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1. ECE 3336 Introduction to Circuits & Electronics Note Set #4 The Node-Voltage Method Fall 2013, TUE&TH4:00-5:300 pm Dr. Wanda Wosik
2. Nodes Important for Node Voltage Method KCL A node is defined as a point where two or more components are connected. We connect components with wires, which do not cause voltage drop (Rwire=0 Ω). NODE VOLTAGE METHOD: we will use KCL in each of the node but it will be done in a systematic way. branches
3. Nodes in the Circuits We have 3 NODES Remember that wires have R≈0Ω no voltage drop along the wires V=0V Nodes – extended definitions: Essential Node – a place where three or more components meet Reference Node – an essential node that we choose as a reference point for voltages
4. The Node-Voltage Method (NVM) The Node-Voltage Method (NVM)allows us to write all the equationsneeded to solve a circuit. Node voltages are defined/measured with respect to a reference node. All other parameters such as current or voltage and finally power can be then determined. It is a very important method for more complicated circuitssince Ohm’s Laws together with KLC and KVL alone would be very cumbersome there. Polarities and consistency in using signs will be very important here.
5. The Node-Voltage Method (NVM) The Node-Voltage Method steps are: Findall essentialnodes. Define one essential node as the reference node. Define the node voltageswith respect to the reference node. Label them. Apply KCL for each non-reference essential node. Write an equation for each current or voltage (independent or dependent) as needed.
6. Kirchhoff’s Current Law (reminder) The algebraic (or signed i.e with directions) summation of currents through a closed surface (such as a node) must equal zero. It means that there is no charge build-up there.
7. KCL in the Example Circuit (reminder) We will use a convention (steady not to flip-flop) + for currents leaving the node – for those that are entering. For this circuit with our notation in KCL, we have the following equation:
8. NVM – 1st Example 1. Find all essential nodes. 2. Define one essential node as the reference node. 3. Define the node voltages with respect to the reference node. 4. Label them. 5. Apply KCL for each non-reference essential node. B A VB VA
9. NVMApply KLC to All the Nodes A Ohm’s Law B Ohm’s Law direction? B For current direction A For current direction direction? VB VA
10. NVM – 2nd Example
11. NVMApply KLC to All the Nodes 1. Find all essential nodes. 2. Define one essential node as the reference node. 3. Define the node voltages with respect to the reference node. 4. Label them. 5. Apply KCL for each non-reference essential node. A B VB VA VC C
12. Solving the Circuit NVM: KCL at 3 nodes give three equations We need two more equations for dependent sources VB VA VC
13. Solving the Circuit For depending sources VB VA We have 5 equations For 5 unknown VC
14. Solving the Circuit For depending sources VB VA VC
15. Number of Node-Voltage Equations in NVM It is veryimportantto determine the number of needed equations before beginning a problem. The rule: For ne essential nodes we write ne-1 equations. This is because we do not write a KCL equation for the reference node. If there are dependent sources(n) in the circuit then for each such source there is one variable i.e one equation. Therefore, total number of equations is ne-1+nequations.
16. NVM for Circuits with Voltage Sources at the Nodes KCL cannot be used directly on a voltage source since we do not know the current there. This current cannot be obtained from Ohm’s Law (unlike for resistors). However, NVM will be used quite easily when voltage sources at the nodes.
17. The Node-Voltage Method (NVM) The Node-Voltage Method steps are: Findall essentialnodes. Define one essential node as the reference node. Define the node voltageswith respect to the reference node. Label them. Apply KCL for each non-reference essential node. Write an equation for each current or voltage (independent or dependent) as needed.
18. Voltage Sources and the NVMSolutions Voltage sources in the circuit will changeNVM steps depending how these sources are connected: In series with another element in the branch Between an essential node and the reference ground Between two non-reference essential nodes.
19. NVM – Voltage Source in the Branch The voltage source vS is in series with the resistor R2. - +
20. NVM – Voltage Source in the Branch 1. Findall essential nodes. 2. Defineone essential node as the reference node 3. Definethe node voltages with respect to the reference node. 4. Labelthem. 5. Apply KCL for each non-reference essential node. Not essential Node A B - + VA VB
21. NVM – Voltage Source in the Branch 1. Define a current ix flowing through the voltage sourceVs (and R2). 2. Make a loop 3. Assign the voltage Vtemp to the nonessential node Vtemp= VB– VS 4. Find the current ix Not essential Node A B - + ix Vtemp VA VB
22. NVM – Voltage Source in the Branch Note that ix has a direction assumed for the source and THAT DOES NOT change when we go to node B for KCL in NVM This is ix<0 but in NVM current is leaving the node Not essential Node A B - + ix Vtemp VA VB
23. NVM – Voltage Source in the Branch The solution has 2 final equations Not essential Node A B - + ix Vtemp VA VB
24. NVM – Voltage Source Between the Reference Node and Another Essential Node The voltage source vS is between two essential nodes.
25. NVM – Vs at the Essential Node 1. Findall essential nodes. 2. Defineone essential node as the reference node 3. Definethe node voltages with respect to the reference node. 4. Labelthem. 5. Apply KCL for each non-reference essential node. A B C VA VB VC
26. NVM – Vs at the Essential Node The goal of NVM is to find the node voltages. Here in Node B we already know the voltage=vs. So the current through vsdoes not need to be calculated (this current would be another unknown). A B C VA VB VC
27. NVM – Vs at the Essential Node Solution A B C VA VB VC
28. NVM – Voltage Source Between Two Non-Reference Essential Nodes - +
29. NVM – Voltage Source b/w Nodes 1. Findall essential nodes. 2. Defineone essential node as the reference node 3. Definethe node voltages with respect to the reference node. 4. Labelthem. 5. Apply KCL for each non-reference essential node. ABC - + VA VB VC
30. NVM – Voltage Source b/w Nodes Define a current (ix) through the voltage source. As before this current will not change direction when we move from node B to C. Write NVM for nodes B and C. ABC - + VA VB VC ix
31. NVM – Voltage Source b/w Nodes Since the current (ix) cannot change (it has + or – sign at B and C, respectively) we can add both equations. no ix here ABC - + VA VB VC ix
32. NVM – Voltage Source b/w Nodes no ix here through the source Currents flow out of the B&C nodes. B&C is called SUPERNODE SUPERNODE equation ABC - + VA VB VC ix
33. NVM – Voltage Source b/w Nodes No explicit ix here through the source Voltage b/w B&C can be found from the loop ABC - + VA VB VC ix
34. Final Solution – NVM for Voltage Source b/w Nodes KCL as in basic NVM SupernodeEquation takes all currents around the viltage source Constraint Equation relates two nodes by voltage source ABC - + VA VB VC ix
35. Number of Equations for Node-Voltage With or Without Voltage Sources Number of equations will be the same w/ or w/o voltage 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 voltage source matters in what steps will be used. Series connection with a resistor allows to find expression for the current flowing also through the source. Location b/w the essential node and reference node gives the node voltage equal to the source (polarity is SUPER important) Location b/w two essential nodes requires a supernodeequation and a constrain equation.