1. Inductors and Capacitors
Topic 15
Inductors and Capacitors
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2. Magnetic Fields and Current
Magnetic fields are created by moving charges
So if we vary the current through these coils
i.e, current!
A loop in the wire concentrates the field
…we have our own antenna to generate voltage
A bunch of loops concentrates it even more
A commercial antenna—its loop generates a voltage across a small gap…
when it intercepts a time-varying magnetic field
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Inductors & Capacitors

3. The Inductor DC Circuits i +v L i + -
An inductor is a device that stores electrical energy in a magnetic field caused by an electrical current
In the same way that a resistor imposes a constraint between…
…the constraint for an inductor is given by:
…the voltage across it…
…and the current through it…
…in the form of Ohm’s Law…
DC Circuits So
The currents and voltages are unchanging
Since v = 0 for any DC current
An inductor is a short circuit at DC
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Inductors & Capacitors

4. The Inductor i v
100 mH + -
A 100mH inductor is driven by a time-varying current source
2e-1=2/e
Let’s make a quick sketch of the time-varying current i t
10 A/s
10t is a straight line that starts at the origin and rises with a slope of 10 A/s
e-5t starts at 1 and decays exponentially i t i t
So their product should look like this
But we need some scale A
0.736
0.2
sec
Which is 0 at .2 secs and infinity
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Inductors & Capacitors

5. The Inductor The voltage jumps up instantaneously to 1 volt +
100mH + -
The voltage jumps up instantaneously to 1 volt
What is the voltage?
secs
volts
1.0
for
Graphing the voltage
0.4
0.2
Note that at
-0.135
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Inductors & Capacitors

6. Current for a Specified Voltage+ -
Suppose we know v(t) but not i. How do we work it out?
t0 is often taken
to be 0 t t0
Some time when the current is known
A later time
Some time in the past
Now
Now
Some time in the future
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Inductors & Capacitors

7. Example 6.2 i v
100 mH + -
Find the current and sketch both voltage and current as a function of time
2e-1 v t
We already plotted this shape
Max at 0.1
We need a scale
.736 .1
From the table of integrals in Appendix G
Now find the current
Replace t with dummy variable τ, which will integrate out
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Inductors & Capacitors

8. Example 6.2 i + v 100 mH - What about a time scale?
secs
amps 2
What about a time scale?
.528
Max voltage occurs at 0.1 secs .2 .3 .4 .1
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Inductors & Capacitors

9. Power and Energy in Inductorsv L + - i So or
So long as di/dt and dw/dt are
finite between t0 and t
Energy so
This condition is satisfied if i(t) is continuous over that interval
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Inductors & Capacitors

10. Power and Energy in Inductorsv L + - i
For well behaved currents
The energy at time t only depends on the current at time t
…plus the initial conditions
If we can find a time, t0, where both w(t0) and i(t0) are 0, then
Can we guarantee to find such a time?
How about when it was built?
So we write more simply
(for i continuous with no initial energy or current)
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Inductors & Capacitors

11. Inductors & Capacitors
Assessment 6.1 ig v
4mH + -
a) Find v(0) So
b) The time >0 when v(t) passes through 0
c) The expression for the power delivered to the inductor
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Inductors & Capacitors

12. Inductors & Capacitors
Assessment 6.1 ig v
4mH + -
c) The expression for the power delivered to the inductor
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Inductors & Capacitors

13. Inductors & Capacitors
d) The time when the power delivered to the inductor is maximum
Let
(d) Plugging .411 mS
into power equation
yields 32.7 watts
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Inductors & Capacitors

14. e) The maximum energy stored
Hey!
We did that already!
e) The maximum energy stored
We found where v went to 0
Max energy occurs for max current
So take derivative of i wrt t and set to 0
From part (b), v (t) went to 0 at 1.54 mS so di/dt must also go to 0 at the same time.
Which is the
answer to part f
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Inductors & Capacitors

15. The Capacitor The capacitor is a device for holding charge q=Cv
The proportionality constant is known as the dielectric constant
A vacuum and many materials have a basic dielectric constant ε0…
C is a measure of how much charge a capacitor can hold for a given voltage—its capacity
But some special materials—known as dielectrics—have a much bigger constant
The bigger the area of the plates the more charge can be held
The closer they are, the more electrical attraction & the more charge can be held
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Inductors & Capacitors

16. Cutaway of an electrolytic capacitor
(big & cluncky, polarized, poor at high frequencies)
An old-fashioned radio air-gap tuning capacitor
A modern tantalum capacitor
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Inductors & Capacitors

17. The Capacitor DC Circuits i + v C-
The constraint on v and i for a capacitor is given by:
DC Circuits
(unchanging currents and voltages)
Since i = 0 for any DC voltage So i
A capacitor is an open circuit at DC
This is hardly surprising….
….where’s a current going to go?
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Inductors & Capacitors

18. Voltage for a Specified Current+ -
Suppose we know i(t) but not v. How do we work it out?
Again, t0 is often
taken to be 0 t t0
Some time when the voltage is known
A later time
Some time in the past
Now
Now
Some time in the future
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Inductors & Capacitors

19. Power and Energy in Capacitorsv C i + - So or
So long as dv/dt and dw/dt are
finite between t0 and t—i.e
v(t) must be continuous
Energy so
For v continuous
and w(t0) & v(t0)
zero or
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Inductors & Capacitors

20. Inductors & Capacitors
Assessment 6.2 v
0.6µF i + -
Find (a) i(0), (b) the power delivered to the capacitor at t = π/80 mS and (c) the energy stored in the cap at that same time
(a)
The text has this wrong
(b)
(c)
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Inductors & Capacitors

21. Combining Inductors v1 v2 v3 i + - + - + - + L1 L3 L2
All the inductors share the same current
i.e., they are in series
Just like resistors!
So inductors in series add
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Inductors & Capacitors

22. Combining Inductors v + - i L1 L2 L3 i1 i2 i3
All the inductors share the same voltage
i.e., they are in parallel
where or
Just like resistors again!
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Inductors & Capacitors

23. Combining Caps v + - i C1 i1 i2 i3 C2 C3
All the capacitors share the same voltage
i.e., they are in parallel
Which is clearly equivalent to
where
This is the opposite
of resistors!
So capacitors in parallel add
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Inductors & Capacitors

24. Combining Caps v + - i C1 v1 v2 v3 C2 C3
All the capacitors share the same current
i.e., they are in series
Again, opposite resistors!
where or
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Inductors & Capacitors

25. Combining Caps v + - i C1 v1 v2 v3 C2 C3 So this v Ceq i + -
Can be replaced by this
where
But it is understood that
That is, C1||C2 stands for a computation, not a circuit configuration
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Inductors & Capacitors

26. Combining Caps v + - i C1 i1 i2 i3 C2 C3
This makes good physical sense
Total capacitance increases
Combining caps in parallel effectively increases their area, increasing their capacity to hold charge
Combining them in series effectively increases their gap, decreasing their capacity to hold charge v + - i C1 v1 v2 v3 C2 C3
Total capacitance decreases
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Inductors & Capacitors