FirstOrder Circuits 6.16.2

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First-Order Circuits (6.1-6.2)

• Prof. Phillips
• March 24, 2003

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1st 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 1st order differential equation.

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Important Concepts

• The differential equation
• Forced and natural solutions
• The time constant
• Transient and steady-state waveforms

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A First-Order RC Circuit

• One capacitor and one resistor
• The source and resistor may be equivalent to a
  circuit with many resistors and sources.

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Applications Modeled bya 1st Order RC Circuit

• 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.

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The Differential Equation(s)

• KVL around the loop
• vr(t) vc(t) vs(t)

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Differential Equation(s)
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What is the differential equation for vc(t)?
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A First-Order RL Circuit

• One inductor and one resistor
• The source and resistor may be equivalent to a
  circuit with many resistors and sources.

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Applications Modeled by a 1st Order LC Circuit

• The windings in an electric motor or generator.

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The Differential Equation(s)

• KCL at the top node

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The Differential Equation
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1st Order Differential Equation

• Voltages and currents in a 1st order circuit
  satisfy a differential equation of the form

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Important Concepts

• The differential equation
• Forced (particular) and natural (complementary)
  solutions
• The time constant
• Transient and steady-state waveforms

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The Particular Solution

• The particular solution vp(t) is usually a
  weighted sum of f(t) and its first derivative.
• That is, the particular solution looks like the
  forcing function
• If f(t) is constant, then vp(t) is constant.
• If f(t) is sinusoidal, then vp(t) is sinusoidal.

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The Complementary Solution

• The complementary solution has the following
  form
• Initial conditions determine the value of K.

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Important Concepts

• The differential equation
• Forced (particular) and natural (complementary)
  solutions
• The time constant
• Transient and steady-state waveforms

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The Time Constant (?)

• The complementary solution for any 1st order
  circuit is
• For an RC circuit, t RC
• For an RL circuit, t L/R

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What Does vc(t) Look Like?
t 10-4
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Interpretation of t

• The time constant, t, is the amount of time
  necessary for an exponential to decay to 36.7 of
  its initial value.
• -1/t is the initial slope of an exponential with
  an initial value of 1.

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Implications of the Time Constant

• Should the time constant be large or small
• Computer RAM
• A sample-and-hold circuit
• An electrical motor
• A camera flash unit

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Important Concepts

• The differential equation
• Forced (particular) and natural (complementary)
  solutions
• The time constant
• Transient and steady-state waveforms

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Transient Waveforms

• The transient portion of the waveform is a
  decaying exponential

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Steady-State Response

• The steady-state response depends on the
  source(s) in the circuit.
• Constant sources give DC (constant) steady-state
  responses.
• Sinusoidal sources give AC (sinusoidal)
  steady-state responses.

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LC Characteristics
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Class Examples