ECA1212 Introduction to Electrical & Electronics Engineering Chapter 2: Circuit Analysis Techniques by Muhazam Mustapha, September 2011

Learning Outcome Understand and perform calculation on circuits with mesh and nodal analysis techniques Be able to transform circuits based on Thevenin’s or Norton’s Theorem as necessary By the end of this chapter students are expected to:

Chapter Content Mesh Analysis Nodal Analysis Source Conversion Thevenin’s Theorem Norton’s Theorem

Mash Analysis Mesh CO2

Mesh Analysis Assign a distinct current in clockwise direction to each independent closed loop of network. Indicate the polarities of the resistors depending on individual loop. [*] If there is any current source in the loop path, replace it with open circuit – apply KVL in the next step to the resulting bigger loop. Use back the current source when solving for current. Steps: CO2

Mesh Analysis Apply KVL on each loop: –Current will be the total of all directions –Polarity of the sources is maintained Solve the simultaneous equations. Steps: (cont) CO2

Mesh Analysis Example: [Boylestad 10 th Ed. E.g modified] R1R1 IaIa IbIb 2V 2Ω2Ω R2R2 1Ω1Ω 6V R3R3 4Ω4Ω a b I1I1 I3I3 I2I2 CO2

Mesh Analysis Example: (cont) Loop a : 2 = 2I a +4(I a −I b ) = 6I a −4I b Loop b : −6 = 4(I b −I a )+I b = −4I a +5I b After solving: I a = −1A, I b = −2A Hence: I 1 = 1A, I 2 = −2A, I 3 = 1A CO2

Noodle Analysis Nodal CO2

Nodal Analysis Determine the number of nodes. Pick a reference node then label the rest with subscripts. [*] If there is any voltage source in the branch, replace it with short circuit – apply KCL in the next step to the resulting bigger node. Apply KCL on each node except the reference. Solve the simultaneous equations. CO2

Nodal Analysis Example: [Boylestad 10 th Ed. E.g modified] R1R1 4A 2Ω2Ω R2R2 6Ω6Ω 2A R3R3 12Ω I1I1 I2I2 I3I3 a b CO2

Nodal Analysis Node a : Node b : After solving: V a = 6V, V b = − 6A Hence: I 1 = 3A, I 2 = 1A, I 3 = −1A Example: (cont) CO2

Mesh vs Nodal Analysis Mesh: Start with KVL, get a system of simultaneous equations in term of current. Nodal: Start with KCL, get a system of simultaneous equations on term of voltage. Mesh: KVL is applied based on a fixed loop current. Nodal: KCL is applied based on a fixed node voltage. CO2

Mesh vs Nodal Analysis Mesh: Current source is an open circuit and it merges loops. Nodal: Voltage source is a short circuit and it merges nodes. Mesh: More popular as voltage sources do exist physically. Nodal: Less popular as current sources do not exist physically except in models of electronics circuits. CO2

Thevenin’s Theorem CO2

Thevenin’s Theorem Statement: Network behind any two terminals of linear DC circuit can be replaced by an equivalent voltage source and an equivalent series resistor Can be used to reduce a complicated network to a combination of voltage source and a series resistor CO2

Calculate the Thevenin’s resistance, R Th, by switching off all power sources and finding the resulting resistance through the two terminals: –Voltage source: remove it and replace with short circuit –Current source: remove it and replace with open circuit Calculate the Thevenin’s voltage, V Th, by switching back on all powers and calculate the open circuit voltage between the terminals. Thevenin’s Theorem CO2

Thevenin’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω 9V Convert the following network into its Thevenin’s equivalent: CO2

Thevenin’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω R Th calculation: CO2

Thevenin’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω 9V V Th calculation: CO2

Thevenin’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 2Ω2Ω 6V Thevenin’s equivalence: CO2

Norton’s Theorem CO2

Norton’s Theorem Statement: Network behind any two terminals of linear DC circuit can be replaced by an equivalent current source and an equivalent parallel resistor Can be used to reduce a complicated network to a combination of current source and a parallel resistor CO2

Calculate the Norton’s resistance, R N, by switching off all power sources and finding the resulting resistance through the two terminals: –Voltage source: remove it and replace with short circuit –Current source: remove it and replace with open circuit Calculate the Norton’s voltage, I N, by switching back on all powers and calculate the short circuit current between the terminals. Norton’s Theorem CO2

Norton’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω 9V Convert the following network into its Norton’s equivalent: CO2

Norton’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω R N calculation: CO2

Norton’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 3Ω3Ω 6Ω6Ω 9V I N calculation: CO2

Norton’s Theorem Example: [Boylestad 10 th Ed. E.g modified] 2Ω2Ω 3A Norton’s equivalence: OR, We can just take the Thevenin’s equivalent and calculate the short circuit current. CO2

Maximum Power Consumption An element is consuming the maximum power out of a network if its resistance is equal to the Thevenin’s or Norton’s resistance. CO2

Source Conversion Use the relationship between Thevenin’s and Norton’s source to convert between voltage and current sources. 2Ω2Ω 3A 2Ω2Ω 6V V = IR CO2