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First-Order Circuits Cont d - PowerPoint PPT Presentation

First-Order Circuits Cont’d. Dr. Holbert April 17, 2006. Introduction. In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations. Real engineers almost never solve the differential equations directly.

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First-Order Circuits Cont’d

Dr. Holbert

April 17, 2006

ECE201 Lect-20

• In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations.

• Real engineers almost never solve the differential equations directly.

• It is important to have a qualitative understanding of the solutions.

ECE201 Lect-20

• The differential equation for the circuit

• Forced (particular) and natural (complementary) solutions

• 1st order circuits: the time constant ()

ECE201 Lect-20

• 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 # of energy storage elements

ECE201 Lect-20

• Equations are linear, constant coefficient:

• The variable x(t) could be voltage or current.

• The coefficients an through a0 depend on the component values of circuit elements.

• The function f(t) depends on the circuit elements and on the sources in the circuit.

ECE201 Lect-20

• Even though there are 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

ECE201 Lect-20

• 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

ECE201 Lect-20

• The forced (particular) solution is the solution to the non-homogeneous equation:

• The particular solution is usually has the form of a sum of f(t) and its derivatives.

• If f(t) is constant, then vp(t) is constant

ECE201 Lect-20

• The natural (or complementary) solution is the solution to the homogeneous equation:

• Different “look” for 1st and 2nd order ODEs

ECE201 Lect-20

• The first-order ODE has a form of

• The natural solution is

• Tau (t) is the time constant

• For an RC circuit, t = RC

• For an RL circuit, t = L/R

ECE201 Lect-20

• 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.

ECE201 Lect-20

• The steady-state response of a circuit is the waveform after a long time has passed, and depends on the source(s) in the circuit.

• Constant sources give DC steady-state responses

• DC SS if response approaches a constant

• Sinusoidal sources give AC steady-state responses

• AC SS if response approaches a sinusoid

• The transient response is the circuit response minus the steady-state response.

ECE201 Lect-20

• Assume solution (only dc sources allowed):

x(t) = K1 + K2 e-t/

• At t=0–, draw circuit with C as open circuit and L as short circuit; find IL(0–) or VC(0–)

• At t=0+, redraw circuit and replace C or L with appropriate source of value obtained in step #2, and find x(0)=K1+K2

• At t=, repeat step #2 to find x()=K1

ECE201 Lect-20

• Find time constant ()

Looking across the terminals of the C or L element, form Thevenin equivalent circuit; =RThC or =L/RTh

• Finish up