Circuits with Energy Storage Elements

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Circuits with Energy Storage Elements
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Introduction

• Circuits with energy storage elements behave
  significantly differently than circuits without
  energy storage elements.

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Without Energy Storage
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With Energy Storage
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Why is This?

• 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.

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Important Concepts

• The order of the circuit
• Forced and natural (homogeneous) responses
• Transient and steady state responses
• 1st order circuits-the time constant
• 2nd order circuits-natural frequency and the
  damping ratio

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The Order of the Circuit

• 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

• Every voltage and current is the solution to a
  differential equation.
• In a circuit of order n, these differential
  equations have order n.
• Equations are linear, constant coefficient

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The Differential Equation

• 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.

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Example RL Circuit
Find the differential equation for v(t)
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Element Currents
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KCL at the Top Node
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Example-RLC Circuit
Find the differential equation for i(t)
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Element Voltages
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KVL Around the Loop
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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
• Roots of the characteristic equation

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The Solution to the Differential Equation

• The solution to any differential equation
  consists of two parts
• v(t) vp(t) vc(t)
• Particular (forced) solution is vp(t)
• Response particular to a given source
• Complementary (natural) solution is vc(t)
• Response common to all sources

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The Particular Solution

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

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Particular Solution
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Thought Question

• What form for the particular solution would you
  expect for the following functions?
• f(t) 10 cos(2p 377t)
• f(t) 20 e-0.001t

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The Complementary Solution

• The complementary solution is the solution to

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Complementary Solution
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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.

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Effect of Initial Conditions

• I computed vc(t) in the IF Amplifier filter in
  response to a step input.
• The initial inductor current is zero.
• Three initial capacitor voltages were used 0V,
  -1V, and 1V.

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Initial Capacitor Voltage of 0V
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Initial Capacitor Voltage of -1V
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Initial Capacitor Voltage of 1V
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Transients and Steady State

• The steady state response of a circuit is the
  waveform after a long time has passed.
• DC SS if response approaches a constant.
• AC SS if response approaches a sinusoid.
• The transient response is the circuit response
  minus the steady state response.

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What are the Transient and Steady State Responses?