Principles of Electric Circuits - Floyd © Copyright 2006 Prentice-Hall Chapter Chapter 9 9 Chapter 9
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PowerPoint Presentation© Copyright 2006 Prentice-Hall
© Copyright 2006 Prentice-Hall
Summary
Summary
Circuit analysis methods in Chapter 9 require use of simultaneous equations.
Simultaneous Equations
To simplify solving simultaneous equations, they are usually set up in standard form. Standard form for two equations with two unknowns is
coefficients
variables
constants
© Copyright 2006 Prentice-Hall
Simultaneous Equations
A circuit has the following equations. Set up the equations in standard form.
Rearrange so that variables and their coefficients are in order and put constants on the right.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Algebraic substitution
© Copyright 2006 Prentice-Hall
Solve for IB in the first equation:
Substitute for IB into the second equation:
Rearrange and solve for IA.
IA = 9.53 mA
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
If you wanted to find IB in the previous example, you can substitute the result of IA back into one of the original equations and solve for IB. Thus,
2.10 mA
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
The method of determinants is another approach to finding the unknowns. The characteristic determinant is formed from the coefficients of the unknowns.
Write the characteristic determinant for the equations. Calculate its value.
1.134
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
To solve for an unknown by determinants, form the determinant for a variable by substituting the constants for the coefficients of the unknown. Divide by the characteristic determinant.
To solve for x2:
© Copyright 2006 Prentice-Hall
9.53 mA
2.10 mA
© Copyright 2006 Prentice-Hall
Summary
Summary
Many scientific calculators allow you to enter a set of equations and solve them “automatically”. The calculator method will depend on your particular calculator, but you will always write the equations in standard form first and then input the number of equations, the coefficients, and the constants. Pressing the Solve key will show the values of the unknowns.
Solving Simultaneous Equations
© Copyright 2006 Prentice-Hall
Summary
Summary
In the branch current method, you can solve for the currents in a circuit using simultaneous equations.
Branch current method
Assign a current in each branch in an arbitrary direction.
Show polarities according to the assigned directions.
Apply KVL in each closed loop.
Apply KCL at nodes such that all branches are included.
Solve the equations from steps 3 and 4.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Assign a current in each branch in an arbitrary direction.
A
+
+
+
Apply KVL in each closed loop. Resistors are entered in kW in this example.
Apply KCL at nodes such that all branches are included.
Solve the equations from steps 3 and 4
(see next slide).
© Copyright 2006 Prentice-Hall
Solving: I1 = 9.53 mA, I2 = 7.43 mA, I3 = -2.10 mA
The negative result for I3 indicates the actual current direction is opposite to the assumed direction.
(Continued)
20.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
In the loop current method, you can solve for the currents in a circuit using simultaneous equations.
Loop current method
Steps:
Assign a current in each nonredundant loop in an arbitrary direction.
Show polarities according to the assigned direction of current in each loop.
Apply KVL around each closed loop.
Solve the resulting equations for the loop currents.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Loop current method
Assign a current in each nonredundant loop in an arbitrary direction.
+
+
+
Apply KVL around each closed loop. Resistors are entered in kW in this example.
+
Loop A
Loop B
Notice that the polarity of R3 is based on loop B and is not the same as in the branch current method.
22.unknown
23.unknown
24.unknown
© Copyright 2006 Prentice-Hall
26.unknown
27.unknown
28.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
The loop current method can be applied to more complicated circuits, such as the Wheatstone bridge. The steps are the same as shown previously.
Loop current method applied to circuits with more than two loops
Loop A
Loop B
Loop C
The advantage to the loop method for the bridge is that it has only 3 unknowns.
31.unknown
32.unknown
33.unknown
34.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
Write the loop current equation for Loop A in the Wheatstone bridge:
Loop A
Loop B
Loop C
© Copyright 2006 Prentice-Hall
Summary
Summary
In the node voltage method, you can solve for the unknown voltages in a circuit using KCL.
Node voltage method
Determine the number of nodes.
Select one node as a reference. Assign voltage designations to each unknown node.
Assign currents into and out of each node except the reference node.
Apply KCL at each node where currents are assigned.
Express the current equations in terms of the voltages and solve for the unknown voltages using Ohm’s law.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
A
Solve the same problem as before using the node voltage method.
There are 4 nodes. A is the one unknown node.
B is selected as the reference node.
Currents are assigned into and out of node A.
Apply KCL at node A (for this case).
Write KCL in terms of the voltages (next slide).
B
40.unknown
© Copyright 2006 Prentice-Hall
© Copyright 2006 Prentice-Hall
Simultaneous equations
The solution of a matrix consisting of an array of coefficients and constants for a set of simultaneous equations.
A closed current path in a circuit.
One current path that connects two nodes.
Key Terms
The junction of two or more components.
A set of n equations containing n unknowns, where n is a number with a value of 2 or more.
46.psd
© Copyright 2006 Prentice-Hall
Quiz
In a set of simultaneous equations, the coefficient that is written a1,2 appears in
a. the first equation
b. the second equation
© Copyright 2006 Prentice-Hall
Quiz
In standard form, the constants for a set of simultaneous equations are written
a. in front of the first variable
b. in front of the second variable
c. on the right side of the equation
d. all of the above
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
To solve simultaneous equations, the minimum number of independent equations must be at least
a. two
b. three
c. four
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
In the equation a1,1x1 +a1,2x2 = b1, the quantity b1 represents
a. a constant
b. a coefficient
c. a variable
© Copyright 2006 Prentice-Hall
a. 4
b. 14
c. 24
d. 34
© Copyright 2006 Prentice-Hall
Quiz
The characteristic determinant for a set of simultaneous equations is formed using
a. only constants from the equations
b. only coefficients from the equations
c. both constants and coefficients from the equations
d. none of the above
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Quiz
A negative result for a current in the branch method means
a. there is an open path
b. there is a short circuit
c. the result is incorrect
d. the current is opposite to the assumed direction
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Quiz
To solve a circuit using the loop method, the equations are first written for each loop by applying
a. KCL
b. KVL
© Copyright 2006 Prentice-Hall
Quiz
A Wheatstone bridge can be solved using loop equations. The minimum number of nonredundant loop equations required is
a. one
b. two
c. three
d. four
© Copyright 2006 Prentice-Hall
Quiz
In the node voltage method, the equations are developed by first applying
a. KCL
b. KVL
© Copyright 2006 Prentice-Hall
© Copyright 2006 Prentice-Hall
Summary
Summary
Circuit analysis methods in Chapter 9 require use of simultaneous equations.
Simultaneous Equations
To simplify solving simultaneous equations, they are usually set up in standard form. Standard form for two equations with two unknowns is
coefficients
variables
constants
© Copyright 2006 Prentice-Hall
Simultaneous Equations
A circuit has the following equations. Set up the equations in standard form.
Rearrange so that variables and their coefficients are in order and put constants on the right.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Algebraic substitution
© Copyright 2006 Prentice-Hall
Solve for IB in the first equation:
Substitute for IB into the second equation:
Rearrange and solve for IA.
IA = 9.53 mA
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
If you wanted to find IB in the previous example, you can substitute the result of IA back into one of the original equations and solve for IB. Thus,
2.10 mA
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
The method of determinants is another approach to finding the unknowns. The characteristic determinant is formed from the coefficients of the unknowns.
Write the characteristic determinant for the equations. Calculate its value.
1.134
© Copyright 2006 Prentice-Hall
Solving Simultaneous Equations
To solve for an unknown by determinants, form the determinant for a variable by substituting the constants for the coefficients of the unknown. Divide by the characteristic determinant.
To solve for x2:
© Copyright 2006 Prentice-Hall
9.53 mA
2.10 mA
© Copyright 2006 Prentice-Hall
Summary
Summary
Many scientific calculators allow you to enter a set of equations and solve them “automatically”. The calculator method will depend on your particular calculator, but you will always write the equations in standard form first and then input the number of equations, the coefficients, and the constants. Pressing the Solve key will show the values of the unknowns.
Solving Simultaneous Equations
© Copyright 2006 Prentice-Hall
Summary
Summary
In the branch current method, you can solve for the currents in a circuit using simultaneous equations.
Branch current method
Assign a current in each branch in an arbitrary direction.
Show polarities according to the assigned directions.
Apply KVL in each closed loop.
Apply KCL at nodes such that all branches are included.
Solve the equations from steps 3 and 4.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Assign a current in each branch in an arbitrary direction.
A
+
+
+
Apply KVL in each closed loop. Resistors are entered in kW in this example.
Apply KCL at nodes such that all branches are included.
Solve the equations from steps 3 and 4
(see next slide).
© Copyright 2006 Prentice-Hall
Solving: I1 = 9.53 mA, I2 = 7.43 mA, I3 = -2.10 mA
The negative result for I3 indicates the actual current direction is opposite to the assumed direction.
(Continued)
20.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
In the loop current method, you can solve for the currents in a circuit using simultaneous equations.
Loop current method
Steps:
Assign a current in each nonredundant loop in an arbitrary direction.
Show polarities according to the assigned direction of current in each loop.
Apply KVL around each closed loop.
Solve the resulting equations for the loop currents.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Loop current method
Assign a current in each nonredundant loop in an arbitrary direction.
+
+
+
Apply KVL around each closed loop. Resistors are entered in kW in this example.
+
Loop A
Loop B
Notice that the polarity of R3 is based on loop B and is not the same as in the branch current method.
22.unknown
23.unknown
24.unknown
© Copyright 2006 Prentice-Hall
26.unknown
27.unknown
28.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
The loop current method can be applied to more complicated circuits, such as the Wheatstone bridge. The steps are the same as shown previously.
Loop current method applied to circuits with more than two loops
Loop A
Loop B
Loop C
The advantage to the loop method for the bridge is that it has only 3 unknowns.
31.unknown
32.unknown
33.unknown
34.unknown
© Copyright 2006 Prentice-Hall
Summary
Summary
Write the loop current equation for Loop A in the Wheatstone bridge:
Loop A
Loop B
Loop C
© Copyright 2006 Prentice-Hall
Summary
Summary
In the node voltage method, you can solve for the unknown voltages in a circuit using KCL.
Node voltage method
Determine the number of nodes.
Select one node as a reference. Assign voltage designations to each unknown node.
Assign currents into and out of each node except the reference node.
Apply KCL at each node where currents are assigned.
Express the current equations in terms of the voltages and solve for the unknown voltages using Ohm’s law.
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
A
Solve the same problem as before using the node voltage method.
There are 4 nodes. A is the one unknown node.
B is selected as the reference node.
Currents are assigned into and out of node A.
Apply KCL at node A (for this case).
Write KCL in terms of the voltages (next slide).
B
40.unknown
© Copyright 2006 Prentice-Hall
© Copyright 2006 Prentice-Hall
Simultaneous equations
The solution of a matrix consisting of an array of coefficients and constants for a set of simultaneous equations.
A closed current path in a circuit.
One current path that connects two nodes.
Key Terms
The junction of two or more components.
A set of n equations containing n unknowns, where n is a number with a value of 2 or more.
46.psd
© Copyright 2006 Prentice-Hall
Quiz
In a set of simultaneous equations, the coefficient that is written a1,2 appears in
a. the first equation
b. the second equation
© Copyright 2006 Prentice-Hall
Quiz
In standard form, the constants for a set of simultaneous equations are written
a. in front of the first variable
b. in front of the second variable
c. on the right side of the equation
d. all of the above
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
To solve simultaneous equations, the minimum number of independent equations must be at least
a. two
b. three
c. four
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
In the equation a1,1x1 +a1,2x2 = b1, the quantity b1 represents
a. a constant
b. a coefficient
c. a variable
© Copyright 2006 Prentice-Hall
a. 4
b. 14
c. 24
d. 34
© Copyright 2006 Prentice-Hall
Quiz
The characteristic determinant for a set of simultaneous equations is formed using
a. only constants from the equations
b. only coefficients from the equations
c. both constants and coefficients from the equations
d. none of the above
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Quiz
A negative result for a current in the branch method means
a. there is an open path
b. there is a short circuit
c. the result is incorrect
d. the current is opposite to the assumed direction
Principles of Electric Circuits - Floyd
© Copyright 2006 Prentice-Hall
Quiz
To solve a circuit using the loop method, the equations are first written for each loop by applying
a. KCL
b. KVL
© Copyright 2006 Prentice-Hall
Quiz
A Wheatstone bridge can be solved using loop equations. The minimum number of nonredundant loop equations required is
a. one
b. two
c. three
d. four
© Copyright 2006 Prentice-Hall
Quiz
In the node voltage method, the equations are developed by first applying
a. KCL
b. KVL
© Copyright 2006 Prentice-Hall
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