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Transcript of 1 ECE 221 Electric Circuit Analysis I Chapter 10 Circuit Analysis 4 Ways Herbert G. Mayer, PSU...
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ECE 221Electric Circuit Analysis I
Chapter 10Circuit Analysis 4 Ways
Herbert G. Mayer, PSUStatus 11/23/2014
For use at Changchun University of Technology CCUT
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Syllabus
Goal Sample Problem1 Solve by Substitution KCL Solve by Cramer’s Rule Solve by Node-Voltage Method Solve by Mesh-Current Method Conclusion Problem1 Same for Problem2
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Goal We’ll analyze simple circuits, named Sample
Problem1 and Sample Problem2
Both with 2 constant voltage sources, 3 resistors
Computing 3 branch currents i1, i2, and i3
First by using conventional algebraic substitution, applying Kirchhoff’s Laws; we’ll need 3 equations
Secondly, we use Cramer’s Rule, with these 3 equations, normalizing them to determinant form
Then, we use the Node-Voltage Method
Finally we compute fictitious currents ia and ib, using the Mesh-Current Method
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Problem 1
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Circuit for Sample Problem1
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Sample Problem1: 3 Equations
KCL at node n1 states:
(1) i1 = i2 + i3
KVL in the left mesh labeled ia yields:
(2) R1*i1 + R3*i3 - V1 = 0
KVL in the right mesh, labeled ib:
(3) R2*i2 + V2 - R3*i3 = 0
i.e. i3 = (R2*i2)/R3 + V2/R3
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Solve Problem1 by Substitution
(1) in (2)
R1*(i2+i3) + R3*i3= V1
R1*i2 + R1*i3 + R3*i3= V1
R1*i2 + i3*(R1+R3)= V1
R1*i2 + (R2*i2 + V2)*(R1+R3)/R3 = V1
. . .
i2*(R1+R2*(R1+R3)/R3) = V1-V2*(R1+R3)/R3
. . .
i2*(100+2*400/3) = 10 - 20*(400/300)
i2 = -45.45 mA
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Solve Problem1 by Substitution
i3 = i2 * R2/R3 + V2/R3 = -0.0303+0.066667
i3 = 0.03636 A
i3 = 36.36 mA
i1 = i2 + i3
i1 = -9.09 mA
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Solve Problem1 by Cramer’s RuleUsing 3 Equations from Substitution
i1= i2 + i3
R1*i1 + R3*i3 - V1 = 0
R2*i2 + V2 - R3*i3 = 0
Normalized:
i1 - i2 - i3= 0
R1*i1 + 0 + R3*i3= V1
0 + R2*i2 - R3*i3= -V2
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Cramer’s Characteristic Determinant
Normalize i1, i2, i3 positions in matrix
| 1 -1 -1| | 0 |
Δ: | R1 0 R3 |,R = | V1 |
| 0 R2 -R3| |-V2 |
| 1 -1 -1|
Δ# = |100 0 300 || 0 200 -300
|
| 1 -1 1|
S = | -1 1 -1|
| 1 -1 1|
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Cramer’s Characteristic Determinant
Δ = 1 | 0 300 | -100 | -1-1 | + 0 =
| 200 -300|| 200 -300 |
Δ = 1*( 0 – 60,000 ) - 100*( 300 + 200 ) =
Δ = -60k - 50k
Δ = -110,000
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Numerator Determinant N1, and i1| 0 -1
-1 |N(i1): | 10 0 300 |
|-20 200 -300 |
N1 = 0 -10 | -1 -1 | -20 |-1 -1 |
| 200 -300 || 0 300 |
N1 = -10 * (300+200) -20 * (-300) =
N1 = -10*500 + 6,000
N1 = 1,000
i1 = 1,000 / -110,000
i1 = -0.00909 A = -9.09 mA
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Numerator Determinant N2, and i2
| 1 0 -1|
N(i2): | 100 10 300 || 0 -20 -300
|
N2 = 1 | 10 300 | -100 | 0 -1 | =
| -20 -300 || -20 -300|
N2 = -3,000 + 6,000 -100 * ( 0 - 20 ) =
N2 = 3,000 + 2,000 = 5,000
i2 = 5,000 / -110,000
i2 = -0.04545 A = -45.45 mA
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Numerator Determinant N3, and i3
| 1 -1 0 |
N(i3): | 100 0 10 || 0 200 -20
|
N3 = 1 | 0 1 | -100 | -1 0 | =
| 200 -20 || 200 -20 |
N3 = -2,000 – 100 * (20 ) = -4,000
i3 = -4,000 / -110,000
i3 = 0.0363636 A = 36.36 mA
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Solve Problem1 by Node-Voltage Method
Ignoring the current or voltage directions from the substitution method, we use the Node-Voltage method at node n1, currents flowing toward reference node n2
We generate 1 equation with unknown V300, voltage at the 300 Ω resistor, generating i3
Once known, we can compute the voltages at R1 and R2, and thus compute the currents i1 and i2, using Ohm’s law
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Problem1 by Node-Voltage Method
3 currents flowing from n1 toward reference node n2:
V300/300 + (V300-10)/100 + (V300-20)/200 = 0
V300 + 3*V300 + V300*2/3 = 30 + 3*20/2
V300*( 1 + 3 + 2/3 ) = 60
V300= 60 * 2 / 11
V300= 10.9090 V
i3 = V300 / 300
i3 = 36.36a mA
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Problem1 by Node-Voltage Method
V(R1) = V1 - V300
V(R1) = 10 - 10.9090 = -0.9090 V
i1 = V(R1) / R1
i1 = -0.9090 / 100
i1 = -9.09 mA
From this follows i2 using KCL:
i2 = i1 - i3
i2 = -9.0909 – 36.3636
i2 = -45.45 mA
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Solve Problem1 by Mesh-Current Method The mesh current is fictitious, one each associated
with each individual mesh
Fictitious in the sense that we act as if it were uniquely tied to a mesh; yet depending on the branch of the mesh, mesh currents from other parts flow though that very mesh as well
Kirchhoff’s current law is trivially satisfied, but mesh currents are not directly measurable with an Ampere meter, when currents from other meshes super-impose
In the Sample Problem we have 2 meshes, with mesh currents indicated as ia and ib
But we must track that, R3 for example, has both flowing though it in opposing directions
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Sample Problem1: Two Mesh Currents
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Solve Problem1 by Mesh-Current Method
KVL for mesh with ia yields:
(1) R1*ia + R3*(ia–ib) = V1
KVL for mesh with ib yields:
(2) R3*(ib-ia) + R2*ib = -V2
From (1) follows:
(1) ib = ( R1*ia + R3*ia - V1 ) / R3
Substitute ib in (2):
(2) -V2 = ib*(R2+R3) - R3*ia
-V2 = ia*(R1+R3)*(R2+R3)/R3 -
V1*(R2+R3)/R3 - R3*ia
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Solve Problem1 by Mesh-Current Method
V1*(R2+R3)/R3 - V2 =
ia*( (R1+R3)*(R2+R3)/R3 – R3)
-20 + 10*5/3 = ia*(400*500/300 – 300)
ia = -10 / 1100
ia = -0.00909 A = -9.09 mA
Since ia = i1:
i1 = -9.09 mA
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Solve Problem1 by Mesh-Current Method
Recall (1):
(1) R1*ia + R3*(ia–ib) = V1
R3*ib = ia*(R1+R3) - V1
ib =ia*(R1+R3)/R3 - V1/R3
ib = -10*400/(1,100*300) - 10/300
ib = -0.04545 A = -45.45 mA
since i2 = ib:
i2 = -45.45 mA
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Conclusion Problem1 via Mesh-Current
Since i3 = i1 - i2, i3 = -9.09 mA - -45.45 mA
it follows:
i3 = 36.36 mA
We see consistency across 4 different methods of circuit analysis
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Problem 2
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Sample Problem2 We’ll analyze a similar circuit, named Sample
Problem2
With 2 constant voltage sources of 3 V and 4 V
Plus 3 resistors at 100, 200, and 300 Ohm
But now the elements are connected differently
Again we compute 3 branch currents i1, i2, and i3
Using 4 methods: substitution method, using Kirchhoff’s Laws
Then we use Cramer’s Rule
Thirdly we use the Node-Voltage Method
Finally the Mesh-Current Method
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Circuit for Sample Problem2
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Sample Problem2: Three Equations
KCL states:
(1) i1= i2 + i3
KVL in the upper mesh labeled ia yields:
(2) i1*100 + i2*200 -3= 0
KVL in the lower mesh, labeled ib yields:
(3) -i2*200 + i3*300 + 4 + 3 =0
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Solve Problem2 by Substitution
-200*i2 + (i1-i2)*300 = -7// (1)in(3)
-500*i2 + 300*i1 =-7 // (3’)
100*i1 + 200*i2 = 3 // (2)*3
300*i1 + 600*i2 = 9 // (2’)
(3’)-(2’)
-500*i2 - 600*i2 =-7 -9 = -16
i2*1,100 =16
i2 = 16 / 1,100
i2 = 14.54 mA
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Solve Problem2 by Substitution
i1*100 + i2*200 = 3
i1*100 = 3-200*(16/1,100)
i1*100 =100/1,100
i1= 1 / 1,100
i1 = 0.91 mA
i3= i1 - i2
i3= -15 / 1,100
i3 = -13.63 mA
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Solve Problem2 by Cramer’s Rule
i1= i2 + i3
i1*100 + i2*200 -3 =0
-i2*200 + i3*300 +4 +3 = 0
Normalized:
i1 - i2 - i3 =0
100*i1 + 200*i2 + 0 = 3
0 - 200*i2 + 300*i3 = -7
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Cramer’s Characteristic Determinant
Normalize i1, i2, i3 positions
| -1 1 1 || 0 |
D# = | 100 200 0 |,R = | 3 |
| 0 -200 300| | -7 |
| 1 -1 1|
S = | -1 1 -1|
| 1 -1 1|
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Cramer’s Characteristic Determinant
Δ = -1 | 200 0 | -100 | 1 1 | + 0 =
| 200 -300||-200 300 |
Δ = -60,000 – 50,000 = -110,000
Δ = - 110 k
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Numerator Determinant N1, and i1| 0 1
1 |N(i1): | 3 200 0 |
| -7 -200 300|
N1 = 0 - 3| 1 1 | -7 | 1 1 |
|-200 300 ||200 0 |
N1 = -3*(300+200) -7*(-200) =
N1 = -1,500 + 1,400
N1 = -100
i1 = -100 / -110,000
i1 = 0.000909 A
i1 = 0.91 mA
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Numerator Determinant N2, and i2
| -1 0 1 |
N(i2): | 100 3 0 || 0 -7
300 |
N2 = -1 | 3 0 | -100 | 0 1 | + 0 =
| -7 300 || -7 300|
N2 = -(900) - 100* (7) = -1,600
i2 = -1,600 / -110,000
i2 = 14.54 mA
From this follows i3 = i1-i2
i3 = -13.63 mA
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Node-Voltage With Sample Problem2
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Solve Problem2 by Node-Voltage Method
Use KCL to compute 3 current from n1 toward reference node n2:
V200/200 + (V200-3)/100 + (V200-3-4)/300 =0
V200*(3/2 + 3 + 1 ) = 9 + 7
V200*11/2 =16
V200 =2.9090 V
i2= V200 / 200
i2 = 14.54 mA
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Solve Problem2 by Node-Voltage Method
KVL in the lower mesh, with V300 being the voltage drop across the 300 resistor, yields:
V300 = -7 + V200 = -7 + 2.9090= -4.091 V
i.e. i3 = V300/300 = -0.013637 mA
i3 = -13.63 mA
i1 = i2 + i3 = 14.54 - 13.63
i1 = 0.91 mA
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Solve Problem2 by Mesh-Current Method
Again we analyze 2 meshes, with fictitious currents ia and ib
The circuit is repeated here for convenience
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Mesh-Current With Sample Problem2
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Solve Problem2 by Mesh-Current Method
KVL for mesh with ia yields:
(1) 100*ia + 200*( ia - ib ) =3
(1) 300*ia - 200*ib= 3
(1) ib = (300*ia-3)/200
KVL for mesh with ib yields:
(2) 300*ib + 200*( ib – ia ) =-7
(2) 500*ib - 200*ia= -7
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Solve Problem2 by Mesh-Current Method
Substitute ib from (1) in (2):
500*(300*ia - 3)/200 - 200*ia = -7
ia = 1/1,100 = 0.91 mA
i1 = ia, hence:
i1 = 0.91 mA
ib = 3*ia/2-3/200 = 3/(1,100 * 2) - 3/200
ib = -13.63 mA
i3 = ib, hence
i3 = -13.63 mA
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Solve Problem2 by Mesh-Current Method
With i2 = i1 – i3, it follows:
i2 = 14.54 mA
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Which Method is easiest?
• It seems the Mesh-Current Method is simplest for these problems
• With the smallest number of equations
• And less chances for students’ sign confusion, as we follow the same sign (or: direction) through the whole mesh
• But for a large number of unknowns Cramer’s Rule is the only way to compute accurately