Network Analysis: Lecture: Source free RL circuit

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Network Analysis: Lecture: Source free RL circuit Powered By Docstoc
					 First Order Circuits: RC and RL
• A zero order circuit has zero energy storage elements.
  (Called a “purely resistive” circuit.)

• A first order circuit has one (irreducible) energy storage
  element.

• The equations that solve it are first order differential
  equations.
• A second order circuit has two (irreducible) energy
  storage elements.

• The equations that solve it are second order differential
  equations.
              Natural response
• It is found by setting the input (forcing function) to zero.
• It is characteristic of the circuit, not of the sources (i.e.
  forcing functions)
• It will have the same number of arbitrary constants as
  the order of the differential equation. (These constants
  are determined from boundary conditions.)
• It provides a transition from the initial values of x (voltage
  or current) to the final value.
• It is also called the transient response.
• It is also called the complementary solution.
• It is also called the free response
       Source Free RL Circuit
• There is no voltage through a inductor if the
  current is not changing with time. A inductor is
  therefore an open circuit to dc.

• A finite amount of energy can be stored in a
  inductor even if the voltage through the inductor
  is zero, when the current across it is constant.

• The inductor never dissipates energy, but only
  stores it.
        Source Free RL Circuit
• Initial current through
  inductor i(0) ≠ 0

• Initial current will
  decay to zero
  overtime i(∞) = 0
            General Solution
• Apply KVL around the
  loop

• Put the equation into
  “standard” form

• Separate the
  variables-i and t
Integrate Both sides
Solve for i(t)
         Source Free RL Circuit




• The time constant  of a circuit is the time required
  for the response to decay by a factor of 1/e or 36.8%
  of its initial value.
• i(t) decays faster for small  and slower for large .
       Source Free RC Circuit
• There is no current through a capacitor if the
  voltage is not changing with time. A capacitor is
  therefore an open circuit to dc.

• A finite amount of energy can be stored in a
  capacitor even if the current through the
  capacitor is zero, when the voltage across it is
  constant.

• The capacitor never dissipates energy, but only
  stores it.
        Source Free RC Circuit
• Initial charge through
  capacitor v(0) ≠ 0

• Initial voltage will
  decay to zero
  overtime v(∞) = 0
            General Solution
• Apply KCL around the
  loop

• Put the equation into
  “standard” form

• Separate the
  variables v and t
Integrate Both sides
Solve for v(t)
                  Time Constant ‫זּ‬
 • The time constant  of a circuit is the time required for
   the response to decay by a factor of 1/e or 36.8% of its
   initial value.
 • v decays faster for small  and slower for large .

                                                Decays more slowly




Time constant   RC                         Decays faster
  Keys for Source-Free Circuit
• RC Circuits       • RL Circuits

• ‫ = זּ‬RC            • ‫ = זּ‬R/L

• Find v(0)         • Find i(0)

• v(t)=v(0)e– t/   • i(t)=i(0)e– t/