Unformatted text preview: ME375 Handouts Electrical Systems
• Basic Modeling Elements
• Interconnection Relationships
• Derive Input/Output Models School of Mechanical Engineering
Purdue University ME375 Electrical  1 1 ME375 Handouts 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]
[Watt]
w : work ( energy ) [J]
wo
[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 2 ME375 Handouts Basic Modeling Elements
• Resistor • Capacitor – Ohms Law
Voltage across is proportional
to the through current.
eR = R i
+
i ⇔
eR i=
− 1
eR
R – Charge collected is proportional
to the voltage across.
– Current is proportional to the rate
of change of the voltage across.
q = C eC i=C Fde I
H dt K
C + eC − R
i – Dissipates energy through heat.
p = R i2 = ⇔ 12
e
R – Analogous to friction elements
in mechanical systems, e.g.
dampers C – Energy supplied is stored in its
electric field and can affect
electric field and can affect
future
future circuit response.
– Analogous to
in
mechanical systems. School of Mechanical Engineering
Purdue University ME375 Electrical  3 3 ME375 Handouts Basic Modeling Elements
• Inductor • Voltage Source
Source – Voltage across is proportional to
the rate of the change of the
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 4 ME375 Handouts Interconnection Laws
• Kirchhoff's Voltage Law
Voltage Law
– The total voltage drop along
any closed loop in the circuit is
zero. ∑e j =0 • Kirchhoff’s Current Law
Current Law
– The algebraic sum of the
currents at any node in the
circuit is zero. Closed
Loop ∑i j =0 Any
Node School of Mechanical Engineering
Purdue University ME375 Electrical  5 5 ME375 Handouts Modeling Steps
• Understand System Function and Identif
Id tify
Input/Output
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.
Check that the Number of Unknown Variables equals the
Check that the Number of Unknown Variables equals the
Number
Number of Equations
– Eliminate Intermediate Variables to Obtain Standard Forms.
School of Mechanical Engineering
Purdue University ME375 Electrical  6 6 ME375 Handouts Example I
Deri the I/O model for the follo
Derive the I/O model for the following
circuit.
circuit. Let voltage ei(t) be the input
and the voltage across the capacitor be
the output.
+
+
ei(t)
_ eR −
R +
i Element Laws:
Laws: eL −
L +
C EOM: eC
_ School of Mechanical Engineering
Purdue University ME375 Electrical  7 7 ME375 Handouts Complex Impedance School of Mechanical Engineering
Purdue University ME375 Electrical  8 8 ME375 Handouts Parallel Connection of Elements School of Mechanical Engineering
Purdue University ME375 Electrical  9 9 ME375 Handouts Series Connection of Elements School of Mechanical Engineering
Purdue University ME375 Electrical  10 10 ME375 Handouts Example II
Obtain the I/O model for the following
Obtain the I/O model for the following
circuit.
circuit. The input is the voltage ei(t) of
the voltage source and the through
current of the inductor is the output. +
ei(t) R1 L
R2 i
C _ Input:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  11 11 ME375 Handouts Example – Complex Impedance
Obtain the I/O model for the following
Obtain the I/O model for the following
circuit.
circuit. The input is the voltage ei(t) of
the voltage source and the through
current of the inductor is the output. +
ei(t) R1 L
R2 i
C _ Input:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  12 12 ...
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 Fall '10
 Meckle
 Mechanical Engineering, Alternating Current, Inductor, Purdue University

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