ECE201 Lect-20 1

First-Order Circuits Cont’d

Dr. Holbert

April 17, 2006

ECE201 Lect-20 2

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.

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

ECE201 Lect-20 3

Important Concepts

• The differential equation for the circuit

• Forced (particular) and natural (complementary) solutions

• Transient and steady-state responses

• 1st order circuits: the time constant ()

ECE201 Lect-20 4

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 # of energy storage elements

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The Differential Equation

• 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.

)()(...)()(

01

1

1 tftxadt

txda

dt

txda

n

n

nn

n

n

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Building Intuition

• 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

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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

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The Forced Solution

• 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

)()(...)()(

01

1

1 tftxadt

txda

dt

txda

n

n

nn

n

n

ECE201 Lect-20 9

The Natural Solution

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

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

0)(...)()(

01

1

1

txadt

txda

dt

txda

n

n

nn

n

n

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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

/)( tc Ketx

0)(1)(

txdt

tdxc

c

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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.

ECE201 Lect-20 12

Transients and Steady State

• 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.

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Step-by-Step Approach

1. Assume solution (only dc sources allowed): x(t) = K1 + K2 e-t/

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

3. 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

4. At t=, repeat step #2 to find x()=K1

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Step-by-Step Approach

5. Find time constant ()Looking across the terminals of the C or L element, form Thevenin equivalent circuit; =RThC or =L/RTh

6. Finish upSimply put the answer together.

ECE201 Lect-20 15

Class Examples

• Learning Extension E7.3

• Learning Extension E7.4

• Learning Extension E7.5