Techniques of Circuit Analysis Qi Xuan Zhejiang University of Technology October 2015 Electric Circuits 1
Structure Terminology Node- Voltage Method Mesh- Current Method Source Transforma:on Thévenin and Norton Equivalents Maximum Power Transfer Superposi:on Electric Circuits 2
Circuit with Realis:c Resistors Sensi&vity analysis: Exploring the effect of a circuit component's value on the circuit's output. Electric Circuits 3
Terminology Planar circuits: the circuits that can be drawn on a plane with no crossing branches. Electric Circuits 4
More Terms for Describing Circuits Name node essen:al node path branch essen:al branch loop mesh planar circuit Defini&on A point where two or more circuit elements join A node where three or more circuit elements join A trace of adjoining basic elements with no elements included more than once A path that connects two nodes A path which connects two essen:al nodes without passing through an essen:al node A path whose last node is the same as the star:ng node A loop that does not enclose any other loops A circuit that can be drawn on a plane with no crossing branches Electric Circuits 5
Example #1 For the circuit in Fig. 4.3, iden:fy: a) all nodes. b) all essen:al nodes. c) all branches. d) all essen:al branches. e) all meshes. f) two paths that are not loops or essen:al branches. g) two loops that are not meshes. Electric Circuits 6
Solu;on for Example #1 Electric Circuits 7
Simultaneous Equa;ons b: number of branches; n: number of nodes Kirchhoff s current law: n-1 independent equa:ons Kirchhoff s voltage law: b-(n-1) independent equa:ons b e : number of essen:al branches n e : number of essen:al nodes Kirchhoff s current law: n e -1 independent equa:ons Kirchhoff s voltage law: b e -(n e -1) independent equa:ons Electric Circuits 8
Example #2 i 1 i 2 Four essen:al nodes: i 3 i 4 i 6 i 5 Seven essen:al branches Three meshes: Electric Circuits 9
Node- Voltage Method 1 2 v 1 v 2 3 Reference node Node 1: Node 2: Electric Circuits 10
Dependent Sources Use the node- voltage method to find the power dissipated in the 5Ω resistor in the circuit. 1 2 v 1 v 2 Node 1: Node 2: Electric Circuits 11
Special Case Electric Circuits 12
Supernode Node 2: Node 3: Electric Circuits 13
Example #3 The circuit has four essen:al nodes: Nodes a and d are connected by an independent voltage source as are nodes b and c. Therefore the problem reduces to finding a single unknown node voltage, because (n e -1)-2 = 1 v b v c Electric Circuits 14
Mesh- Current Method b e =3 n e =2 b e -(n e -1)=2 Electric Circuits 15
Dependent Sources Mesh 1: Mesh 2: Mesh 3: Constraint: Electric Circuits 16
Special Case Mesh a: Mesh c: Mesh b: Constraint: Electric Circuits 17
Supermesh Electric Circuits 18
Node- Voltage vs. Mesh- Current Does one of the methods result in fewer simultaneous equa;ons to solve? Does the circuit contain supernodes? If so, using the node- voltage method will permit you to reduce the number of equa:ons to be solved. Does the circuit contain supermeshes? If so, using the mesh- current method will permit you to reduce the number of equa:ons to be solved. Will solving some por:on of the circuit give the requested solu:on? If so, which method is most efficient for solving just the per:nent por:on of the circuit? Electric Circuits 19
Source Transforma:ons i L R L i L R L Electric Circuits 20
Example #4 (a) Use source transforma:ons to find the voltage v o in the circuit. (b) Find the power developed by the 250 V voltage source. (c) Find the power developed by the 8 A current source. Electric Circuits 21
Solu;on for Example #4 Electric Circuits 22
Thévenin and Norton Equivalents Thévenin and Norton equivalents are circuit simplifica:on techniques that focus on terminal behavior and thus are extremely valuable aids in analysis. Thevenin equivalent circuit: an independent voltage source V Th in series with a resistor R Th, which replaces an interconnec:on of sources and resistors. Norton equivalent circuit: consists of an independent current source in parallel with the Norton equivalent resistance, Electric Circuits 23
Thévenin Equivalent v oc V Th = v oc Thévenin voltage V Th equals to the open- circuit voltage in the original circuit. i sc R Th = V Th /i sc = v oc /i sc Thévenin resistance R Th is the ratio of the open-circuit voltage to the short-circuit current. Electric Circuits 24
Example #5 V Th = v 1 = 32 V R Th = V Th /i sc = 8 A Thévenin Equivalents Electric Circuits 25
Norton Equivalent Electric Circuits 26
More on Thevenin Equivalent A voltage source is deac:vated by replacing it with a short circuit. A current source is deac:vated by replacing it with an open circuit. Electric Circuits 27
Maximum Power Transfer Power Transfer l The first emphasizes the efficiency of the power transfer. u Power utility systems are a good example of this type because they are concerned with the generation, transmission, and distribution of large quantities of electric power. l The second basic type of system emphasizes the amount of power transferred. l Communication and instrumentation systems are good examples because in the transmission of information, or data, via electric signals, the power available at the transmitter or detector is limited Electric Circuits 28
p is maximized when the derivative is zero, thus we have Electric Circuits 29
Sensi:ve Analysis Electric Circuits 30
Summary Basic terms: node, essen:al node, path, branch, essen:al branch, mesh, and planar circuit. Node- voltage and mesh- current methods Source transforma:ons Thévenin and Norton equivalents Maximum power transfer superposi:on Electric Circuits 31