3: Nodal Analysis. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 1 / 12. 3: Nodal Analysis
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- Charla Griffith
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1 Current Floating Voltage Dependent E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 1 / 12
2 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
3 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. There are two ways to do this: (1) Nodal Analysis - systematic; always works (2) Circuit Manipulation - ad hoc; but can be less work and clearer E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
4 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. There are two ways to do this: (1) Nodal Analysis - systematic; always works (2) Circuit Manipulation - ad hoc; but can be less work and clearer Reminders: A node is all the points in a circuit that are directly interconnected. We assume the interconnections have zero resistance so all points within a node have the same voltage. Five nodes: A,,E. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
5 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. There are two ways to do this: (1) Nodal Analysis - systematic; always works (2) Circuit Manipulation - ad hoc; but can be less work and clearer Reminders: A node is all the points in a circuit that are directly interconnected. We assume the interconnections have zero resistance so all points within a node have the same voltage. Five nodes: A,,E. Ohm s Law: V BD = IR 5 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
6 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. There are two ways to do this: (1) Nodal Analysis - systematic; always works (2) Circuit Manipulation - ad hoc; but can be less work and clearer Reminders: A node is all the points in a circuit that are directly interconnected. We assume the interconnections have zero resistance so all points within a node have the same voltage. Five nodes: A,,E. Ohm s Law: V BD = IR 5 KVL:V BD = V B V D E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
7 Aim of Nodal Analysis Current Floating Voltage Dependent The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need. There are two ways to do this: (1) Nodal Analysis - systematic; always works (2) Circuit Manipulation - ad hoc; but can be less work and clearer Reminders: A node is all the points in a circuit that are directly interconnected. We assume the interconnections have zero resistance so all points within a node have the same voltage. Five nodes: A,,E. Ohm s Law: V BD = IR 5 KVL:V BD = V B V D KCL: Total current exiting any closed region is zero. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 2 / 12
8 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
9 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
10 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
11 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
12 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
13 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
14 Nodal Analysis Stage 1: Current Floating Voltage Dependent To find the voltage at each node, the first step is to label each node with its voltage as follows (1) Pick any node as the voltage reference. Label its voltage as0v. (2) If any fixed voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then go back to step (2) until all nodes are labelled. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 3 / 12
15 Nodal Analysis Stage 2: Current Floating Voltage Dependent The second step is to write down a KCL equation for each node labelled with a variable by setting the total current flowing out of the node to zero. For a circuit withn nodes ands voltage sources you will haven S 1 simultaneous equations to solve. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 4 / 12
16 Nodal Analysis Stage 2: Current Floating Voltage Dependent The second step is to write down a KCL equation for each node labelled with a variable by setting the total current flowing out of the node to zero. For a circuit withn nodes ands voltage sources you will haven S 1 simultaneous equations to solve. We only have one variable: X 8 1 k + X 0 2 k + X ( 2) 3 k = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 4 / 12
17 Nodal Analysis Stage 2: Current Floating Voltage Dependent The second step is to write down a KCL equation for each node labelled with a variable by setting the total current flowing out of the node to zero. For a circuit withn nodes ands voltage sources you will haven S 1 simultaneous equations to solve. We only have one variable: X 8 1 k + X 0 2 k + X ( 2) 3 k = 0 Numerator for a resistor is always of the formx V N wherev N is the voltage on the other side of the resistor. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 4 / 12
18 Nodal Analysis Stage 2: Current Floating Voltage Dependent The second step is to write down a KCL equation for each node labelled with a variable by setting the total current flowing out of the node to zero. For a circuit withn nodes ands voltage sources you will haven S 1 simultaneous equations to solve. We only have one variable: X 8 1 k + X 0 2 k + X ( 2) 3 k = 0 (6X 48)+3X +(2X +4) = 0 Numerator for a resistor is always of the formx V N wherev N is the voltage on the other side of the resistor. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 4 / 12
19 Nodal Analysis Stage 2: Current Floating Voltage Dependent The second step is to write down a KCL equation for each node labelled with a variable by setting the total current flowing out of the node to zero. For a circuit withn nodes ands voltage sources you will haven S 1 simultaneous equations to solve. We only have one variable: X 8 1 k + X 0 2 k + X ( 2) 3 k = 0 (6X 48)+3X +(2X +4) = 0 11X = 44 X = 4 Numerator for a resistor is always of the formx V N wherev N is the voltage on the other side of the resistor. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 4 / 12
20 Current Current Floating Voltage Dependent Current sources cause no problems. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
21 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
22 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
23 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
24 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X andy. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
25 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X andy. (3) Write equations X X 2 + X Y 3 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
26 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X andy. (3) Write equations X X 2 + X Y 3 = 0 Y X 3 +( 1) = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
27 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X andy. (3) Write equations X X 2 + X Y 3 = 0 Y X 3 +( 1) = 0 Ohm s law works OK if all resistors are inkω and all currents inma. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
28 Current Current Floating Voltage Dependent Current sources cause no problems. (1) Pick reference node. (2) Label nodes: 8,X andy. (3) Write equations X X 2 + X Y 3 = 0 Y X 3 +( 1) = 0 Ohm s law works OK if all resistors are inkω and all currents inma. (4) Solve the equations: X = 6, Y = 9 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 5 / 12
29 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
30 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
31 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
32 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8,X E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
33 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8,X andx +2since it is joined tox via a voltage source. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
34 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8,X andx +2since it is joined tox via a voltage source. (3) Write KCL equations but count all the nodes connected via floating voltage sources as a single super-node giving one equation X X 2 + (X+2) 0 3 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
35 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8,X andx +2since it is joined tox via a voltage source. (3) Write KCL equations but count all the nodes connected via floating voltage sources as a single super-node giving one equation X X 2 + (X+2) 0 3 = 0 Ohm s law always involves the difference between the voltages at either end of a resistor. (Obvious but easily forgotten) E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
36 Floating Voltage Current Floating Voltage Dependent Floating voltage sources have neither end connected to a known fixed voltage. We have to change how we form the KCL equations slightly. (1) Pick reference node. (2) Label nodes: 8,X andx +2since it is joined tox via a voltage source. (3) Write KCL equations but count all the nodes connected via floating voltage sources as a single super-node giving one equation X X 2 + (X+2) 0 3 = 0 (4) Solve the equations: X = 4 Ohm s law always involves the difference between the voltages at either end of a resistor. (Obvious but easily forgotten) E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 6 / 12
37 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
38 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
39 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 Or using conductances: (X V 1 )G 1 +(X V 2 )G 2 +(X V 3 )G 3 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
40 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 Or using conductances: (X V 1 )G 1 +(X V 2 )G 2 +(X V 3 )G 3 = 0 X(G 1 +G 2 +G 3 ) = V 1 G 1 +V 2 G 2 +V 3 G 3 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
41 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 Or using conductances: (X V 1 )G 1 +(X V 2 )G 2 +(X V 3 )G 3 = 0 X(G 1 +G 2 +G 3 ) = V 1 G 1 +V 2 G 2 +V 3 G 3 X = V 1G 1 +V 2 G 2 +V 3 G 3 G 1 +G 2 +G 3 = Vi G i Gi E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
42 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 Or using conductances: (X V 1 )G 1 +(X V 2 )G 2 +(X V 3 )G 3 = 0 X(G 1 +G 2 +G 3 ) = V 1 G 1 +V 2 G 2 +V 3 G 3 X = V 1G 1 +V 2 G 2 +V 3 G 3 G 1 +G 2 +G 3 = Vi G i Gi VoltageX is the average ofv 1, V 2, V 3 weighted by the conductances. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
43 Weighted Average Circuit Current Floating Voltage Dependent A very useful sub-circuit that calculates the weighted average of any number of voltages. KCL equation for nodex: X V 1 R 1 + X V 2 R 2 + X V 3 R 3 = 0 Still works ifv 3 = 0. Or using conductances: (X V 1 )G 1 +(X V 2 )G 2 +(X V 3 )G 3 = 0 X(G 1 +G 2 +G 3 ) = V 1 G 1 +V 2 G 2 +V 3 G 3 X = V 1G 1 +V 2 G 2 +V 3 G 3 G 1 +G 2 +G 3 = Vi G i Gi VoltageX is the average ofv 1, V 2, V 3 weighted by the conductances. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 7 / 12
44 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value 6 in decimal. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
45 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
46 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
47 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. X = 1 2 V V V G 2 = 1 R 2 = 1 2k = 1 2 ms,... E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
48 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. X = 1 2 V V V = 1 7 (4V 2 +2V 1 +V 0 ) G 2 = 1 R 2 = 1 2k = 1 2 ms,... E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
49 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. X = 1 2 V V V = 1 7 (4V 2 +2V 1 +V 0 ) butv i = 5 b i since it connects to either0 V or5 V G 2 = 1 R 2 = 1 2k = 1 2 ms,... E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
50 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. X = 1 2 V V V = 1 7 (4V 2 +2V 1 +V 0 ) butv i = 5 b i since it connects to either0 V or5 V = 5 7 (4b 2 +2b 1 +b 0 ) = 5 7 b G 2 = 1 R 2 = 1 2k = 1 2 ms,... E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
51 Digital-to-Analog Current Floating Voltage Dependent A 3-bit binary number,b, has bit-weights of4,2and1. Thus110 has a value6in decimal. If we label the bitsb 2 b 1 b 0, thenb = 4b 2 +2b 1 +b 0. We useb 2 b 1 b 0 to control the switches which determine whetherv i = 5V orv i = 0V. ThusV i = 5b i. Switches shown forb = 6. X = 1 2 V V V = 1 7 (4V 2 +2V 1 +V 0 ) butv i = 5 b i since it connects to either0 V or5 V = 5 7 (4b 2 +2b 1 +b 0 ) = 5 7 b G 2 = 1 R 2 = 1 2k = 1 2 ms,... So we have made a circuit in whichx is proportional to a binary numberb. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 8 / 12
52 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
53 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
54 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
55 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
56 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
57 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) (4) Write KCL equations: X U 10 + X 10 + X Y 20 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
58 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) (4) Write KCL equations: X U 10 + X 10 + X Y 20 = 0 Y X 20 +I S + Y 15 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
59 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) (4) Write KCL equations: X U 10 + X 10 + X Y 20 = 0 Y X 20 +I S + Y 15 = 0 (5) Solve all three equations to findx,y andi S in terms ofu: E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
60 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) (4) Write KCL equations: X U 10 + X 10 + X Y 20 = 0 Y X 20 +I S + Y 15 = 0 (5) Solve all three equations to findx,y andi S in terms ofu: X = 0.1U, Y = 1.5U, I S = 0.18U E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
61 Dependent Current Floating Voltage Dependent A dependent voltage or current source is one whose value is determined by voltages or currents elsewhere in the circuit. These are most commonly used when modelling the behaviour of transistors or op-amps. Each dependent source has a defining equation. In this circuit: I S = 0.2W ma wherew is in volts. (1) Pick reference node. (2) Label nodes: 0, U,X andy. (3) Write equation for the dependent source,i S, in terms of node voltages: I S = 0.2(U X) (4) Write KCL equations: X U 10 + X 10 + X Y 20 = 0 Y X 20 +I S + Y 15 = 0 (5) Solve all three equations to findx,y andi S in terms ofu: X = 0.1U, Y = 1.5U, I S = 0.18U Note that the value ofu is assumed to be known. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 9 / 12
62 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
63 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
64 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
65 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5,X, X +3 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
66 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5,X, X +3and X +V S. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
67 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5,X, X +3and X +V S. (3) Write equation for the dependent source,v S, in terms of node voltages: V S = 10J = 10 X+V S V S = X 5 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
68 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5,X, X +3and X +V S. (3) Write equation for the dependent source,v S, in terms of node voltages: V S = 10J = 10 X+V S V S = X 5 (4) Write KCL equations: all nodes connected by floating voltage sources and all components connecting these nodes are in the same super-node X+V S X 5 + X+3 5 = 0 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
69 Dependent Voltage Current Floating Voltage Dependent The value of the highlighted dependent voltage source isv S = 10J Volts wherej is the indicated current inma. (1) Pick reference node. (2) Label nodes: 0, 5,X, X +3and X +V S. (3) Write equation for the dependent source,v S, in terms of node voltages: V S = 10J = 10 X+V S V S = X 5 (4) Write KCL equations: all nodes connected by floating voltage sources and all components connecting these nodes are in the same super-node X+V S X 5 + X+3 5 = 0 (5) Solve the two equations: X = 1 andv S = 2 E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 10 / 12
70 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
71 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
72 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
73 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
74 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
75 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
76 Universal Nodal Analysis Current Floating Voltage Dependent (1) Pick any node as the voltage reference. Label its voltage as0v. Label any dependent sources withv S, I S,... (2) If any voltage sources are connected to a labelled node, label their other ends by adding the value of the source onto the voltage of the labelled end. (3) Pick an unlabelled node and label it withx, Y,..., then loop back to step (2) until all nodes are labelled. (4) For each dependent source, write down an equation that expresses its value in terms of other node voltages. (5) Write down a KCL equation for each normal node (i.e. one that is not connected to a floating voltage source). (6) Write down a KCL equation for each super-node. A super-node consists of a set of nodes that are joined by floating voltage sources and includes any other components joining these nodes. (7) Solve the set of simultaneous equations that you have written down. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 11 / 12
77 Summary Current Floating Voltage Dependent Nodal Analysis Simple Circuits (no floating or dependent voltage sources) E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 12 / 12
78 Summary Current Floating Voltage Dependent Nodal Analysis Simple Circuits (no floating or dependent voltage sources) Floating Voltage use supernodes: all the nodes connected by floating voltage sources (independent or dependent) E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 12 / 12
79 Summary Current Floating Voltage Dependent Nodal Analysis Simple Circuits (no floating or dependent voltage sources) Floating Voltage use supernodes: all the nodes connected by floating voltage sources (independent or dependent) Dependent Voltage and Current Label each source with a variable Write extra equations expressing the source values in terms of node voltages Write down the KCL equations as before E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 12 / 12
80 Summary Current Floating Voltage Dependent Nodal Analysis Simple Circuits (no floating or dependent voltage sources) Floating Voltage use supernodes: all the nodes connected by floating voltage sources (independent or dependent) Dependent Voltage and Current Label each source with a variable Write extra equations expressing the source values in terms of node voltages Write down the KCL equations as before Mesh Analysis (in most textbooks) Alternative to nodal analysis but doesn t work for all circuits No significant benefits ignore it E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 12 / 12
81 Summary Current Floating Voltage Dependent Nodal Analysis Simple Circuits (no floating or dependent voltage sources) Floating Voltage use supernodes: all the nodes connected by floating voltage sources (independent or dependent) Dependent Voltage and Current Label each source with a variable Write extra equations expressing the source values in terms of node voltages Write down the KCL equations as before Mesh Analysis (in most textbooks) Alternative to nodal analysis but doesn t work for all circuits No significant benefits ignore it For further details see Hayt et al. Chapter 4. E1.1 Analysis of Circuits ( ) Nodal Analysis: 3 12 / 12
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