# Network Analysis: Lecture: Source free RL circuit by bilalhasan

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```									 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/

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