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 Variables
•
•
•
•
•
•
•
• q : charge [C] (Coulomb)
i : current [A]
e : voltage [V]
R : resistance [Ω]
C : capacitance [Farad]
L : inductance [H] (Henry)
p : power [Watt]
w : work ( energy ) [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
to the through current.
eR = R i
+ ⇔
eR i= – Charge collected is proportional
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.
1 2
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
through current.
eL = L
+
i F d iI
H dt K eL − e(t) + − • Current Source L – 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. – Maintain specified current,
regardless of the required
voltage. 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
zero. ∑e j =0 • Kirchhoff’s Current Law
Kirchhoff’
– 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 Modeling Steps
• Understand System Function and Identify
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
Number of Equations
– Eliminate Intermediate Variables to Obtain Standard Forms.
School of Mechanical Engineering
Purdue University ME375 Electrical  6 3 ME375 Handouts In Class Exercise
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.
+
+ eR −
R +
i Element Laws: eL −
L +
C ei(t) No. of Unknowns: eC
_ _ Simplify: School of Mechanical Engineering
Purdue University ME375 Electrical  7 Loop Approach vs. Node Approach
• Loop Approach:
– Identify pertinent current loops and label the currents
– Write elemental voltages in terms of these currents • Node Approach:
– Identify pertinent voltage nodes and label these voltages
– Identify and label all distinct currents
– Write elemental equations in terms of these voltages and
currents School of Mechanical Engineering
Purdue University ME375 Electrical  8 4 ME375 Handouts Example – Loop Approach
Obtain the I/O model for the following
circuit. The input is the voltage ei(t) of
the voltage source and the through
current of the inductor is the output.
+
+ eL − eR1 − + R1 + L
eR2 R2 ei(t) i + C eC _ _ _ Input:
Input:
Output:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  9 Example – Node Approach
Obtain the I/O model for the following
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:
Input:
Output:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  10 5 ME375 Handouts 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
current of the inductor is the output. +
ei(t) R1 L
R2 i
C _ Input:
Input:
Output:
Output:
School of Mechanical Engineering
Purdue University ME375 Electrical  11 6 ...
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 Spring '10
 Meckle
 Mechanical Engineering, Alternating Current, Purdue University, School of Mechanical Engineering

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