coe288_Lecture3.pptx

CIRCUIT THEORY COE 288 Dr. Bright Yeboah- Akowuah  NODAL ANALYSIS-MESH AND LOOPS  Ohm’s Law In a resistor, the voltage across a resistor is directly proportional to the current flowing through it. V = IR The resistance of an element is measured in unit...

CIRCUIT THEORY COE 288 Dr. Bright Yeboah- Akowuah  NODAL ANALYSIS-MESH AND LOOPS  Ohm’s Law In a resistor, the voltage across a resistor is directly proportional to the current flowing through it. V = IR The resistance of an element is measured in units of Ohms, The higher the resistance, the less current will flow through for a given voltage. Ohm’s law requires conforming to the passive sign convention.  Kirchoff’s Laws  Ohm’s law is not sufficient for circuit analysis.  Kirchoff’s laws complete the needed tools.  There are two laws:  Current law.  Voltage law.  KCL Kirchhoff’s current law is based on conservation of charge. It states that the algebraic sum of currents entering a node (or a closed boundary) is zero. It can be expressed as:  KCL 𝑖 𝐴 +¿ 𝑖 𝐵 + ( − 𝑖𝐶 ) + (− 𝐼 𝐷 ) =0 ¿ (− 𝑖¿¿ 𝐴)+ ( − 𝑖 𝐵 ) + ( 𝐼 𝐶 ) +(𝑖 𝐷)=0 ¿ 𝑖 𝐴 +𝑖 𝐵 =𝑖𝑐 +𝑖 𝐷  KV L Kirchoff’s voltage law is based on conservation of energy. It states that the algebraic sum of currents around a closed path (or loop) is zero. It can be expressed as: Example: Determine  Solution: Finding We apply KVL around the loop Applying Ohm’s law to resistor gives  Nodes Branches and Loops Circuit elements can be interconnected in multiple ways. To understand this, we need to be familiar with some network topology concepts. A branch represents a single element such as a voltage source or a resistor. A node is the point of connection between two or more branches. A loop is any closed path in a circuit.  Network Topology A loop is independent if it contains at least one branch not shared by any other independent loops. Two or more elements are in series if they share a single node and thus carry the same current. Two or more elements are in parallel if they are connected to the same two nodes and thus have the same voltage.  Nodal Analysis If instead of focusing on the voltages of the circuit elements, one looks at the voltages at the nodes of the circuit, the number of simultaneous equations to solve for can be reduced. Given a circuit with n nodes, without voltage sources, the nodal analysis is accomplished via three steps: 1. Select a node as the reference node. Assign voltages v1,v2,…vn to the remaining n-1 nodes, voltages are relative to the reference node. 2. Apply KCL to each of the n − 1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of node voltages. 3. Solve the resulting n − 1 simultaneous equations to obtain the unknown node voltages. The reference, or datum, node is commonly referred to as the ground since its voltage is by default zero.  Applying Nodal Analysis Let’s apply nodal analysis to this circuit to see how it works. This circuit has a node that is designed as ground. We will use that as the reference node (node 0). The remaining two nodes are designed 1 and 2 and assigned voltages v1 and v2. Now apply KCL to each node: At node 1. At node 2.  Apply Nodal Analysis II We can now use OHM’s law to express the unknown currents i1, i2, and i3 in terms of node voltages. In doing so, keep in mind that current flows from high potential to low. From this we get: The last step is to solve the system of equations.  Including voltage sources  Depending on what nodes the source is connected to, the approach varies.  Between the reference node and a non-reference mode:  Set the voltage at the non-reference node to the voltage of the source.  In the example circuit v1 = 10V.  Between two non-reference nodes.  The two nodes form a supernode.  Supernode  A supernode is formed by enclosing a voltage source (dependant or independent) connected between two non- reference nodes and any elements connected in parallel with it.  Why?  Nodal analysis requires applying KCL.  The current through the voltage source cannot be known in advance (Ohm’s law does not apply).  By lumping the nodes together, the current balance can still be described.  In the example circuit node 2 and 3 form a supernode.  The current balance would be:  Or this can be expressed as:  Analysis with a supernode In order to apply KVL to the supernode in the example, the circuit is redrawn as shown. Going around this loop in the clockwise direction gives: Note the following properties of a supernode: 1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages. 2. A supernode has no voltage of its own. 3. A supernode requires the application of both KCL and KVL.  Mesh Analysis  Another general procedure for analyzing circuits is to use the mesh currents as the circuit variables.  Recall:  A loop is a closed path with no node passed more than once.  A mesh is a loop that does not contain any other loop within it.  Mesh analysis uses KVL to find unknown currents.  Mesh analysis is limited in one aspect: It can only apply to circuits that can be rendered planar.  A planar circuit can be drawn such that there are no crossing branches.  Planar versus Nonpalanar  The figure on the left is a nonplanar  The figure on the right is a circuit: The branch with the planar circuit: It can be resistor prevents the circuit from being redrawn to avoid crossing drawn without crossing branches branches  Mesh Analysis Steps  Mesh analysis follows these steps: 1. Assign mesh currents i1,i2,…in to the n meshes. 2. Apply KVL to each of the n mesh currents. 3. Solve the resulting n simultaneous equations to get the mesh currents.  Mesh Analysis Example The above circuit has two paths that are meshes (abefa and bcdeb). The outer loop (abcdefa) is a loop, but not a mesh. First, mesh currents i1 and i2 are assigned to the two meshes. Applying KVL to the meshes: Mesh Analysis with Current Sources The presence of a current source makes the mesh analysis simpler in that it reduces the number of equations. If the current source is located on only one mesh, the current for that mesh is defined by the source. For example:  Supermesh Similar to the case of nodal analysis where a voltage source shared two non-reference nodes, current sources (dependent or independent) that are shared by more than one mesh need special treatment. The two meshes must be joined together, resulting in a supermesh. The supermesh is constructed by merging the two meshes and excluding the shared source and any elements in series with it. A supermesh is required because mesh analysis uses K VL. But the voltage across a current source cannot be known in advance. Intersecting supermeshes in a circuit must be combined to for a larger supermesh.  Creating a Supermesh In this example, a 6A current course is shared between mesh 1 and 2. The supermesh is formed by merging the two meshes. resistor in series with it The current source and the are removed.  Supermesh Example Using the circuit from the last slide: Apply KVL to the supermesh. We next apply KCL to the node in the branch where the two meshes intersect. Solving these two equations we get: Note that the supermesh required using both KVL and KCL.  Cramer’s rule  In circuit analysis, we often encounter a set of simultaneous equations having the form: where  Cramer’s rule…  Equation (1) from the previous slide could be written in matric form as : = ….(2) There several methods of solving such equations, one which is Cramer’s rule  Cramer’s rule…  Cramer’s rule states that the solution to equation 2 from the previous slide is : ……  Cramer’s rule… where ’s are determinants given by  Cramer’s rule…  Notice that is the determinant of matrix A and is the determinant of the matrix formed by replacing the kth column of A by B. It is evident that Cramer’s rule applies only when When , the set of equations has no unique solution, because the equations are linearly dependent. Cramer’s rule: Example 1 Cramer’s rule: Example 1… Cramer’s rule: Example 2… Cramer’s rule: Example 2…  Cramer’s rule: Example 2… We apply this formular to find the determinant This requires that we repeat the first two rows of the matrix Cramer’s rule: Example 2… Cramer’s rule: Example 2… Cramer’s rule: Example 2… Cramer’s rule: Example 3  Determine the voltages at the nodes in circuit below Cramer’s rule: Example 3 solution  Determine the voltage Cramer’s rule: Example 3 solution… Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. Using Cramer’s rule to solve the three equations simultaneously where , and are the determinants to be calculated as follows. Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. Cramer’s rule: Example 3 solution.. THANK YOU