Transcript Step Response - Virginia Tech

Parallel RLC Network
Objective of Lecture
 Derive the equations that relate the voltages across and the
currents flowing through a resistor, an inductor, and a
capacitor in parallel as:
 the unit step function associated with voltage or current
source changes from 0 to 1 or
 a switch connects a voltage or current source in the circuit.
 Describe the solution to the 2nd order equations when the
condition is:
 Overdamped
 Critically Damped
 Underdamped
Parallel RLC Network
 With a current source switched into the circuit at t= to.
Boundary Conditions
 You must determine the initial condition of the
inductor and capacitor at t < to and then find the final
conditions at t = ∞s.
 Since the voltage source has a magnitude of 0V at t < to
 iL(to-) = 0A and v(to-) = vC(to-) = 0V
 vL(to-) = 0V and iC(to-) = 0A
 Once the steady state is reached after the switch has
opened at t > to, replace the capacitor with an open
circuit and the inductor with a short circuit.


iL(∞s) = Is and v(∞s) = vC(∞s) = 0V
vL(∞s) = 0V and iC(∞s) = 0A
Selection of Parameter
 Initial Conditions
 iL(to-) = 0A and v(to-) = vC(to-) = 0V
 vL(to-) = 0V and iC(to-) = 0A
 Final Conditions
 iL(∞s) = Is and v(∞s) = vC(∞s) = oV
 vL(∞s) = 0V and iC(∞s) = 0A
 Since the current through the inductor is the only
parameter that has a non-zero boundary condition,
the first set of solutions will be for iL(t).
Kirchhoff’s Current Law
iR (t )  iL (t )  iC (t )  iS (t )
v(t )  vR (t )  vL (t )  vC (t )
dvC (t )
vR (t )
 iL (t )  C
 IS
R
dt
diL (t )
vL (t )  v(t )  L
dt
d 2iL (t ) L diL (t )
LC

 iL (t )  I S
2
dt
R dt
d 2iL (t ) 1 diL (t ) iL (t ) I S



2
dt
RC dt
LC LC
iL (t )  it (t )  iss (t )
Set of Solutions when t > to
 Similar to the solutions for the natural response, there
are three different solutions. To determine which one
to use, you need to calculate the natural angular
frequency of the parallel RLC network and the term a.
1
o 
LC
1
a
2 RC
Transient Solutions when t > to
 Overdamped response
iL (t )  A1e
s1t
 A2e
s2t
 Critically damped response
 Underdamped response
iL (t )  ( A1  A2t )e
at
iL (t )  [ A1 cos(d t )  A2 sin(d t )]e
where
t  t  to
at
Steady State Solutions when t > to
 The final condition of the current through the
inductor is the steady state solution.
 iL(∞s) = Is
Complete Solution when t > to
 Overdamped response
iL (t )  A1e
s1t
s2 t
 Is
at
 Is
 A2e
 Critically damped response
iL (t )  ( A1  A2t )e
 Underdamped response
iL (t )  [ A1 cos(d t )  A2 sin(d t )]eat  Is
Other Voltages and Currents
 Once the current through the inductor is known:
diL (t )
vL (t )  L
dt
vL (t )  vC (t )  vR (t )
dvC (t )
iC (t )  C
dt
iR (t )  vR (t ) / R
Summary
 The set of solutions when t > to for the current through the inductor in
a RLC network in parallel was obtained.
 There are two components to the solution.
 The transient component, which has the same form as the transient solution for
the natural response of a parallel RLC circuit.
 The steady state component, which is the final condition for the current flowing
through the inductor is the steady state solution.
 Selection of equations is determine by comparing the natural
frequency o to a.
 Coefficients are found by evaluating the equation and its first
derivation at t = to-.
 The current through the inductor is equal to the initial condition when
t < to
 Using the relationships between current and voltage, the voltage across
the inductor and the voltages and currents for the capacitor and
resistor can be calculated.