1. Initial Conditions on L's and C's
ECE - UBC
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Basic assumption
Currents and voltages, all have finite values.
Also, energy cannot be “moved” or disposed of instantaneously.
10/05/2018
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3. Initial Conditions on L’s
Current in an inductor cannot change “suddenly” (that is, no finite change in zero time)…that would imply that its voltage is infinite!
10/05/2018
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4. Initial Conditions in an inductor
At any point in time t = a, the inductor current “right after ‘a’” is the same as “right before ‘a’”:
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5. Initial Conditions in an inductor
The current in an inductor cannot change instantaneously!
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…I.C. in an inductor
At any point in time t = a, the inductor current “right after ‘a’” is the same as “right before ‘a’”:
And … changes
In voltage?
Careful!
In general…
10/05/2018
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7. Initial Conditions in an inductor
In particular, at t = 0
If we know the current in the inductor before a switch operates ...
… that’ll be the current right after the switch operates (at least for an instant)
10/05/2018
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…I.C. in an inductor
Right after t = a, and only for an infinitesimal instant, the inductor behaves like a stubborn current source…
10/05/2018
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9. Tutorial Time What are the currents in the elements right
after the switch closes
at t = 5s?
The source is DC,
and a long time …
means it’s in SS.
(homework) Right after the switch closes, what is the voltage in the inductor?
At t = 5- (right before 5),
we were in DCSS!.
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Tutorial Time
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Tutorial Time
(1)
REF
Snapshot valid ONLY at
The moment t = 5+ sec
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12. Initial Conditions on C’s
Voltage in a capacitor cannot change “suddenly” (that is, no finite change in zero time)…lest its current becomes infinite!!!
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…I.C. in a capacitor
At any point in time t = a, the capacitor voltage “right after ‘a’” is the same as “right before ‘a’”:
Achtung!
In general…
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…I.C. in a capacitor
Right after t = a, and only for an infinitesimal instant, the capacitor behaves like a stubborn voltage source…
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Tutorial Time
The switch has been closed for a long time …
… it’s in steady state!
The source is DC …
… it’s DC-SS
At t = 5s, the switch opens. Determine the voltage in the capacitor right before the switch opens, and the current in the capacitor right after the switch opens.
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Snapshot at t = 5-
The switch has been closed for a long time …
… it’s in steady state!
The source is DC …
… it’s DC-SS
The capacitor is modeled as an open circuit, and the circuit solved for vc
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Snapshot at t = 5-
A voltage divider…
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Snapshot at t = 5+
The capacitor is represented by a V-src…
… with the voltage it had before the switch moved!
Now, the current in C…
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The Expo function
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20. Exponential functions
It appears often in the solution of circuits in transient conditions.
t is the time constant. At t = t, the function is only 37% of its original amplitude. At t = 5t …
…the value is so small that by engineering standards we consider that i(t) is zero!
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21. …exponential function
The derivative of an exponential function is proportional to itself.
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Exercise (homework)
A current i(t) is known to be exponential. Its value at t = 2.5 ms is 27 mA, its value at t = 8 ms is 5 mA.
Determine A, and “tau” (the time constant) in …
How long before the current decays by engineering standards?
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