ECE 1004 – Lecture 24

Exam #2 Review
 ECE 1004

v Today’s lecture:

Ø Exam 2 review

Lab Day on Friday

Exam 2 is Monday

2
 Exam 2
• 5 problems with some multiple choice/short-
answer questions
• On Chapters 3, 7 up to 7.4, and class notes.
• It will be held just like Exam 1
• Don’t forget to bring your calculator and a pencil
• Redo all the homework and in-class problems for
practice
Capacitors
 Capacitance Equations
Capacitor equations
t

C
q equations
These
=current to () =ò ()
q allow qfortcapacitor voltage
t dt + and/or
V be= calculated at any point in time: t 0
i q ( )
V C t0
Initial condition, if any.
t t
q (qt )(t ) 11
V (tV)(=t ) = = = òò i(it(t))dt
dt ++ v(t00)) integral form
C C CCt0 t 0

dV (t
dV (t ) )
integral form i (t )
Differential form i (t ) = C= C
Differential form
dt
dt
 Properties of Capacitors
• Ideal capacitors all have these characteristics:
• When the voltage is not changing, the current through the
cap is zero.
• This means that with DC applied to the terminals, no current
will flow – cap acts like an open circuit.
• The voltage on the capacitor’s plates can’t change
instantaneously.
• An abrupt change in voltage would require an infinite current!
• This means if the voltage on the cap does not equal the
applied voltage, charge will flow and the voltage will finally
reach the applied voltage.
 Common types of capacitors

Can be expensive
Common in audio

Dry electrolytic

Tantalum,OSCON,Electrolytic are polarized.

 Nonidealities of real capacitors
The “working voltage” is the voltage limit that
cannot be exceeded.
The conductive plates
The dielectric is not a of the capacitor have
perfect insulator non-zero resistance
à Rleakage à RESR

We will discuss
inductors later

Changing current leads to inductance

This is just something to keep à LESL
in mind. We will assume ideal (we will learn about inductances next)
capacitors in the class
problems.
 Types of capacitors in the real-world
• Power supply caps
• Often called “ripple capacitors”
• Large electrolytics are used for this purpose
• Typically in the 1,000 to 100,000 μF range
• Coupling caps
• block DC and pass AC, used in series
• Film capacitors are best, typically 0.1 to 10 μF
• Decoupling caps
• Reduce noise, used in parallel with larger ones
• Often called “bypass” capacitors
• A variety can be used, typically 0.1 to 1000 μF
http://www.richtek.com/en/Design%20Support/Technical%20Document/AN038
 Capacitances in
Parallel
dV
i1 = C1
dt
dV
i2 = C2 The same voltage across all capacitors
dt
dV This the standard way of connecting
i3 = C3 capacitors. They are commonly used as
dt
“banks” of parallel caps to increase Farads

- i + i1 + i2 + i3 = 0 Apply KCL to the top node

dV dV dV dV dV
i = i1 + i2 + i3 = C1 + C2 + C3 = (C1 + C2 + C3 ) = Ceq
dt dt dt dt dt
Ceq = C1 + C2 + C3
 Capacitances in
Series
the same current through all
capacitors
Apply KVL: v1+v2+v3-v=0
t t t
1 1 1
v = v1 + v2 + v2 = ò
C1 0
i (t )dt +
C2 ò0 i(t )dt + C3 ò0 i(t )dt
t t t t
Q 1 1 1 1
v= =
Ceq Ceq ò0 i(t )dt = C1 ò0 i(t )dt + C2 ò0 i(t )dt + C3 ò0 i(t )dt =
é1 1 1 ù
t
The only time this is done is to increase
=ê + + ú ò i (t ) dt the working voltage of the capacitors
ë C1 C2 C3 û 0 (e.g., two 250V caps in series can handle
1 1 1 1 500V across them)
= + +
Ceq C1 C2 C3
 Energy Stored in a Capacitor
Calculate power p(t) first: q(t ) q(t )
C= v(t ) =
dv (t ) v(t ) C
p(t ) = v(t )i (t ) = v(t )C
dt
t t v (t )
dv
w(t ) = ò p(t )dt =C ò v dt = C ò vdv =
t0 t0
dt 0
2
1 2 1 q (t )
= Cv (t ) = v(t )q(t ) = 1 2
2 2 2C w = Cv
2
P3.25: Equivalent Capacitance

z z

z
 Energy Stored in a Capacitor
Calculate power p(t) first: q(t ) q(t )
C= v(t ) =
dv (t ) v(t ) C
p(t ) = v(t )i (t ) = v(t )C
dt
t t v (t )
dv
w(t ) = ò p(t )dt =C ò v dt = C ò vdv =
t0 t0
dt 0
2
1 2 1 q (t )
= Cv (t ) = v(t )q(t ) = 1
2 2 w = Cv 2C 2

2
Inductors
 Magnetic Field Lines

Magnetic fields can The flux density

be visualized as lines vector B is tangent
of flux that form to the lines of flux
closed paths

B = Magnetic flux density

 Inductors
• If a current is passed through an inductor, the voltage
across it is directly proportional to the time rate of
change in current
𝑑𝑖
𝑣! = 𝐿
𝑑𝑡
• Where, L, is the unit of inductance, measured in
Henries, H.
• One Henry is 1 volt-second per ampere.
• The voltage developed tends to oppose a changing
flow of current.
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 Current in an Inductor
• The current voltage relationship for an inductor
!
is: 1
𝐼= % 𝑣 𝜏 𝑑𝜏 + 𝑖(𝑡" )
𝐿
!!
• The power delivered to the inductor is:
𝑑𝑖
𝑝 = 𝑣𝑖 = 𝐿 𝑖
𝑑𝑡
• The energy stored is:
1 #
𝑤 = 𝐿𝑖 Very similar to capacitor formula
2
18
 Properties of Inductors
• If the current through an inductor is constant, the
voltage across it is zero
• With DC applied, inductor acts like a short circuit
• The current through an inductor cannot change
instantaneously
• If this did happen, the voltage across the inductor
would be infinity!
• This is an important consideration if an inductor is to
be turned off abruptly; it will produce a high voltage
• Basically the opposite of a capacitor
 Equivalent Inductance
(5H )(5H ) 25H 2
Leq = = = 2.5H Leq = 2H + 3H = 5H
5H + 5H 10H

It’s just like resistors

Leq = 1H + 2.5H = 3.5H

 Inductors and Capacitors under
DC conditions
• If a circuit is DC, then nothing is time-varying
• Therefore the di/dt and dv/dt are zero
• This means inductor voltage = 0 (looks like a short)
• And capacitor current = 0 (looks like an open circuit)
Representation of data
Decimal conversion to new bases
1. Separate into integer and fractional parts.
2. Convert the integer part
1. Repeatedly divide the integer part by the new
radix, or base.
2. The digits for the new radix are the remainders in
reverse order of computation.
3. Convert fractional part
1. Repeatedly multiply the fraction by the new radix.
2. The integer digits for the new radix fraction are the
integer digits of the product in order of
computation.
Example: Convert 34210 to Binary
Example: Convert 0.39210 to binary
Conversion from Binary to Decimal
• For integer part
– Use positive powers of 2 starting at 0 and add up the
powers.
• For the decimal part
– Use negative powers of 2 starting at -1 and add up the
powers.
• Examples:
– 101 = 1x22 + 0x21 + 1x20 = 4 + 0 + 1 = 5
– 1010 = 1x23 + 0x22 + 1x21 + 0x20 = 8 + 0 + 2 + 0 = 10
– 1.1 = 1x20 + 1x2-1 = 1+1/2 = 1.5
– 1.011 = 1x20 + 0x2-1 + 1x2-2 + 1x2-3 = 1+0+1/4+1/8 = 1.375
Octal is generally older, was
convenient for 7 segment
displays, but is still used in
some Unix applications, and
a few others as well.

Hex is the default in

addressing today due to its
compact representation
 Converting Binary to Octal
• Converting Binary to Octal is easy to do.
• To do this, you group bits in groups of 3.
– If bits are not a multiple of 3, add leading zeros
• Convert the 3 bits to a decimal number.
• The number in octal is the combination of these
numbers.
– 101101.010110
2

5 5 2 6
– 55.268
 Convert Binary to Hex
• Groups bits in groups of 4 (nibbles)
• Quick and easy to do.
• You can use hex as shorthand for binary.
– FFF vs. 111111111111
 2’s Complement Representation
• The most common way to represent a signed
binary integer on a computer is 2’s
complement.
• Pros
– Fast to compute the 2’s complement.
– No special hardware needed to compute basic
math.
• Cons
– Range is limited due to sign bit
 How to convert a number to 2’s
complement:
• Positive numbers are not changed.
• Negative numbers are converted in the
following way:
– Flip all the bits (called 1’s complement)
• 0s become 1s
• 1s become 0s
– Add 1 to the flipped bits (called 2’s complement)
– Ignore any overflow.
• Overflow is a carry that would require more bits to
store than the number of bits available
 Example

1110 0001 Carry bits

33 0010 0001 Minuend
+ -27 + 1110 0101 Subtrahend
6 0000 0110

the subtrahend is 27 in 2s complement. 27 is 00011011

Boolean algebra
 Logic gates
and their truth
tables
 Two Approaches for Synthesizing
Two approaches
Logic for synthesizing logic
Circuits
circuits

• Suppose we have a truth table for a circuit.

• Suppose we want to find a logic circuit
corresponding to the truth table.
• We can use either of the following:
– Sum of products
– Product of sums
 Sum of Products

• We look for the rows with a corresponding 1

in the output column.
• We produce the minterms for of the rows
where there is a 1 in the output
• The minterm is a product of all the inputs
• The total logic equation is the sum of
products.
 Product of Sums
• An alternative way is to look for the rows with
a corresponding 0 in the output column.
• Write a sum for each row where D is 0.
• Sum term that contains all input variables are
maxterms.
• The maxterm is a sum of all the inputs
• The total logic equation is the product of
sums.
Output high: Rows 0, 2, 6, 7
represent the SOP

Output low: Rows 1, 3, 4, 5

represent the POS
Products

Sum
Sums

Product
Diodes
27

Diode symbol and structure

http://conceptselectronics.com/diodes/structure-pn-junction-diode/
43

Diode (continued)

http://volga.eng.yale.edu/index.php/main/semiconductors
 Junction voltage drop
Anode Cathode

The diode always “drops” 0.7V across it.

When you apply a voltage greater than 0.7V to a
diode’s anode, it will turn on and conduct current.
 Biasing LEDs

• The current has to be limited to 20 mA DC for long life

• Or a PWM signal can be used to extend the life (covered next)
• A resistor is necessary.
• To calculate resistor, you must take into account the 2 V drop!
https://www.daenotes.com/electronics/industrial-electronics/LED-light-emitting-diode

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 Pulse-Width Modulation

• PWM is a pulse “train” of digital “0 or 1” voltage pulses

• It allows one to easily dim the LEDs, and reduces power loss
• We make them blink thousand of times per second to extend life
• It has become the standard way of controlling motors and LEDs
See you on LAB DAY a.k.a. Friday!

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