boundary conditions. objective of lecture demonstrate how to determine the boundary conditions on...
TRANSCRIPT
![Page 1: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/1.jpg)
Boundary Conditions
![Page 2: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/2.jpg)
Objective of LectureDemonstrate how to determine the boundary
conditions on the voltages and currents in a 2nd order circuit. These boundary conditions will be used when
calculating the transient response of the circuit.
![Page 3: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/3.jpg)
2nd Order CircuitsA second order differential equation is
required to solve for the voltage across or the current flowing through a component.The circuit will contain at least one resistor
and the equivalent of two energy storage elements 2 capacitors, 2 inductors, or a capacitor and an
inductor
![Page 4: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/4.jpg)
Boundary ConditionsSteady state
For step response functions u(t- to) for all times betweent = +/- ∞ except for some time period after t = to
Capacitors are opens Inductors are short circuits
During the transition at the step t = to
Voltage across a capacitor is continuous vC(to
+) = vC (to -)
Current through an inductor is continuous iL(to
+) = iL(to -)
![Page 5: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/5.jpg)
Initial ConditionRedraw the circuit at t < to
Determine the value of all voltage and current sources at t< to
Make the appropriate substitutions for the energy storage devices.Substitute an open circuit (∞ resistor) for all
capacitors. Note: IC(t < to ) = 0A.
Substitute an short circuit (0 resistor) for all inductors. Note: VL(t < to ) = 0V.
Calculate VC(t < to ) and IL(t < to ).
![Page 6: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/6.jpg)
Final ConditionRedraw the circuit at t =∞ sDetermine the value of all voltage and current
sources at t =∞ sMake the appropriate substitutions for the
energy storage devices.Substitute an open circuit (∞ resistor) for all
capacitors. Note: iC(t =∞ s) = 0A.
Substitute an short circuit (0 resistor) for all inductors. Note: vL(t =∞ s) = 0V.
Calculate vC(t =∞ s) and iL(t =∞ s).
![Page 7: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/7.jpg)
Example #1
![Page 8: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/8.jpg)
Example #1 (con’t)Initial Condition: The circuit is:
![Page 9: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/9.jpg)
Example #1 (con’t)
∞
![Page 10: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/10.jpg)
iL (-∞) = iL (to-) = 0A vL (-∞) = vL (to
-) = 0V
iC (-∞) = iC (to-) = 0A vC (-∞) = vC (to
-) = [R2/(R1+R2)]V
∞
Example #1 (con’t)
![Page 11: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/11.jpg)
Example #1 (con’t)Final Condition: The switch opens,
which removes V1 and R1 from the circuit.
![Page 12: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/12.jpg)
Example #1 (con’t)The energy stored in the inductor and
capacitor will be dissipated through R2 and R3 as t increased from t= to.
![Page 13: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/13.jpg)
Example 1 (con’t)At time t = ∞s, the energy stored in the
inductor and in the capacitor will be completely released to the circuit.
∞
![Page 14: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/14.jpg)
Example #1 (con’t)
iL (∞s) = 0A vL (∞s) = 0V
iC (∞s) = 0A vC (∞s) = 0V
∞
![Page 15: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/15.jpg)
Example #1 (con’t)
For to < t << ∞s
iL (t) ≠ 0 vL t) ≠ 0
iC (t) ≠ 0 vC (t) ≠ 0
![Page 16: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/16.jpg)
Electronic ResponseDraw the circuits when t < to and t = ∞s for
the following circuit:
![Page 17: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/17.jpg)
Example #2 (con’t)
![Page 18: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/18.jpg)
Example #2 (con’t)
iL (-∞s) = 0.3mA vL (-∞s) = 0V
iC (-∞s) = 0A vC (-∞s) = 3.5V
![Page 19: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/19.jpg)
Example #2 (con’t)
iL (∞s) = 0A vL (∞s) = 0V
iC (∞s) = 0A vC (∞s) = 5V
![Page 20: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/20.jpg)
Example #3
![Page 21: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/21.jpg)
Example #3 (con’t)
iL1 (-∞s) = -1mA vL1 (-∞s) = 0V
iL2 (-∞s) = 1mA vL1 (-∞s) = 0V
![Page 22: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/22.jpg)
Example #3 (con’t)
![Page 23: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/23.jpg)
Example #3 (con’t)
iL1 (∞s) = -1mA vL1 (∞s) = 0V
iL2 (∞s) = 1.4mA vL2(∞s) = 0V
![Page 24: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/24.jpg)
Example #4
![Page 25: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/25.jpg)
iL1 (-∞s) =- 1 mAvL1 (-∞s) = 0V
iC1 (-∞s) = viC2 (-∞s) = 0A
vC1 (-∞s) = vC2 (-∞s) = 4V
Example #4 (con’t)
![Page 26: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/26.jpg)
Example # 4 (con’t)
iL1 (∞s) = 0mA vL1 (∞s) = 0V
vC1 (∞s) = vC2 (∞s) = 1V iC1 (∞s) = iC2 (∞s) = 0A
![Page 27: Boundary Conditions. Objective of Lecture Demonstrate how to determine the boundary conditions on the voltages and currents in a 2 nd order circuit. These](https://reader033.vdocuments.mx/reader033/viewer/2022051210/5516d9925503469d338b4853/html5/thumbnails/27.jpg)
Summary