1st Order Circuits

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1st Order Circuits
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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 by a 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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More Applications

• The low-pass filter for an envelope detector in a
  superhetrodyne AM receiver.
• A sample-and-hold circuit for a PCM encoder
• The capacitor is charged to the voltage of a
  waveform to be sampled.
• The capacitor holds this voltage until an A/D
  converter can convert it to bits.

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The Differential Equation(s)

• KCL 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.
• 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
• What value must t have to give a solution to

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Complementary Solution

• How do I choose the value of K?
• The 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 LC circuit, t L/R

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

• 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
• The low-pass filter for the envelope detector
• The sample-and-hold circuit
• The electrical motor

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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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Computer RAM

• Voltage across a memory capacitor may look like
  this

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Low Pass Filter

• Voltage in the filter may look like this

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Sample and Hold

• The voltage in the sample and hold circuit might
  look like this