step response
DESCRIPTION
Step Response. 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 0 to 1 or - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/1.jpg)
Series RLC Network
![Page 2: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/2.jpg)
Objective of LectureDerive 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 0 to 1 ora switch connects a voltage or current source
into the circuit.Describe the solution to the 2nd order
equations when the condition is:OverdampedCritically DampedUnderdamped
![Page 3: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/3.jpg)
Series RLC NetworkWith a step function voltage source.
![Page 4: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/4.jpg)
Boundary ConditionsYou 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
-) = 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. i(∞s) = iL(∞s) = 0A and vC(∞s) = Vs vL(∞s) = 0V and iC(∞s) = 0A
![Page 5: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/5.jpg)
Selection of ParameterInitial Conditions
i(to-) = iL(to
-) = 0A and vC(to-) = 0V
vL(to-) = 0V and iC(to
-) = 0AFinal Conditions
i(∞s) = iL(∞s) = 0A and vC(∞s) = Vs 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).
![Page 6: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/6.jpg)
Kirchhoff’s Voltage Law
oossotoC
SC
CC
SCCC
CL
CC
SLL
C
tttvttvttvLC
Vtv
LCdt
tdv
L
R
dt
tvd
Vtvdt
tdvRC
dt
tvdLC
titidt
tdvCti
VRidt
tdiLtv
tv
when t )()()(
)(1)()(
)()()(
)()(
)()(
0)(
)(
0)(
2
2
2
2
![Page 7: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/7.jpg)
Set of Solutions when t > toSimilar 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 series RLC network and the term .
L
RLC
o
2
1
![Page 8: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/8.jpg)
Transient Solutions when t > toOverdamped response ( > o)
where t-to = t
Critically damped response ( = o)
Underdamped response ( < o)
20
22
20
21
2121)(
s
s
eAeAtv tstsC
tC etAAtv )()( 21
22
21 )]sin()cos([)(
od
tddC etAtAtv
![Page 9: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/9.jpg)
Steady State Solutions when t > toThe final condition of the voltages across the
capacitor is the steady state solution.vC(∞s) = Vs
![Page 10: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/10.jpg)
Complete Solution when t > toOverdamped response
Critically damped response
Underdamped response
VseAeAtv tstsC 21
21)(
VsetAAtv tC )()( 21
VsetAtAtv tddC )]sin()cos([)( 21
ottt where
![Page 11: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/11.jpg)
Other Voltages and CurrentsOnce the voltage across the capacitor is
known, the following equations for the case where t > to can be used to find:
)()(
)()(
)()()()(
)()(
tRitvdt
tdiLtv
titititidt
tdvCti
RR
LL
RLC
CC
![Page 12: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/12.jpg)
SummaryThe set of solutions when t > to for the voltage
across the capacitor in a RLC network in series was obtained.The final condition for the voltage across the capacitor
is the steady state solution.Selection of equations is determine by comparing the
natural frequency oto 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.
![Page 13: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/13.jpg)
Parallel RLC Network
![Page 14: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/14.jpg)
Objective of LectureDerive 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 0 to 1 ora switch connects a voltage or current source
into the circuit.Describe the solution to the 2nd order
equations when the condition is:OverdampedCritically DampedUnderdamped
![Page 15: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/15.jpg)
Parallel RLC NetworkWith a current source switched into the
circuit at t= to.
![Page 16: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/16.jpg)
Boundary ConditionsYou 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 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) = Is and v(∞s) = vC(∞s) = 0V vL(∞s) = 0V and iC(∞s) = 0A
![Page 17: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/17.jpg)
Selection of ParameterInitial Conditions
iL(to-) = 0A and v(to
-) = vC(to-) = 0V
vL(to-) = 0V and iC(to
-) = 0AFinal Conditions
iL(∞s) = Is and v(∞s) = vC(∞s) = oVvL(∞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).
![Page 18: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/18.jpg)
Kirchhoff’s Current Law
)()()(
)()(1)(
)()()(
)()()(
)()(
)(
)()()()(
)()()()(
2
2
2
2
tititi
LC
I
LC
ti
dt
tdi
RCdt
tid
Itidt
tdi
R
L
dt
tidLC
dt
tdiLtvtv
Idt
tdvCti
R
tv
tvtvtvtv
titititi
sstL
SLLL
SLLL
LL
SC
LR
CLR
SCLR
![Page 19: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/19.jpg)
Set of Solutions when t > toSimilar 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 .
RC
LCo
2
1
1
![Page 20: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/20.jpg)
Transient Solutions when t > toOverdamped response
Critically damped response
Underdamped response
where
tstsL eAeAti 21
21)(
tL etAAti )()( 21
tddL etAtAti )]sin()cos([)( 21
ottt
![Page 21: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/21.jpg)
Other Voltages and CurrentsOnce the current through the inductor is
known:
Rtvtidt
tdvCti
tvtvtvdt
tdiLtv
RR
CC
RCL
LL
/)()(
)()(
)()()(
)()(
![Page 22: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/22.jpg)
Complete Solution when t > toOverdamped response
Critically damped response
Underdamped response
IseAeAti tstsL 21
21)(
IsetAAti tL )()( 21
IsetAtAti tddL )]sin()cos([)( 21
![Page 23: Step Response](https://reader030.vdocuments.us/reader030/viewer/2022032805/5681333e550346895d9a34e4/html5/thumbnails/23.jpg)
SummaryThe set of solutions when t > to for the current
through the inductor in a RLC network in parallel was obtained.The final condition for the current through the inductor
is the steady state solution.Selection of equations is determine by comparing the
natural frequency oto 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.