1. Topics to be Discussed Steady State and Transient Response.
The Simple RL Circuit.
Concept of Time Constant.
Meaning of Time Constant.
Growth of Current in Series RL Circuit.
The Simple RC Circuit.
Discharging of a Capacitor.
Charging of a Capacitor.
Comparison between RC and RL Circuits.
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Ch. 8 DC Transients
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2. Steady State
Both the inductance and capacitance are energy-storing elements.
When connected to a dc source, energy starts flowing to these elements.
Initially the rate of flow of energy is high, but as more and more energy is stored, the rate of flow decreases.
When maximum possible energy has been stored, the flow of energy stops altogether. We say that the circuit has reached its ‘steady state’.
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Ch. 8 DC Transients
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3. Transient Response
If we switch off the source, or switch over the network to another source, the circuit starts attaining another ‘steady state’.
The time taken by the circuit to change over from one steady-state condition to another steady-state condition is called transient time.
The response of the circuit during this time is known as transient response.
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4. The Simple RL Circuit
At t = 0-, a steady current that has been flowing in the circuit,
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5. For t > 0+, applying KVL,
Re-writing the equation to separate variables and then integrating,
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6. Decaying Exponential Curve
At t = 0+, the current is I0. As time increases, the current decreases and approaches zero.
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7. Concept of Time Constant
From equation, we see that with larger L/R ratio, the current takes longer to decay.
By doubling L/R, the “width” of the curve also doubles.
The “width” is proportional to L/R.
Instead of “width”, we use the concept of “time constant (τ).
It is the time that would be required for the current to drop to zero if it continued to drop at its initial rate.
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8. The initial rate of decay = the slope of line AB
From triangle OAB,
The ratio L/R must have the units of time.
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9. Meaning of Time Constant
Determining the value of i(t)/I0 at t = τ, we have
Thus, in one time constant the response drops to 36.8 % of its initial value. Hence,
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10. How long does it take for the current to decay to zero ?
Ans. : To answer this question, let us calculate
Click
Above values are shown on the graph.
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11. It takes about five time constants for the current to decay to zero.
i(3τ) = I0,
i(4τ) = I0,
i(5τ) = I0,
… … …
It takes about five time constants for the current to decay to zero.
At the end of this time interval, the current is less than one percent of its original value.
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12. Example 1
The circuit has been in the condition shown for a long time. The switch is opened at t = 0.
Determine the current i(0+) = I0.
Find vR across 20-Ω resistor at the instant just after the switch is opened.
Find vL across the inductor immediately after the switch is opened.
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13. The current supplied by the 24-V source,
Solution : (i) Under steady-state condition, the voltage drop across an inductor is zero and it behaves as a short-circuit. The equivalent resistance faced by the 24-V source is
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The current supplied by the 24-V source,
Click
By current division,
Click
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14. The circuit after the switch is opened (t > 0).
Immediately after the switch is opened, the current through the inductor remains the same. Hence,
The circuit after the switch is opened (t > 0).
(ii) The voltage across the 20-Ω resistor,
Click
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15. Because of this change, there will be an induced emf in the inductor.
(iii)
Immediately after the switch is opened the current starts decaying at the initial rate.
Because of this change, there will be an induced emf in the inductor.
This emf in the inductor will have a polarity (as shown) so as to oppose the decay in current.
The induced emf is given as
Click
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Ch. 8 DC Transients
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16. Growth of Current in Series RL Circuit
Since the current in an inductor cannot change by a finite amount in zero time, we must have i(0+) = 0.
After t = 0, the current slowly increases and approaches its steady state value I0 = V/R.
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Ch. 8 DC Transients
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17. The response to this circuit for t > 0 can be found as
The value of i(t)/I0 at t = τ,
Click
Thus, in one time constant the response rises to 63.2 % of its final value.
It takes about five time constants for the current to grow to its final steady state value.
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Ch. 8 DC Transients
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18. Rate of Growth of Current
The initial rate of growth of current is given by the slope of the curve at the origin.
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Thus, the smaller the value of L, the faster the current rises to its final value.
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19. Example 2
A coil having an inductance of 14 H and a resistance of 10 Ω is connected to a dc voltage source of 140 V, through a switch.
(a) Calculate the value of current in the circuit at an instant 0.4 s after the switch has been closed.
(b) Once the current reaches its final steady state value, how much time it would take the current to drop to 8 A after the switch is opened ?
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20. Solution : The time constant,
(a) The final steady state value of the current,
Click
The value of current at t = 0.4 s is given by
Click
Click
(b) For decaying current,
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21. The Simple RC Circuit (Discharging of a Capacitor)
(a) The circuit (b) The voltage response.
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22. The solution of this equation is
The total current leaving the node at the top of the circuit diagram must be zero. Therefore,
The solution of this equation is
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The time constant, τ = RC
As per definition of the time constant τ, the slope of the curve at t = 0 is given as –V0/ τ.
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23. Charging of a Capacitor
(a) The circuit. (b) The voltage and current response.
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24. The response of the circuit can be determined as
Since the voltage across a capacitor cannot change instantaneously, we must have
It means that initially the capacitor behaves as a short-circuit. The value of the initial current in the circuit,
The response of the circuit can be determined as
Wednesday, November 21, 2018
Ch. 8 DC Transients
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25. Example 3
The single-pole double-throw switch S has been in position b for a long time so that the 5-μF capacitor is fully discharged. Now, at t = 0, the switch is thrown to position a. Determine
(a) v(0+),
(b) i(0+),
(c) time constant τ,
(d) v and i at t = 15 ms.
Wednesday, November 21, 2018
Ch. 8 DC Transients
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26. Solution :
(a)
Since the voltage across a capacitor cannot change instantaneously, we have v(0+) = v(0) = 0 V
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(b)
(c)
Click
(d)
At t = 15 ms :
Click
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27. Single-Capacitor RC Circuit and Single-Inductor RL Circuit
The time constant for
For LR circuit,
For CR circuit,
Here, RTh is Thevenin resistance as “seen” by the capacitor or inductor.
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30. The voltages and currents approach their final values asymptotically.
It means that they never actually reach them.
However, after five time-constants they change by 99.3 % of their total change.
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31. (For solving Problems)
Important Note
(For solving Problems)
If immediately after switching,
v(0+) and i(0+) are initial values
and v(∞) and i(∞) are final values.
Then, the expressions for all the voltages and currents in the circuit for any time t are given as
Wednesday, November 21, 2018
Ch. 8 DC Transients
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32. Comparison between RC and RL Circuits
Though both give similar response, but we prefer RC over RL circuit, because
Inductors are not as nearly ideal as capacitors.
Inductors are relatively bulky, heavy and difficult to fabricate, especially using integrated-circuit techniques.
Inductors are relatively costlier.
The magnetic field emanating from the inductors can induce unwanted voltages in other components.
Wednesday, November 21, 2018
Ch. 8 DC Transients
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33. Review Steady State and Transient Response. The Simple RL Circuit.
Concept of Time Constant.
Meaning of Time Constant.
Growth of Current in Series RL Circuit.
The Simple RC Circuit.
Discharging of a Capacitor.
Charging of a Capacitor.
Comparison between RC and RL Circuits.
Wednesday, November 21, 2018
Ch. 8 DC Transients
Next