Transient Analysis DC Steady-State

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Transient AnalysisDC Steady-State

• ELEC 308
• Elements of Electrical Engineering
• Dr. Ron Hayne
• Images Courtesy of Allan Hambley and Prentice-Hall

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Transient Analysis

• Scope of study
• Circuits that contain sources, switches,
  resistances, inductances, and capacitances
• Transients
• Time-varying currents and voltages resulting from
  sudden application of sources, usually due to
  switching
• Transient Analysis
• Involves using circuit concepts from Chapters 1
  2
• Current-voltage relationships for inductances and
  capacitances involve derivatives and integrals
• Create circuit equations that are differential
  equations

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First-Order RC or RL Circuits

• First-Order RC Circuits
• Contains DC sources, resistances, and a SINGLE
  capacitance
• First-Order RL Circuits
• Contains DC sources, resistances, and a SINGLE
  inductance
• Frequently used in timing applications
• Due to time constant

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First-Order Circuit Algorithm

• Apply KCL, KVL, and/or Ohms Law to write the
  circuit equation.
• If equation contains integrals, differentiate
  each term to produce a PURE differential
  equation.
• Assume a solution of the form K1K2est.
• Substitute the solution into the diff. eqn. to
  determine the values of K1 and s.
• Use the initial conditions to determine the value
  of K2.
• Write the final solution.

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Discharge of a Capacitance
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Discharge of a Capacitance
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Discharge of a Capacitance
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Discharge of a Capacitance
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Time Constant

• Time interval t RC is called the time constant
  of the circuit
• After t5t, vC(t)0

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Charging a Capacitance
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Charging a Capacitance
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Charging a Capacitance
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Charging a Capacitance

• First term is STEADY-STATE RESPONSE
• Or FORCED RESPONSE
• Second term is TRANSIENT RESPONSE

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Charging a Capacitance
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DC Steady State

• Transient terms in the expressions for current
  and voltages in RLC circuits decay to zero with
  time
• For DC sources, steady-state currents and
  voltages are CONSTANT
• For steady-state conditions with DC sources
• CAPACITANCES behave like OPEN circuits
• INDUCTANCES behave like SHORT circuits

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Capacitance in DC Steady-State

• Remember current through a capacitance
• If voltage is constant, current is _________.
• CAPACITANCE behaves just like an ____ circuit

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Inductance in DC Steady-State

• Remember voltage across an inductance
• If current is constant, voltage is _________.
• INDUCTANCE behaves just like a ______ circuit

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Steady-State DC Analysis

• Find vx and ix for t gtgt 0

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Exercise 4.3

• Find va and ia for t gtgt 0

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RL Transient Analysis

• Find i(t) and v(t)

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RL Transient Analysis

• Time interval t L/R is called the time constant
  of the circuit
• After t5t, i (t)2

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Example 4.3

• Find i(t) and v(t)

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Example 4.3
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Exercise 4.6

• Find i(t) and v(t)

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Summary

• Transient Analysis
• First Order RC Circuits
• First Order RL Circuits
• DC Steady State