Download Electrical system models

Lecture 8:
Modeling Electrical Systems
1. Elements making up an electrical system
2. First-principles modeling of electrical
systems in the time domain
3. Modeling in the Laplace domain (next
time)
ME 431, Lecture 8
determine the mathematical models that
capture the behavior of an electrical system
1
Modeling Electrical Systems
• Current (i) – is a measure of the rate of flow of charge
(electrons) through a circuit (i=dq/dt), current has
direction
ME 431, Lecture 8
• Voltage (e) – is a measure of the force that causes
electrons to move through a circuit (a potential
measured w.r.t. a ground)
2
Electrical systems consist
of three basic types of
elements
1. Resistance elements
2. Capacitance elements
3. Inductance elements
ME 431, Lecture 8
Modeling Electrical Systems
3
Modeling Electrical Systems
e  iR
(Ohm's law)
• Dissipate energy (like a damper)
• Resistance has units of an Ohm (Ω)
ME 431, Lecture 8
• Resistance Elements
4
Modeling Electrical Systems
q
1
1
C   e  q   idt
e
C
C
• Capacitance is measured as charge stored per unit
voltage
• If you apply a voltage across a capacitor a
potential builds up that is then released if the
voltage is removed … in other words, capacitors
store potential energy (like a spring)
• Capacitance has units of a Farad (F)
ME 431, Lecture 8
• Capacitance Elements
5
Modeling Electrical Systems
di
eL
dt
• An inductor is a coil of wire such that current
through the coil generates a magnetic field which
induces a voltage that is proportional to how fast
the current is changing
• If power is disconnected, the induced voltage will
make the current continue to flow (like an inertia)
• Inductance elements store kinetic energy
• Inductance has units of a Henry (H)
ME 431, Lecture 8
• Inductance Elements
6
Electrical Circuits
e  e1  e2  e3  iR1  iR2  iR3
ME 431, Lecture 8
• Resistors in series
 i( R1  R2  R3 )
Requiv
e
  R1  R2  R3
i
7
Electrical Circuits
• Resistors in parallel
Requiv
e 1
1
1
  
 
i  R1 R2 R3 
ME 431, Lecture 8
i  i1  i2  i3
e
e
e



R1 R2 R3
1
1
1
 e 
 
 R1 R2 R3 
1
8
Electrical Circuits
• Kirchoff’s Current Law (node law)
i1  i3  i2  i4  i5
ME 431, Lecture 8
• Current in to a node is conserved
9
Electrical Circuits
• Sum of voltages around a loop equals
zero
di 1
e  iR  L   (i )dt  0
dt C
ME 431, Lecture 8
• Kirchoff’s Voltage Law (loop law)
10
Electrical Circuits
L
ei +
_
di1 1
loop 1: ei  L
  (i1  i2 )dt  0
dt C
1
loop 2:  i2 R   (i2  i1 )dt  0
C
C
i1
R
eo
ME 431, Lecture 8
• Use one equation for each loop
• Assume a direction for current, if solution is
negative, know direction is opposite
i2
11
Electrical Circuits
• Equations can be rewritten in terms of
1
charge q
loop 1: e  Lq  (q  q )  0
1
1
2
C
1
loop 2:  q2 R  (q2  q1 )  0
C
• A mechanical analog exists for each circuit
• What are the state variables?
energy storage element
capacitor
inductor
state variable
q1  q2
ME 431, Lecture 8
i
x1  q1
x2  q2
x3  q1
12
Electrical Circuits
x1  x3
1
1
x2  
x2 +
x1
CR
CR
1
1
1
x3  
x1 +
x2  ei
CL
CL
L
y  x3
ME 431, Lecture 8
• Putting into state space form where ei is the input and i1
is the output
13
Electrical Circuits

 0
 x1  
x    1
 2   CR
 x3  
1

 CL

 
1
  x1   0 
 



0  x2    0  u

  x3   1 
 
0
L

0
1

CR
1
CL
 x1 
y   0 0 1  x2    0 u
 x3 
ME 431, Lecture 8
• Putting into matrix form
14
Example
• Find the transfer function Eo(s)/Ei(s)
ei +
_
C
i1
R
i2
eo
ME 431, Lecture 8
L
15
ME 431, Lecture 8
Example (con’t)
16