Transcript Simultaneous Equations • Can use matrix maths to solve linear simultaneous equations

Simultaneous Equations
• Can use matrix maths to solve linear
simultaneous equations
• Arise from:
– Electric Circuits
• Kirchoff’s laws
– Fluid Dynamics
Systems of equations #1
Kirchoff’s Laws
ε0
R0
ε1
R2
R1
Kirchoff’s Laws
• Kirchoff’s Node Law
– The sum of currents at a node is equal to zero
– Current does not build up at a node
• Kirchoff’s Loop Law
– The sum of potentials around a loop is equal
to zero
Kirchoff’s Laws
Apply to circuit #1
I0
ε0
ε1
I1
I2
R0
Loop1
R2
Loop2
R1
Kirchoff’s Laws
Apply to circuit #2
• Node Law
– I0-I2-I1 = 0
• Loop1
– I0R0-e0+I2R2=0
• Loop2
– -I2R2-e1+I1R1=0
Kirchoff’s Laws
Apply to circuit #3
• Get rid of I2 & rewrite
I0
I2
I1 0 solve I2
I0
I1
• Loop 1
I0 R0
R2 I0
R2 I1 E0
• Loop 2
R2 I0
R2 I1
I1 R1 E1
Kirchoff’s Laws & Simultaneous
Equations
• Ohms law
Re sis tan ce  Current  Voltage
• Also works with
Coeffs unknowns  Re sults
arrays of resistances
& vectors of voltages
Re sis tan ces  Currents  Voltages & currents
Kirchoff’s Law
I0 R0
R2 I0
R0
R2 I0
R2 I1
R2
R2
R2
R1
R2
R2 I1 E0• Loop1 &
• Loop2 from before
I1 R1 E1

• Write in matrix form
I0
E0
I1
E1
Kirchoff’s Laws
Solution step
E0
7 V
R0
4  ohm
R0
R2
3 V
E1
R2
R1
R2
R1
• Give values to
resistance & voltages
R2
5  ohm
1

E0
E1
8  ohm
R2

1.25
1
• Form solution step
A
Systems of equations #2
• Solution Step
Coeffs unknowns  Re sults
Coeffs1  Coeffs unknowns  Coeffs1  Re sults 
unknowns  Coeffs1  Re sults 
Systems of equations #3
3x0  1x1  5 x2  20
2 x0  3x1  1x2  5
 1x0  4 x1  7
Coeffs unknowns  Re sults
3 1 5
2 3
x0
20
1  x1
5
1 4 0
x2
7
Systems of equations #4
3 1 5
1
20
1
1
 5
 2
1 4 0
7
3
2 3
unknowns  Coeffs1  Re sults 
3 1 5
Coeff
2 3
1
1 4 0
20
Res
From matrix toolbar
5
7
1
1
Unknown Coeff  Res
Unknown 2
3