Unformatted text preview: ME375 Handouts Electrical Systems
• Basic Modeling Elements
Modeling Elements
• Interconnection Relationships
• Derive Input/Output Models School of Mechanical Engineering
Purdue University ME375 Electrical  1 Variables
•
•
•
•
•
•
•
• q : charge [C] (Coulomb)
[C]
i : current [A]
[A]
e : voltage [V]
[V]
R : resistance [Ω]
C : capacitance [Farad]
[Farad]
L : inductance [H] (Henry)
[H]
p : power [Watt]
power [Watt]
w : work ( energy ) [J]
[J]
1 [J] (Joule) = 1 [VAsec]
[V d
q=i
dt
q (t1 ) = q (t0 ) +
p = e⋅i
w (t1 ) = w (t0 ) + School of Mechanical Engineering
Purdue University = w ( t0 ) + z
z
z t1 t0 i(t ) dt t1 t0
t1 t0 p(t ) dt
( e ⋅ i ) dt ME375 Electrical  2 1 ME375 Handouts Basic Modeling Elements
• Resistor • Capacitor – Ohms Law
Voltage across is proportional
Voltage across is proportional
to
to the through current.
eR = R i
+ ⇔
eR i=
− – Charge collected is proportional
to the voltage across
to the voltage across.
– Current is proportional to the rate
of change of the voltage across. 1
eR
R q = C eC i=C Fde I
H dt K
C + eC − R i ⇔ i – Dissipates energy through heat.
12
p= Ri =
e
R C – Energy supplied is stored in its
electric field and can affect
future circuit response.
– Analogous to
in
mechanical systems. 2 – Analogous to friction elements
in mechanical systems, e.g.
dampers School of Mechanical Engineering
Purdue University ME375 Electrical  3 Basic Modeling Elements
• Inductor • Voltage Source – Voltage across is proportional to
the rate of the change of the
the rate of the change of the
through
through current.
eL = L
+
i F d iI
H dt K eL − e(t) + − • Current Source L – Maintain specified current,
regardless of the required
voltage. – Energy supplied is stored in its
magnetic field.
w= – Maintain specified voltage
across two points, regardless of
the required current. 1
L i2
2 – Analogous to
mechanical systems. in
i(t)
School of Mechanical Engineering
Purdue University ME375 Electrical  4 2 ME375 Handouts Interconnection Laws
• Kirchhoff's Voltage Law
– The total voltage drop along
any closed loop in the circuit is
th
zero. ∑e j =0 • Kirchhoff’s Current Law
– The algebraic sum of the
currents at any node in the
th
circuit is zero. Closed
Loop ∑i j =0 Any
Node School of Mechanical Engineering
Purdue University ME375 Electrical  5 Modeling Steps
• Understand System Function and Identify
Input/Output Variables
Input/Output Variables
• Draw Simplified Schematics Using Basic Elements
• Develop Mathematical Model
–
–
–
– Label Each Element and the Corresponding Voltages.
Label Each Node and the Corresponding Currents.
Apply Interconnection Laws.
Interconnection Laws
Check that the Number of Unknown Variables equals the
Number of Equations
– Eliminate Intermediate Variables to Obtain Standard Forms.
School of Mechanical Engineering
Purdue University ME375 Electrical  6 3 ME375 Handouts Example I
Derive the I/O model for the following
circuit. Let voltage ei(t) be the input
and the voltage across the capacitor be
the output.
+
+
ei(t)
_ eR −
R +
i Element Laws: eL −
L +
C EOM: eC
_ School of Mechanical Engineering
Purdue University ME375 Electrical  7 Complex Impedance School of Mechanical Engineering
Purdue University ME375 Electrical  8 4 ME375 Handouts Example II
Obtain the I/O model for the following
circuit. The input is the voltage ei(t) of
the voltage source and the through
the voltage source and the through
current
current of the inductor is the output. + R1 L
R2 ei(t) i
C _ Input:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  9 Example – Complex Impedance
Obtain the I/O model for the following
circuit. The input is the voltage ei(t) of
the voltage source and the through
the voltage source and the through
current
current of the inductor is the output. +
ei(t) R1 L
R2 i
C _ Input:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  10 5 ME375 Handouts Parallel Connection of Elements School of Mechanical Engineering
Purdue University ME375 Electrical  11 Series Connection of Elements School of Mechanical Engineering
Purdue University ME375 Electrical  12 6 ...
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 Fall '10
 Meckle
 Mechanical Engineering, Alternating Current, Inductor, Purdue University

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