Transcript Natural_Response_RLC

Series RLC Network
Objective of Lecture
 Derive the equations that relate the voltages across a
resistor, an inductor, and a capacitor in series as:
 the unit step function associated with voltage or current
source changes from 1 to 0 or
 a switch disconnects a voltage or current source into the
circuit.
 Describe the solution to the 2nd order equations when
the condition is:
 Overdamped
 Critically Damped
 Underdamped
Series RLC Network
 With a step function voltage source.
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
 i(to-) = iL(to-) = 0A and vC(to-) = Vs
 vL(to-) = 0V and iC(to-) = 0A
 Once the steady state is reached after the voltage source
has a magnitude of Vs at t > to, replace the capacitor with
an open circuit and the inductor with a short circuit.


i(∞s) = iL(∞s) = 0A and vC(∞s) = 0V
vL(∞s) = 0V and iC(∞s) = 0A
Selection of Parameter
 Initial Conditions
 i(to-) = iL(to-) = 0A and vC(to-) = Vs
 vL(to-) = 0V and iC(to-) = 0A
 Final Conditions
 i(∞s) = iL(∞s) = 0A and vC(∞s) = oV
 vL(∞s) = 0V and iC(∞s) = 0A
 Since the voltage across the capacitor is the only
parameter that has a non-zero boundary condition,
the first set of solutions will be for vC(t).
Kirchhoff’s Voltage Law
 v(t )  0
diL (t )
vC (t )  L
 Ri L  0
dt
dvC (t )
iC (t )  C
dt
iL (t )  iC (t )
d 2 vC (t )
dvC (t )
LC
 RC
 vC (t )  0
2
dt
dt
d 2 vC (t ) R dvC (t ) 1


vC (t )  0
2
dt
L dt
LC
vC (t  to )  vt (t  to ) when t  to
General Solution
Let vC(t) = AesDt
AR sDt
A sDt
As e 
se 
e 0
L
LC
R
1
sDt
2
Ae ( s  s 
)0
L
LC
R
1
2
s  s
0
L
LC
2 sDt
General Solution (con’t)
R
1
s  s
0
L
LC
2
2
R
1
 R 
s1  
   
2L
LC
 2L 
2
R
1
 R 
s2  
   
2L
LC
 2L 
General Solution (con’t)
s1      
2
2
o
s 2      
2
R

2L
o 
2
o
s  2s    0
2
2
o
1
LC
General Solution (con’t)
vC1 (t )  A1e
s1Dt
vC 2 (t )  A2 e
s2 Dt
vC (t )  vC1 (t )  vC 2 (t )  A1e
s1Dt
 A2 e
s2 Dt
Solve for Coefficients A1 and A2
 Use the boundary conditions at to- and t = ∞s to solve
for A1 and A2.

vC (to )  VS
 Since the voltage across a capacitor must be a
continuous function of time.




vC (to )  vC (to )  vC1 (to )  vC 2 (to )  VS
A1e s1 0 s   A2 e s2 0 s   A1  A2  VS
 Also know that
dvC (to ) d
iC (to )  C
 vC1 (to )  vC 2 (to )  0
dt
dt
s1 A1e s1 0 s   s2 A2 e s2 0 s   s1 A1  s2 A2  0
Overdamped Case
   o
 implies that C > 4L/R2
s1 and s2 are negative and real numbers
vC (t )  A1e s1Dt  A2 e s2 Dt
Critically Damped Case
   o
 implies that C = 4L/R2
s1 = s2 = -  = -R/2L
vC (t )  A1e Dt  A2 DteDt
Underdamped Case
  < o
 implies that C < 4L/R2
s1     2   o2    j d
s 2     2   o2    j d
 d   o2   2
 j
 1 , i is used by the mathematicians for
imaginary numbers
vC (t )  e Dt ( A1e jd Dt  A2 e  jd Dt )
e j  cos   j sin 
e  j  cos   j sin 
vC (t )  e Dt [ A1 (cos d Dt  j sin d Dt )  A2 (cos d Dt  j sin d Dt )]
vC (t )  e Dt [( A1  A2 ) cos d Dt  j ( A1  A2 ) sin d Dt ]
vC (t )  e Dt [ B1 cos d Dt  jB2 sin d Dt ]
B1  A1  A2
B2  A1  A2
Angular Frequencies
 o is called the undamped natural frequency
 The frequency at which the energy stored in the
capacitor flows to the inductor and then flows back to
the capacitor. If R = 0W, this will occur forever.
 d is called the damped natural frequency
 Since the resistance of R is not usually equal to zero,
some energy will be dissipated through the resistor as
energy is transferred between the inductor and
capacitor.

 determined the rate of the damping response.
Properties of RLC network
 Behavior of RLC network is described as damping,
which is a gradual loss of the initial stored energy
 The resistor R causes the loss
  determined the rate of the damping response

If R = 0, the circuit is loss-less and energy is shifted back and
forth between the inductor and capacitor forever at the
natural frequency.
 Oscillatory response of a lossy RLC network is possible
because the energy in the inductor and capacitor can be
transferred from one component to the other.

Underdamped response is a damped oscillation, which is
called ringing.
Properties of RLC network
 Critically damped circuits reach the final steady state
in the shortest amount of time as compared to
overdamped and underdamped circuits.
 However, the initial change of an overdamped or
underdamped circuit may be greater than that obtained
using a critically damped circuit.
Set of Solutions when t > to
 There are three different solutions which depend on
the magnitudes of the coefficients of the dvC (t ) and
dt
the vC (t ) terms.
 To determine which one to use, you need to calculate the
natural angular frequency of the series RLC network and
the term .
o 
1
LC
R

2L
Transient Solutions when t > to
s1Dt
s2 Dt
v
(
t
)

A
e

A
e
 Overdamped response ( > o) C
1
2
where Dt  t  to
s1     2  02
s2     2  02
 Critically damped response ( = o)
vC (t )  ( A1  A2 Dt )e Dt
 Underdamped response ( < o)
vC (t )  [ A1 cos(d Dt )  A2 sin( d Dt )]e Dt
 d  o 2   2
Find Coefficients
 After you have selected the form for the solution based
upon the values of o and 
 Solve for the coefficients in the equation by evaluating
the equation at t = to- and t = ∞s using the initial and
final boundary conditions for the voltage across the
capacitor.


vC(to-) = Vs
vC(∞s) = oV
Other Voltages and Currents
 Once the voltage across the capacitor is known, the
following equations for the case where t > to can be
used to find:
dvC (t )
iC (t )  C
dt
i (t )  iC (t )  iL (t )  iR (t )
diL (t )
vL (t )  L
dt
vR (t )  Ri R (t )
Solutions when t < to
 The initial conditions of all of the components are the
solutions for all times -∞s < t < to.
 vC(t) = Vs
 iC(t) = 0A
 vL(t) = 0V
 iL(t) = 0A
 vR(t) = 0V
 iR(t) = 0A
Summary
 The set of solutions when t > to for the voltage across the
capacitor in a RLC network in series was obtained.
 Selection of equations is determine by comparing the natural
frequency o to .
 Coefficients are found by evaluating the equation and its first
derivation at t = to- and t = ∞s.
 The voltage across the capacitor is equal to the initial
condition when t < to
 Using the relationships between current and voltage, the
current through the capacitor and the voltages and
currents for the inductor and resistor can be calculated.
Parallel RLC Network
Objective of Lecture
 Derive the equations that relate the voltages across a
resistor, an inductor, and a capacitor in parallel as:
 the unit step function associated with voltage or current
source changes from 1 to 0 or
 a switch disconnects a voltage or current source into the
circuit.
 Describe the solution to the 2nd order equations when
the condition is:
 Overdamped
 Critically Damped
 Underdamped
RLC Network
 A parallel RLC network where the current source is
switched out of 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-) = Is and v(to-) = vC(to-) = 0V
 vL(to-) = 0V and iC(to-) = 0A
 Once the steady state is reached after the voltage source
has a magnitude of Vs at t > to, replace the capacitor with
an open circuit and the inductor with a short circuit.


iL(∞s) = 0A and v(∞s) = vC(∞s) = 0V
vL(∞s) = 0V and iC(∞s) = 0A
Selection of Parameter
 Initial Conditions
 iL(to-) = Is and v(to-) = vC(to-) = 0V
 vL(to-) = 0V and iC(to-) = 0A
 Final Conditions
 iL(∞s) = 0A 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).
Kirchoff’s Current Law
iR (t )  iL (t )  iC (t )  0
v(t )  vR (t )  vL (t )  vC (t )
dvC (t )
vR (t )
 iL (t )  C
0
R
dt
diL (t )
vL (t )  v(t )  L
dt
d 2iL (t ) L diL (t )
LC

 iL (t )  0
2
dt
R dt
d 2iL (t ) 1 diL (t ) iL (t )


0
2
dt
RC dt
LC
General Solution
1
1
s 
s
0
RC
LC
2
2
1
1
 1 
s1  
 
 
2 RC
 2 RC  LC
2
1
1
 1 
s2  
 
 
2 RC
 2 RC  LC
s1      
2
1

2 RC
1
o 
LC
2
o
s 2     2   o2
s  2s    0
2
2
o
Note that the equation for the natural frequency of the RLC
circuit is the same whether the components are in series or in
parallel.
Overdamped Case
   o
 implies that L > 4R2C
s1 and s2 are negative and real numbers
iL1 (t )  A1e
s1Dt
iL 2 (t )  A2 e
s2 Dt
Dt  t  to
iL (t )  iL1 (t )  iL 2 (t )  A1e
s1Dt
 A2 e
s2 Dt
Critically Damped Case
   o
 implies that L = 4R2C
s1 = s2 = -  = -1/2RC
iL (t )  A1e
Dt
 A2 Dte
Dt
Underdamped Case
  < o
 implies that L < 4R2C
s1     2  o2    jd
s2     2  o2    jd
d  o2   2
iL (t )  e Dt [ A1 cos d Dt  A2 sin d Dt ]
Other Voltages and Currents
 Once 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.
 Selection of equations is determine by comparing the natural
frequency o to .
 Coefficients are found by evaluating the equation and its first
derivation at t = to- and t = ∞s.
 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.