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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

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

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

ECE201 Lect-20

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.

ECE201 Lect-20

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

ECE201 Lect-20

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

ECE201 Lect-20

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

ECE201 Lect-20

The Natural Solution

- The natural (or complementary) solution is the solution to the homogeneous equation:
- Different “look” for 1st and 2nd order ODEs

ECE201 Lect-20

First-Order Natural Solution

- 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

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

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.

ECE201 Lect-20

Step-by-Step Approach

- 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

Step-by-Step Approach

- Find time constant ()

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

- Finish up

Simply put the answer together.

ECE201 Lect-20

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