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![Page 1: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/1.jpg)
Series RLC Network
![Page 2: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/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 1 to 0 ora switch disconnects a voltage or current source
into the circuit.Describe the solution to the 2nd order
equations when the condition is:OverdampedCritically DampedUnderdamped
![Page 3: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/3.jpg)
Series RLC NetworkWith a step function voltage source.
![Page 4: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/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
-) = 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
![Page 5: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/5.jpg)
Selection of ParameterInitial Conditions
i(to-) = iL(to
-) = 0A and vC(to-) = Vs
vL(to-) = 0V and iC(to
-) = 0AFinal 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).
![Page 6: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/6.jpg)
Kirchhoff’s Voltage Law
ootoC
CCC
CCC
CL
CC
LL
C
tttvttv
tvLCdt
tdv
L
R
dt
tvd
tvdt
tdvRC
dt
tvdLC
titidt
tdvCti
Ridt
tdiLtv
tv
when t )()(
0)(1)()(
0)()()(
)()(
)()(
0)(
)(
0)(
2
2
2
2
![Page 7: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/7.jpg)
General SolutionLet vC(t) = Aest
01
0)1
(
0
2
2
2
LCs
L
Rs
LCs
L
RsAe
eLC
Ase
L
AReAs
ts
tststs
![Page 8: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/8.jpg)
LCL
R
L
Rs
LCL
R
L
Rs
1
22
1
22
2
2
2
1
012 LC
sL
Rs
General Solution (con’t)
![Page 9: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/9.jpg)
LC
L
R
o
12
222
221
o
o
s
s
02 22 oss
General Solution (con’t)
![Page 10: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/10.jpg)
tstsCCC
tsC
tsC
eAeAtvtvtv
eAtv
eAtv
21
2
1
2121
22
11
)()()(
)(
)(
General Solution (con’t)
![Page 11: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/11.jpg)
Solve for Coefficients A1 and A2Use the boundary conditions at to
- and t = ∞s
to solve for A1 and A2.
Since the voltage across a capacitor must be a continuous function of time.
Also know that
SoC Vtv )(
S
ssss
SoCoCoCoC
VAAeAeA
Vtvtvtvtv
210
20
1
21
21
)()()()(
0
0)()()(
)(
22110
220
11
21
21
AsAseAseAs
tvtvdt
d
dt
tdvCti
ssss
oCoCoC
oC
![Page 12: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/12.jpg)
Overdamped Case
implies that C > 4L/R2
s1 and s2 are negative and real numbers
tstsC eAeAtv 21
21)(
![Page 13: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/13.jpg)
Critically Damped Case
implies that C = 4L/R2
s1 = s2 = - = -R/2L
ttC teAeAtv
21)(
![Page 14: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/14.jpg)
Underdamped Case
implies that C < 4L/R2
, i is used by the mathematicians for imaginary numbers
22
222
221
od
do
do
js
js
1j
![Page 15: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/15.jpg)
]sincos[)(
]sin)(cos)[()(
)]sin(cos)sin(cos[)(
sincos
sincos
)()(
21
2121
21
21
tjBtBetv
tAAjtAAetv
tjtAtjtAetv
je
je
eAeAetv
ddt
C
ddt
C
ddddt
C
j
j
tjtjtC
dd
211 AAB 212 AAB
![Page 16: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/16.jpg)
Angular Frequencieso 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 = 0, this will occur forever.
d is called the damped natural frequencySince 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.
![Page 17: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/17.jpg)
![Page 18: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/18.jpg)
Properties of RLC networkBehavior of RLC network is described as damping,
which is a gradual loss of the initial stored energyThe 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.
![Page 19: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/19.jpg)
Properties of RLC networkCritically 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.
![Page 20: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/20.jpg)
Set of Solutions when t > toThere are three different solutions which
depend on the magnitudes of the coefficients of the and the terms. 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
)(tvC dt
tdvC )(
![Page 21: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/21.jpg)
Transient Solutions when t > toOverdamped response ( > o)
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
ottt where
![Page 22: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/22.jpg)
Find CoefficientsAfter 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
![Page 23: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/23.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 24: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/24.jpg)
Solutions when t < toThe 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
![Page 25: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/25.jpg)
SummaryThe 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 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 26: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/26.jpg)
Parallel RLC Network
![Page 27: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/27.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 1 to 0 ora switch disconnects a voltage or current source
into the circuit.Describe the solution to the 2nd order
equations when the condition is:OverdampedCritically DampedUnderdamped
![Page 28: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/28.jpg)
RLC NetworkA parallel RLC network where the current
source is switched out of the circuit at t = to.
![Page 29: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/29.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
-) = 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
![Page 30: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/30.jpg)
Selection of ParameterInitial Conditions
iL(to-) = Is and v(to
-) = vC(to-) = 0V
vL(to-) = 0V and iC(to
-) = 0AFinal Conditions
iL(∞s) = 0A 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 31: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/31.jpg)
Kirchoff’s Current Law
0)()(1)(
0)()()(
)()()(
0)(
)()(
)()()()(
0)()()(
2
2
2
2
LC
ti
dt
tdi
RCdt
tid
tidt
tdi
R
L
dt
tidLC
dt
tdiLtvtv
dt
tdvCti
R
tv
tvtvtvtv
tititi
LLL
LLL
LL
CL
R
CLR
CLR
![Page 32: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/32.jpg)
General Solution
LCRCRCs
LCRCRCs
1
2
1
2
1
1
2
1
2
1
2
2
2
1
0112 LC
sRC
s
![Page 33: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/33.jpg)
LC
RC
o
12
1
222
221
o
o
s
s
02 22 oss
Note that the equation for the natural frequency of the RLC circuit is the same whether the components are in series or in parallel.
![Page 34: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/34.jpg)
Overdamped Case
implies that L > 4R2Cs1 and s2 are negative and real numbers
tstsLLL
o
tsL
tsL
eAeAtititi
ttt
eAti
eAti
21
2
1
2121
22
11
)()()(
)(
)(
![Page 35: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/35.jpg)
Critically Damped Case
implies that L = 4R2Cs1 = s2 = - = -1/2RC
ttL teAeAti
21)(
![Page 36: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/36.jpg)
Underdamped Case
implies that L < 4R2C
]sincos[)( 21
22
222
221
tAtAeti
js
js
ddt
L
od
do
do
![Page 37: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/37.jpg)
Other Voltages and CurrentsOnce current through the inductor is known:
Rtvtidt
tdvCti
tvtvtvdt
tdiLtv
RR
CC
RCL
LL
/)()(
)()(
)()()(
)()(
![Page 38: 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.](https://reader038.fdocuments.us/reader038/viewer/2022110323/56649d8d5503460f94a75c0b/html5/thumbnails/38.jpg)
SummaryThe 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 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.