Unit 3
Mathematical Modeling of Physical Systems
Lect05: Objectives
At the end of the lecture, I should be able to:
• Find the transfer function for linear, time-
invariant electrical networks
2
Approaches to System
Modeling
First principles
Based on known laws
Based on known laws
Physics, Queueing theory
Difficult to do for complex system
Experimental (System ID)
Statistical/Data-driven Models
Requires data
Is there a good ‘training set’?
3
Electrical Network T.F.
Voltage-Current, Voltage-Charge and Impedance
Relationships for Capacitor, Resistors and Inductors
4
Electrical Network T.F.
5
6
Electrical Network T.F.
Example 9: Find the transfer function relating
the capacitor voltage, VC(s), to the input
voltage, V(s) in the RLC network below:
7
Electrical Network T.F.
Solution: In any problem, you must first
decide what the input and output should
be. The problem statement in this case
is clear: we are to treat the capacitor
voltage as the output and the applied
voltage as the input.
Summing the voltages around the loop,
di(t ) 1
assuming zero initial conditions, yields L Ri (t ) i(t )dt v(t )
the differential equation for this network dt C0
as:
d 2 q(t ) dq(t ) 1
Changing variables from current to L R q(t ) v(t )
charge using i(t ) dq(t ) yields: dt 2
dt C
dt
8
Electrical Network T.F.
d 2 q(t ) dq(t ) 1
Changing variables from current to
charge using i(t ) dq(t ) yields:
L 2
R q(t ) v(t )
dt
dt dt C
From the voltage-charge relationship
for a capacitor in Table 2, q(t ) CvC.(t ) d 2vC (t ) dvC (t )
Substituting in the above eq. yields:
LC 2
RC vC (t ) v(t )
dt dt
Taking the Laplace transform
assuming zero initial conditions, ( LCs 2 RCs 1)VC (s) V (s)
rearranging terms and simplifying
yields:
1
Solving for the transfer function VC ( s) LC
V ( s) s 2 R s 1
L LC 9
Electrical Network T.F.
Let us now develop a technique for simplifying the solution
for future problems. First, take the Laplace transform of
the eq. in the voltage-current column of Table 2 assuming
zero initial conditions.
For the capacitor For the resistor, For the inductor,
1
V ( s) I ( s) V (s) RI (s) V (s) LsI (s)
Cs
Now we define the following transfer V ( s)
function Z ( s)
I ( s) 10
Electrical Network T.F.
Let us now demonstrate how the concept of impedance
simplifies the solution for the transfer function.
t
di(t ) 1
L Ri (t ) i( )d v(t )
dt C0
Taking the Laplace transform:
1
( Ls R ) I ( s) V ( s)
Cs
Which is in the form of:
[sum of impedance]I(s) = [sum of applied voltage]
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Electrical Network T.F.
Example 10: Repeat example 9 using mesh
analysis and transform methods without writing
a differential eq.
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Electrical Network T.F.
Using the transformed figure, writing a
1
mesh eq. using the impedance as we ( Ls R ) I ( s) V ( s)
would use resistor values in a purely Cs
resistive circuit, we obtain:
I (s) 1
Solving for I(s)/V(s):
V ( s) Ls R 1
Cs
1
VC ( s) I ( s)
But the voltage across capacitor is the
product of current and the impedance.
Thus, Cs
Solving the I(s), substituting into the
previous eq. and simplifying yields the
same result as example 9.
13
Problem 2.2 (b)
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