Equations Overview and Relationships

Relation	Resistor (R)	Capacitor (C)	Inductor(L)
$v-i$:	$v=iR$	$v=\frac{1}{C}{\int }_{t_0}^{t}i(t)dt+v(t_0)$	$v=L\frac{di}{dt}$
$i-v:$	$I=v/R$	$i=C\frac{dv}{dt}$	$I=\frac{1}{L}{\int }_{t_0}^{t}v(t)dt+i(t_0)$
P or w	$p=I^{2}R=\frac{v^2}{R}$	$w=\frac{1}{2}C{v}^2$ ($w$ is the energy)	$w=\frac{1}{2}L{i}^2$ ($w$ is the energy)
Series	$R_{eq}=R_1+R_2$	$C_{eq}=\frac{C_1{C}_2}{C_1+C_2}$	$L_{eq}=L_1+L_2$
Parallel	$R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$	$C_{eq}=C_1+C_2$	$L_{eq}=\frac{L_1{L}_2}{L_1+L_2}$
Circuit variable that cannot change abruptly:	N/A	$V$ (Capacitors smooth voltage)	$i$ (Inductors smooth current)
* Other equations are also applicable but shown in each section

Capacitors

Overview

Types

Power Supply Caps

• "Ripple Capacitors"
• Generally used to convert AC to DC, big pulses of current
• Large electrolytic are used for this purpose
• Typically in the 1,000 to 100,000 $\mu F$ range

Coupling Caps

• Block DC and pass Ac, used in series
• Film capacitors are best typically 0.1 to 10 $\mu F$

Decoupling Caps

• Reduce noise, used in parallel with larger ones, can be used with microprocessors
• Often called "bypass" capacitors
• A variety can be used, typically 0.1 to 1,000 $\mu F$

Capacitors in Parallel

• Called banks

General Formulas

Relationship

$q=Cv$

Current

$I=C\frac{dv}{dt}$

Charge After Time

$q(t)={\int }_{t_0}^{t}i(t)dt+d(t_0)$

Dielectric constant

${ϵ}_0=8.85\times 10^{-12}F/m$

$C=\frac{ϵA}{d}$

Capacitors can have different dielectric constants, leading to proportionally increased or decreased capacitance.

Equivalent

Opposite manner of resistors.

Inductors

Properties

• If the current through an inductor is constant the voltage across it is zero
• Ideal inductors are treated as having zero resistance.
• With DC applied, inductor acts like 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
• Opposite of capacitor

Equivalent

Same manner as resistors.

Synthesis of Logic Circuits

Sum of Products

Row	A	B	C	D	Equation
0	0	0	0	1	(A'B'C')
1	0	0	1	0
2	0	1	0	1	(A'BC')
3	0	1	1	0
4	1	0	0	0
5	1	0	1	0
6	1	1	0	1	(ABC')
7	1	1	1	1	(ABC)

Product of Sums

Row	A	B	C	D	Equation
0	0	0	0	1
1	0	0	1	0	A+B+C'
2	0	1	0	1
3	0	1	1	0	A+B'+C'
4	1	0	0	0	A'+B+C
5	1	0	1	0	A'+B+C'
6	1	1	0	1
7	1	1	1	1

Diode

Properties

• Diodes generally have a voltage drop of 0.7V. Applied voltage has to be greater than 0.7V.
• P type is doped with group 3 (3 electrons in outside shell). Typically boron.
• To make N-type, add electrons, add group 5. Typically phosphorus.
• Anode = positive
• Cathode = negative

Number Conversions

Two's Complement

Overflow Rule

• If 2 Two's Complement numbers are added, and they both have the same sign (both positive or both negative), then overflow occurs if and only if the result has the opposite sign. Overflow never occurs when adding operands with different signs.
  • Adding two positive numbers must give a positive result
  • Adding two negative numbers must give a negative result