COROLLARY THEOREMS - ELECTRONINC DESIGN NOTES: Resistors

ELECTRONIC DESIGN NOTES #3

Resistors

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In Electronic Design Notes 1 it was mentioned we can control the current passing through electrical circuits using resistors. Of course, there are other means to achieve good current control, but resistors are simple, cheap, and very efficient electronic components.

Control is the true magic word in electronics. No matter what we do, the purpose is to take control of voltages, currents, frequencies, and of various functions. Once we achieve total control, then we can do whatever we can think of. Resistors help us control all electrical characteristics that are calculated based on resistance: that is, current, voltage, frequency, etc.

There are few basic things we need to know about resistors, therefore they are listed summarily as follows:
1. Types of Resistors
2. Useful Formulas
3. Resistors Color Code Chart
4. Equivalent Series and Parallel Resistors
5. Controlling Voltages and Currents
6. Detecting Current Sense
7. Frequency Control

NOTE
The basic notions highlighted in this page are related to few electronic design topics presented in the first part, Hardware Design, of LEARN HARDWARE FIRMWARE AND SOFTWARE DESIGN.

TYPES OF RESISTORS

There is an incredible variety of resistors, though in this page are presented only few main groups. First of all, resistors are groped in:
A. fixed
B. variable

Depending on the way we insert them into electrical circuits, resistors can be:
a. through hole (TH)
b. surface mount (SM)
c. various mechanical fixtures

Further, resistors are built as:
1. carbon composition
2. metal film
3. carbon film
4. wirewound

Of course we could differentiate resistors based on the power (current) they can safely handle, but there are way too many types available.

Variable resistors come in few particular types. They are important to note here because each type requires specific schematics:
1. potentiometers
2. rheostats
3. decades
4. programmable potentiometers (details of hardware schematics and firmware programming source code are in LEARN HARDWARE FIRMWARE AND SOFTWARE DESIGN)

Fig1: Rheostat and Potentiometer wiring Schematics

The Rheostat circuit is used to control currents (Iv)

The Potentiometer circuit is used to control voltages (Uv)

USEFUL FORMULAS

In order to work with resistors we need few formulas, as follows:
1. Resistance
2. Ohm's Law in DC circuits
3. Ohm's Law in AC circuits
4. Impedance
5. Resistance value at current temperature
6. Conductance
7. Electrical conductivity
8. DC power
9. Horse power conversion
10. Kilowatt to hp conversion
11. DC Energy
12. Efficiency

RESISTORS FORMULAS
FORMULA	NAME
*R [Ω]= ρ L / A**	Resistance (details are in Design Notes 1)
R [Ω] = U [V] / I [A]	Ohm's Law in DC circuits (details are in Design Notes 1)
Z [Ω] = U [V] / I [A]	Ohm's Law in AC circuits (details are in Design Notes 1)
Z = √[R² + (X_L - X_c)²]	Impedance (details are in Design Notes 1)
*X_L[Ω] = 2PIf [Hz]L [H]**	*Inductive Reactance*
*X_c[Ω]= 1/2PIf [Hz]C [C]**	*Capacitive Reactance*
*R_t = R_o (1 + α * t)**	R_t = Resistance at current temperature R_o = resistance at 0 Celsius t = actual temperature α = temperature coefficient of resistivity
G [Siemens] = 1 / R	*Conductance*
σ = 1/ρ	*Electrical conductivity*
P [W] = U [V] * I [A] P [W] = I²[A] * R [Ω]	*DC power*
1 [hp] = 746 [W]	*Conversion to horsepower*
1 [kW] = 1.34 [hp]	*Conversion to kilowatt*
*W [J] = U [V] I [A] * T [s]**	*DC Energy*
η = Pout / Pin	*Efficiency*

RESISTORS COLOR CODE CHART

Fixed resistors of the "through hole" type are marked using a special color code.

The new SM (surface mount) types are marked using numbering systems specific to each manufacturer--please consult their Data Sheets.

RESISTOR COLOR CODE
	Color	1st Band	2nd Band	3rd Band	Multiplier	Tolerance
	Black	0	0	0	10⁰	-
	Brown	1	1	1	10¹	(+/-)1%
	Red	2	2	2	10²	(+/-)2%
	Orange	3	3	3	10³	(+/-)3%
	Yellow	4	4	4	10⁴	(+/-)4%
	Green	5	5	5	10⁵	(+/-)0.5%
	Blue	6	6	6	10⁶	(+/-)0.25%
	Violet	7	7	7	10⁷	(+/-)0.1%
	Gray	8	8	8	10⁸/10^-2	-
	White	9	9	9	10⁹/10^-1	-
	Gold	-	-	-	10^-1	(+/-)5%
	Silver	-	-	-	10^-2	(+/-)10%
	None	-	-	-	-	(+/-)20%

NOTE
The third band is missing in most instances.

EQUIVALENT SERIES AND PARALLEL RESISTORS

The equivalent of series resistors is calculated with: R_T = Σ R_i
The equivalent of parallel resistors is calculated with: 1/R_T = Σ 1/R_i

Calculation examples for three resistors:

	Fig 2: The equivalent resistance of 3 series resistors R_T = R₁ + R₂ + R₃ R_T = 2K +3K + 4K = 9K The series equivalent R_T is greater than the greatest component
	Fig 3: The equivalent resistance of 3 resistors in parallel 1/R_T = 1/R₁ +1/R₂ + 1/R₃ R_T = R1R2R3 / (R2R3 + R1R3 +R1*R2) R_T = 24 / (12 + 8 + 6) = 24 / 26 = 0.923K The parallel equivalent R_T is always smaller than the smallest component

CONTROLLING VOLTAGES AND CURRENTS

First thing, please take a look at Figs 4 and 5.

	Fig 4: Voltage control circuit *Voltage divider formula: U_i = (U R_i) / R_T** U = 12 V = total voltage applied R_i = 3Ω = incremental resistance R_T = 1Ω + 3Ω = 4Ω = total resistance U_i= (12 * 3) / 4 = 9 V = incremental voltage
	Fig 5: Current control circuit I_max = U / R₁ = 12 / 3 = 4 mA

In Fig 4, we control the voltage U_i using a "voltage divider" schematic. We can even adjust U_i if we use a potentiometer schematic as we did in in Fig 1. The voltage divider schematic allows for 2, 3,..n precise voltage levels to be supplied to the "Load" circuit. In Fig 4 are the formulas needed to calculate the Ui value.

In Fig 5, R1 plays the role of a "current limiter". The meaning of that schematic is, the maximum current supplied to Load is 4 mA. Even if Load becomes a short-circuit, the maximum current will not exceed 4 mA. Please note: using a variable resistor wired as in Fig 1, the Rheostat, we can adjust the maximum/minimum current supplied to Load.

In both of the above schematics, Figs 4 and 5, we could use programmable potentiometers. Details about the simplest and the most efficient ways of working with programmable potentiometers are presented in LEARN HARDWARE FIRMWARE AND SOFTWARE DESIGN.

DETECTING CURRENT SENSE

Please take a look at Fig 6.

Fig 6: Detecting current sense

Possible cases:
V1 > V2 Load is drawing power
V1 = V2 Load is Open
V1 < V2 Load is generating power
V2 = 0 Load is short-circuited

In normal conditions V1 = 12 V, and V2 = 11.988 V (R1 = 1 KΩ). These values are sent to the Analog-to-Decimal channels 1 and 2 (randomly chosen) of the PIC controller. Next, we transform the analog voltages into their decimal equivalents, then we compare them mathematically. The result is one of the following instances:

V1 > V2 (Load is drawing power)
V1 = V2 (Load is Open)
V1 < V2 (Load is generating power)
V2 = 0 (Load is short-circuited)

If an accident happens and Load becomes a generator, the current will change its sense, therefore V2 will become greater than V1:

V2 > V1

Not only we are able to detect the sense of the current, but we know precisely how much current Load is drawing in each moment.

Details about working with Microchip dsPIC controllers and about AD conversion are presented in LEARN HARDWARE FIRMWARE AND SOFTWARE DESIGN.

FREQUENCY CONTROL

Frequency control is presented in Filters Design Notes.

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