Civil Engineering Reference
In-Depth Information
(as defined in Section 2.2)‚ with stamps given by Equation (2.20)‚ then‚ using
the MNA formulation from Chapter 2‚ an RLC system may be represented as:
where
G
and
C
are constant matrices whose entries depend on the values of
the RLC elements,
X
is the vector of unknowns, consisting of node voltages
and currents in Type 2 elements such as inductors and voltage sources, and
E
is the excitation vector.
If we represent
X
in terms of its moments, we have:
Note that the entry of vector is simply the moment of the variable.
We will use this to find the impulse response of the system‚ i.e.‚ the response
of the system to a
excitation. Under this condition‚ we have
a
constant vector in the
domain. Substituting these two into Equation (3.33)‚
Matching the powers of
on either side of the equation‚ we obtain the following
relations:
The above set of equations has a very useful physical interpretation‚ related to
the original system that is described by Equation (3.33).
The solution of Equation (3.36) is identical to that of the original system
when an impulse excitation is applied‚ and is set to zero. In other words‚
it is the response of the original circuit to a unit impulse at DC‚ i.e.‚ with
all capacitors open-circuited and all inductors short-circuited.
Equation (3.37) corresponds to Equation (3.33) if is set to zero‚ the original
excitation is set to zero‚ and a new excitation of is applied instead.
This implies that the original circuit is modified as follows:
(a)
(b)
All voltage sources are short-circuited and current sources open-circuited.
Each capacitor is replaced by a current source of value
where
is the
moment of the voltage
across the capacitor.
(c)
Each self or mutual inductance
is replaced by a voltage source of
value
on line
where
is the
moment of
the current through inductor
Example - An RC Line:
We will demonstrate the above ideas on the RC
line shown in Figure 3.3. To simplify our calculations‚ we will assume that all re-
sistors are of value
and all capacitors have the (unrealistic) value of 1F. The
