Differential Equation Solutions of Transient Circuits Dr Holbert
![Differential Equation Solutions of Transient Circuits Dr. Holbert March 3, 2008 Lect 12 EEE Differential Equation Solutions of Transient Circuits Dr. Holbert March 3, 2008 Lect 12 EEE](https://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-1.jpg)
Differential Equation Solutions of Transient Circuits Dr. Holbert March 3, 2008 Lect 12 EEE 202 1
![1 st Order Circuits • Any circuit with a single energy storage element, an 1 st Order Circuits • Any circuit with a single energy storage element, an](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-2.jpg)
1 st Order Circuits • Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1 • Any voltage or current in such a circuit is the solution to a 1 st order differential equation Lect 12 EEE 202 2
![RLC Characteristics Element Resistor V/I Relation DC Steady-State V=IR Capacitor I = 0; open RLC Characteristics Element Resistor V/I Relation DC Steady-State V=IR Capacitor I = 0; open](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-3.jpg)
RLC Characteristics Element Resistor V/I Relation DC Steady-State V=IR Capacitor I = 0; open Inductor V = 0; short ELI and the ICE man Lect 12 EEE 202 3
![A First-Order RC Circuit + vr(t) – R vs(t) + – + vc(t) C A First-Order RC Circuit + vr(t) – R vs(t) + – + vc(t) C](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-4.jpg)
A First-Order RC Circuit + vr(t) – R vs(t) + – + vc(t) C – • One capacitor and one resistor in series • The source and resistor may be equivalent to a circuit with many resistors and sources Lect 12 EEE 202 4
![The Differential Equation + vr(t) – R vs(t) + – + C – vc(t) The Differential Equation + vr(t) – R vs(t) + – + C – vc(t)](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-5.jpg)
The Differential Equation + vr(t) – R vs(t) + – + C – vc(t) KVL around the loop: vr(t) + vc(t) = vs(t) Lect 12 EEE 202 5
![RC Differential Equation(s) From KVL: Multiply by C; take derivative Multiply by R; note RC Differential Equation(s) From KVL: Multiply by C; take derivative Multiply by R; note](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-6.jpg)
RC Differential Equation(s) From KVL: Multiply by C; take derivative Multiply by R; note vr=R·i Lect 12 EEE 202 6
![A First-Order RL Circuit + is(t) R L v(t) – • One inductor and A First-Order RL Circuit + is(t) R L v(t) – • One inductor and](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-7.jpg)
A First-Order RL Circuit + is(t) R L v(t) – • One inductor and one resistor in parallel • The current source and resistor may be equivalent to a circuit with many resistors and sources Lect 12 EEE 202 7
![The Differential Equations + is(t) R L v(t) – KCL at the top node: The Differential Equations + is(t) R L v(t) – KCL at the top node:](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-8.jpg)
The Differential Equations + is(t) R L v(t) – KCL at the top node: Lect 12 EEE 202 8
![RL Differential Equation(s) From KCL: Multiply by L; take derivative Lect 12 EEE 202 RL Differential Equation(s) From KCL: Multiply by L; take derivative Lect 12 EEE 202](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-9.jpg)
RL Differential Equation(s) From KCL: Multiply by L; take derivative Lect 12 EEE 202 9
![1 st Order Differential Equation Voltages and currents in a 1 st order circuit 1 st Order Differential Equation Voltages and currents in a 1 st order circuit](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-10.jpg)
1 st Order Differential Equation Voltages and currents in a 1 st order circuit satisfy a differential equation of the form where f(t) is the forcing function (i. e. , the independent sources driving the circuit) Lect 12 EEE 202 10
![The Time Constant ( ) • The complementary solution for any first order circuit The Time Constant ( ) • The complementary solution for any first order circuit](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-11.jpg)
The Time Constant ( ) • The complementary solution for any first order circuit is • For an RC circuit, = RC • For an RL circuit, = L/R • Where R is the Thevenin equivalent resistance Lect 12 EEE 202 11
![What Does vc(t) Look Like? = 10 -4 Lect 12 EEE 202 12 What Does vc(t) Look Like? = 10 -4 Lect 12 EEE 202 12](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-12.jpg)
What Does vc(t) Look Like? = 10 -4 Lect 12 EEE 202 12
![Interpretation of • The time constant, , is the amount of time necessary for Interpretation of • The time constant, , is the amount of time necessary for](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-13.jpg)
Interpretation of • The time constant, , is the amount of time necessary for an exponential to decay to 36. 7% of its initial value • -1/ is the initial slope of an exponential with an initial value of 1 Lect 12 EEE 202 13
![Applications Modeled by a 1 st Order RC Circuit • The windings in an Applications Modeled by a 1 st Order RC Circuit • The windings in an](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-14.jpg)
Applications Modeled by a 1 st Order RC Circuit • The windings in an electric motor or generator • Computer RAM – A dynamic RAM stores ones as charge on a capacitor – The charge leaks out through transistors modeled by large resistances – The charge must be periodically refreshed Lect 12 EEE 202 14
![Important Concepts • The differential equation for the circuit • Forced (particular) and natural Important Concepts • The differential equation for the circuit • Forced (particular) and natural](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-15.jpg)
Important Concepts • The differential equation for the circuit • Forced (particular) and natural (complementary) solutions • Transient and steady-state responses • 1 st order circuits: the time constant ( ) • 2 nd order circuits: natural frequency (ω0) and the damping ratio (ζ) Lect 12 EEE 202 15
![The Differential Equation • Every voltage and current is the solution to a differential The Differential Equation • Every voltage and current is the solution to a differential](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-16.jpg)
The Differential Equation • Every voltage and current is the solution to a differential equation • In a circuit of order n, these differential equations have order n • The number and configuration of the energy storage elements determines the order of the circuit • n number of energy storage elements Lect 12 EEE 202 16
![The Differential Equation • Equations are linear, constant coefficient: • The variable x(t) could The Differential Equation • Equations are linear, constant coefficient: • The variable x(t) could](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-17.jpg)
The Differential Equation • Equations are linear, constant coefficient: • The variable x(t) could be voltage or current • The coefficients an through a 0 depend on the component values of circuit elements • The function f(t) depends on the circuit elements and on the sources in the circuit Lect 12 EEE 202 17
![Building Intuition • Even though there an infinite number of differential equations, they all Building Intuition • Even though there an infinite number of differential equations, they all](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-18.jpg)
Building Intuition • Even though there an infinite number of differential equations, they all share common characteristics that allow intuition to be developed: – Particular and complementary solutions – Effects of initial conditions Lect 12 EEE 202 18
![Differential Equation Solution • The total solution to any differential equation consists of two Differential Equation Solution • The total solution to any differential equation consists of two](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-19.jpg)
Differential Equation Solution • The total solution to any differential equation consists of two parts: x(t) = xp(t) + xc(t) • Particular (forced) solution is xp(t) – Response particular to a given source • Complementary (natural) solution is xc(t) – Response common to all sources, that is, due to the “passive” circuit elements Lect 12 EEE 202 19
![Forced (or Particular) Solution • The forced (particular) solution is the solution to the Forced (or Particular) Solution • The forced (particular) solution is the solution to the](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-20.jpg)
Forced (or Particular) Solution • The forced (particular) solution is the solution to the non-homogeneous equation: • The particular solution usually has the form of a sum of f(t) and its derivatives – That is, the particular solution looks like the forcing function – If f(t) is constant, then x(t) is constant – If f(t) is sinusoidal, then x(t) is sinusoidal Lect 12 EEE 202 20
![Natural/Complementary Solution • The natural (or complementary) solution is the solution to the homogeneous Natural/Complementary Solution • The natural (or complementary) solution is the solution to the homogeneous](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-21.jpg)
Natural/Complementary Solution • The natural (or complementary) solution is the solution to the homogeneous equation: • Different “look” for 1 st and 2 nd order ODEs Lect 12 EEE 202 21
![First-Order Natural Solution • The first-order ODE has a form of • The natural First-Order Natural Solution • The first-order ODE has a form of • The natural](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-22.jpg)
First-Order Natural Solution • The first-order ODE has a form of • The natural solution is • Tau ( ) is the time constant • For an RC circuit, = RC • For an RL circuit, = L/R Lect 12 EEE 202 22
![Second-Order Natural Solution • The second-order ODE has a form of • To find Second-Order Natural Solution • The second-order ODE has a form of • To find](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-23.jpg)
Second-Order Natural Solution • The second-order ODE has a form of • To find the natural solution, we solve the characteristic equation: which has two roots: s 1 and s 2 • The complementary solution is (if we’re lucky) Lect 12 EEE 202 23
![Initial Conditions • The particular and complementary solutions have constants that cannot be determined Initial Conditions • The particular and complementary solutions have constants that cannot be determined](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-24.jpg)
Initial Conditions • The particular and complementary solutions have constants that cannot be determined without knowledge of the initial conditions • The initial conditions are the initial value of the solution and the initial value of one or more of its derivatives • Initial conditions are determined by initial capacitor voltages, initial inductor currents, and initial source values Lect 12 EEE 202 24
![2 nd Order Circuits • Any circuit with a single capacitor, a single inductor, 2 nd Order Circuits • Any circuit with a single capacitor, a single inductor,](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-25.jpg)
2 nd Order Circuits • Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2 • Any voltage or current in such a circuit is the solution to a 2 nd order differential equation Lect 12 EEE 202 25
![A 2 nd Order RLC Circuit i (t) R vs(t) + – C L A 2 nd Order RLC Circuit i (t) R vs(t) + – C L](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-26.jpg)
A 2 nd Order RLC Circuit i (t) R vs(t) + – C L The source and resistor may be equivalent to a circuit with many resistors and sources Lect 12 EEE 202 26
![The Differential Equation i(t) + vr(t) – R vs(t) + – + vc(t) C The Differential Equation i(t) + vr(t) – R vs(t) + – + vc(t) C](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-27.jpg)
The Differential Equation i(t) + vr(t) – R vs(t) + – + vc(t) C – vl(t) + – L KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) Lect 12 EEE 202 27
![RLC Differential Equation(s) From KVL: Divide by L, and take the derivative Lect 12 RLC Differential Equation(s) From KVL: Divide by L, and take the derivative Lect 12](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-28.jpg)
RLC Differential Equation(s) From KVL: Divide by L, and take the derivative Lect 12 EEE 202 28
![The Differential Equation Most circuits with one capacitor and inductor are not as easy The Differential Equation Most circuits with one capacitor and inductor are not as easy](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-29.jpg)
The Differential Equation Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form: Lect 12 EEE 202 29
![Class Examples • Drill Problems P 6 -1, P 6 -2 • Suggestion: print Class Examples • Drill Problems P 6 -1, P 6 -2 • Suggestion: print](http://slidetodoc.com/presentation_image_h/ba25b0b24693d0ae5b61c3b7be1303a4/image-30.jpg)
Class Examples • Drill Problems P 6 -1, P 6 -2 • Suggestion: print out the two-page “First and Second Order Differential Equations” handout from the class webpage Lect 12 EEE 202 30
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