 =0 It results from energy conservation We will always go around mesh (loop) clockwise. positive sign assigned for a voltage drop negative sign assigned to a voltage rise. 11

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KVL an Example (from Set #2) 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 Entering Negative See mnemonics 12

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Number of equations in MCM Determine that first i.e. before beginning a problem. For n m meshes, we need to write n m 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 n m +v equations. 13

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Solving Circuits by MCM This circuit is already drawn in planar form. Most circuits (here) will be like that. Ensure that the circuit is planar (redraw if necessary) Write an equation for each current or voltage dependent source (if any). Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Identify all meshes. 14

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MCM – 1 st Example Write an equation for each current or voltage dependent source (if any). Define meshes currents: labels and direction i.e. polarity for each mesh current. Apply KVL for each mesh. Identify all meshes. 15 iAiA iBiB iCiC Mesh current notation. Reference polarity is clockwise (choice is arbitrary) Choose your convention and do not change it.

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MCM – 1 st Example 16 iAiA iBiB iCiC 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 R 1, two mesh currents, i A and i B, flow. In resistor R 3, two mesh currents, i B and i C, flow. The mesh currents are not real (cannot be measured). They give the net value and direction of the current (this can be measured).

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17 iAiA iBiB iCiC Here, we have labeled the branch currents and voltages for each term of the equation. MCM - mesh A i2i2 i1i1 Apply KVL for each mesh. 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, i 2 = i A, i 1 = (i A – i B )

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A Solving Circuits by MCM (mesh A) Ensure that the circuit is planar (redraw if necessary) Write an equation for each current or voltage for dependent sources. Define the mesh currents: labels and directions i.e. polarity for each mesh current. Apply KVL for each mesh. (3 equations) 18 iAiA iBiB iCiC Ohms Law & KVL In branches net currents ONLY

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Apply KVL for mesh B. Solving Circuits by MCM (mesh B and C) Apply KVL for mesh C. 19 iAiA iBiB iCiC BC

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Solving Circuits by MCM (all meshes: A, B, C) We have the same number of equations (3) as unknowns (3). solve 20 iAiA iBiB iCiC

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Find the current i x Circuit is planar 21 MCM – 2 nd Example

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Circuit is planar 22 MCM – 2 nd Example Defined the mesh currents for the three meshes. Use clockwise directions for all meshes. iAiA iBiB iCiC

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KVL equations for meshes A, B, and C. 3 eqs. 5 unknowns? Dependent sources 23 Solving circuit - 2 nd Example iAiA iBiB iCiC

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Now, we have: 5 equations for 5 unknowns. Complete MCM eqs. Dependent sources 24 iAiA iBiB iCiC

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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. 25

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Sequence of Steps in MCM with Current Sources 26 Write an equation for each current or voltage dependent source (if any). 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. Identify all meshes. Solution depends on where in the circuit the current source will appear: as a part of one mesh as a part of two meshes.

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MCM with Current Source in One Mesh w/o Sharing 27 iAiA iBiB iCiC iDiD Define meshes currents: labels and direction i.e. polarity for each mesh current. Identify all meshes. Apply KVL for each mesh. Problem: The voltage across current sources can be anything. V is1 =?V is2 =? Solution: KVL in meshes A and D are not needed. The goal was to find mesh currents and we already know them.

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Solution Equations: Mesh current i A is equal to the current source i S1, Mesh current i D is equal to but opposite in sign of the current source i S iAiA iBiB iCiC iDiD

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29 MCM with Current Source Shared by Two Meshes iAiA iBiB iCiC 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 i B and i C are not equal to i S. Define meshes in the circuit

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Define the voltage across the current source to be v X. 30 iAiA iBiB iCiC MCM with Current Source Use a Voltage on i s +-+- vxvx Write KVL equations for meshes B and C, using v X.

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Eliminate the new variable v X by using KVL in mesh B and C. Add the B equation to the C equation to get: 31 MCM with Current Source Eliminate the Voltage v x iAiA iBiB iCiC +-+- vxvx Supermesh Equation Supermesh

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The current source determines the difference between i B and i C. From KCL in Node 1 we have i C -i S -i B =0 32 MCM with Current Source Find i s as functions of Mesh Currents i B and i C iAiA iBiB iCiC 1 Constraint Equation Supermesh Equation

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That gives us four equations for four unknowns. 33 iAiA iBiB iCiC MCM with Current Source Complete Equations One dependent source

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34 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.

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