Weatstone Bridge

The Wheatstone Bridge circuit consists of two potential dividers connected between the supply and ground. One divider is formed from R₁ and R₂, the other from R₃ and R₄. The voltage at the point between R₁ and R₂, and that between R₃ and R₄ will vary according to the resistor value. V_bridge is the voltage between these points, as shown in the diagram below:

When the voltages are equal V_bridge will be zero, at which point the bridge is said to be balanced. At this point the value of R₄ can be found from:

R₄ = (R₃ * R₂) / R₁

A typical application for a Wheatstone Bridge is for measuring resistance; if R₁ is variable and is adjusted it until V_bridge = 0 the value of R₄ can then be found from the above equation.
If all four resistor values (R₁ to R₄) and the supply voltage (V) are known the voltage across the bridge (V_bridge) can be found by working out the voltage from each potential divider and subtracting one from the other. The equation for this is:

V_bridge = (R₄ / (R₃ + R₄) * V) - (R₂ / (R₁ + R₂) * V)

This can be simplified to:

V_bridge = ((R₄ / (R₃ + R₄)) - (R₂ / (R₁ + R₂))) * V

If the bridge is balanced the equivalent resistance of the circuit between V and GND is:



R₁ + R₂ in parallel with R₃ + R₄
R_E = ((R₁ + R₂) * (R₃ + R₄)) / (R₁ + R₂ + R₃ + R₄)

WEEE / RoHS Directives

2 new European directives are due to come over the next 2 years, both of which will have an impact on the electronics industry and hobbyists alike. Now is the time to understand the implications of these directives and prepare for them.

WEEE (Waste Electrical and Electronic Equipment):

The WEEE directive will make manufacturers responsible for disposal and recycling of products they produce. A few products will be exempt from WEEE, but manufacturers of those that are not must make provisions for products to be returned to them at the end of their life cycle. The manufacturers must then bear the cost of recycling these products. It is expected that legislation will be introduced by the end of 2004, with compliance by August 2005.

Manufacturers must think about how they will provide for this, as well as designing products to minimise the costs and environmental impact. Currently over 6 million tons of waste electronic equipment is dumped each year, WEEE aims to reduce this by encouraging environmentally sound design and shifting responsibility for recycling onto manufacturers.

RoHS (Restriction of certain Hazardous Substances):

This directive will probably have a bigger impact than WEEE on hobbyists. From 2006 the use of 6 substances commonly used in electronics will be restricted. These are lead, cadmium, mercury, hexavalent chromium, polybrominated biphenyls and polybrominated diphenyl ethers. In practice RoHS is linked to WEEE, as the environmental impact of these substances is greater than the health risks these pose.



RoHS will mean standard tin/lead solder can no longer be used, this in itself presents big problems. Firstly it is not just the solder you assemble boards with that is affected - components and PCBs must be lead-free too. Secondly lead-free solders have a higher melting point, increasing the risk of thermal damage to components and boards. RoHS could also affect the flux used in the solder too, again affecting the solder's properties. Thirdly there are a number of lead-free alternatives, often incompatible with one another, and as yet no set standard. If the solder you are using is not compatible with that used to tin component leads or the PCB you may run into problems. That said I have recently used lead-free PCBs with lead solder and experienced no problems.

See the May 2004 edition of EPE magazine for an excellent article on lead-free soldering and it's impact. If you plan to stock up on components or solder you should check whether they are lead free. Check this in catalogues and on manufacturers websites. Lead-free components should be kept separate from those containing lead. Businesses will need to keep documents from suppliers showing components are compliant in case a product's compliance is contested. More information about both directives can be found on the RS and Farnell websites, as well as Envirowise.



It must be remembered RoHS doesn't only affect solder, it will also affect other components that contain banned substances, such as Ni-Cad batteries. However in most cases alternatives are readily available. Controlled use of hazardous substances in some products will be permitted, most notably mercury in fluorescent tubes.

SI Units

 

Quantity	Symbol	Units
Resistance	R	ohms (W)
Capacitance	C	Farads (F)
Conductance	G	Siemens (S)
Current	I	amperes (A)
Charge	Q	Coulombs (C)
Energy	W	joules (J)
Power	P	watts (W)
E.M.F	E	volts (V)
Mass	M	kilograms (Kg)
Length	L	metres (m)
Time	T	seconds (s)
Velocity	V	metres/second (m/s)
Angular Velocity	w	radians/second (r/s)

Series Combinations of R, C & L in A.C. Circuits

R and L in series:

The voltages dropped across the inductor and the resistor must be added together using phasor summation, i.e. adding the two voltages as they would be shown on a phasor diagram, if the second phasor is drawn starting from the end of the first phasor. Since V_L  is at right angles to V_R then the components are added using pythagoras's theorem, a² = b² + c².


The triangle this makes shows the voltages V_R and V_L, and the total voltage V on the hypotenuse (see below).

Since V = IZ, dividing all the sides by I gives R, X_L and Z, the impedances.

Similarly, multiplying all sides by I gives I²R, I²X_L and I²Z , the power dissipated in the components.
The term impedance (Z) is used for the combination of component resistances and reactances, e.g. the 'circuit impedance' refers to the total impedance of a circuit containing a number of resistors, inductors and capacitors whose resistances and reactances have been combined using phasor summation.

Phase Angle:

The angle between the adjacent and the hypotenuse of the triangles is the phase angle of the combined components. I lags V.

f = cos^-1 (R/Z)

R and C in series:

As for R and L in series, except that for the phase angle of the two combined components, I leads V.

R, L and C in series:

This can be visualised from drawing all 3 components on a phasor diagram. The resistance is in phase, therefore it's phasor will be horizontal. The inductance will then have it's phasor going upwards, but the capacitance phasor will come down again from this point. The hypotenuse of the triangle is from the end of the capacitance phasor back to the start. This hypotenuse is the resultant voltage drop.

Again, the values can be divided and multiplied by I to give impedance and power respectively.

V² = (V_R² + (V_L - V_C)²)



Take the square root of these to get the voltage dropped, V.

RMS Values

The Root Mean Square (R.M.S) value of a voltage or current is:

V_RMS = V_P / Ö2  and I_RMS = I_P / Ö2

where V_RMS and I_RMS are the R.M.S. values, and V_P and I_P are the peak values.

Ö2 » 1.414

Similarly, multiply R.M.S. by Ö2 to get the peak value.



Values quoted are R.M.S. unless otherwise stated - normally the R.M.S. subscript is left off.
E.g. 240V A.C. mains has a peak value of 240 X Ö2 = 340V peak.

R, C & L in A.C. Circuits

Here are some general formulae for working out values in an A.C. circuit:

Resistors:

Resistors are exactly the same as in D.C. circuits

I = V / R           P = I²R

Capacitors:

I = V / X_C

X_C is the capacitive reactance = 1 / (2pfC)
f is the frequency, C is the capacitance
(Remember Voltage lags Current by 90^o)

Inductors:

I = V / X_L

X_L is the inductive reactance = 2pfL
f is the frequency, L is the inductance
(Remember Voltage leads Current by 90^o)

Resistor Theory

Resistors provide an obstruction to the flow of electricity around a circuit. A voltage is dropped across a resistor, dependant on the current flowing through the resistor. Here are some useful formulae:

Voltage Dropped across a Resistor:

V = IR, where I is in Amps, and R in Ohms

This can be re-arranged to give Current or Resistance if Voltage is already known (see ohms law).

Power Dissipated in a Resistor:

P = I²R or V²/R where P is in Watts

Always make sure the resistor's power rating is not exceeded!

Resistors in Series:

Putting resistors in series increases the total resistance:
R = R₁ + R₂ = R₃ .....
Note that the total resistance should be greater than that of any of the individual resistors.

Resistors in Parallel:

Putting resistors in parallel reduces the total resistance:
(1/R) = (1/R₁) + (1/R₂) + (1/R₃) .....
If you have only 2 resistors in parallel you can use:
R = (R₁ R₂) / (R₁ + R₂)
Note that the total resistance will be less than that of any of the individual resistors.

Preferred Resistor Values

Resistors are available in a number of standard ranges, often called 'preferred values' These ranges, or series, are set out by the Electronic Industries Association (EIA), and are E3, E6, E12, E24, E48, E96 and E192. The number after the 'E' denotes the number of values the series contains per decade. The E3 and E6 series aren't really used these days; the most common series is probably E24. The series are logarithmic and are derived from the resistor tolerance; resistors with a tighter tolerance can have more values in the series that won't overlap one another. The series are sometimes referred to by the tolerance, the two being related as follows:



E3: 50% tolerance
E6: 20% tolerance
E12: 10% tolerance
E24: 5% tolerance
E48: 2% tolerance
E96: 1% tolerance
E192: less than 1% tolerance

When designing a circuit you need to round calculated resistor values to a preferred value. Normally this would be the value closest to the calculated one for the series you are using. However in some circumstances its better to round up to the next higher value, such as with current limiting resistors where using a value smaller than calculated could overload a component. The tables below show the values for some of the more common preferred series.

E12 Series:

1R0	10R	100R	1K0	10K	100K	1M0	10M
1R2	12R	120R	1K2	12K	120K	1M2
1R5	15R	150R	1K5	15K	150K	1M5
1R8	18R	180R	1K8	18K	180K	1M8
2R2	22R	220R	2K2	22K	220K	2M2
2R7	27R	270R	2K7	27K	270K	2M7
3R3	33R	330R	3K3	33K	330K	3M3
3R9	39R	390R	3K9	39K	390K	3M9
4R7	47R	470R	4K7	47K	470K	4M7
5R6	56R	560R	5K6	56K	560K	5M6
6R8	68R	680R	6K8	68K	680K	6M8
8R2	82R	820R	8K2	82K	820K	8M2

E24 Series:

1R0	10R	100R	1K0	10K	100K	1M0	10M
1R1	11R	110R	1K1	11K	110K	1M1
1R2	12R	120R	1K2	12K	120K	1M2
1R3	13R	130R	1K3	13K	130K	1M3
1R5	15R	150R	1K5	15K	150K	1M5
1R6	16R	160R	1K6	16K	160K	1M6
1R8	18R	180R	1K8	18K	180K	1M8
2R0	20R	200R	2K0	20K	200K	2M0
2R2	22R	220R	2K2	22K	220K	2M2
2R4	24R	240R	2K4	24K	240K	2M4
2R7	27R	270R	2K7	27K	270K	2M7
3R0	30R	300R	3K0	30K	300K	3M0
3R3	33R	330R	3K3	33K	330K	3M3
3R6	36R	360R	3K6	36K	360K	3M6
3R9	39R	390R	3K9	39K	390K	3M9
4R3	43R	430R	4K3	43K	430K	4M3
4R7	47R	470R	4K7	47K	470K	4M7
5R1	51R	510R	5K1	51K	510K	5M1
5R6	56R	560R	5K6	56K	560K	5M6
6R2	62R	620R	6K2	62K	620K	6M2
6R8	68R	680R	6K8	68K	680K	6M8
7R5	75R	750R	7K5	75K	750K	7M5
8R2	82R	820R	8K2	82K	820K	8M2
9R1	91R	910R	9K1	91K	910K	9M1

E48 Series:

The values of the decade 100-1000 for the E48 series are:

100, 105, 110, 115, 121, 127, 133, 140, 147, 154, 162, 169, 178, 187, 196, 205, 215, 226, 237, 249, 261, 274, 287, 301, 316, 332, 348, 365, 383, 402, 422, 442, 464, 487, 511, 536, 562, 590, 619, 649, 681, 715, 750, 787, 825, 866, 909, 953

Multiply or divide the values by 10 for other decades.

E96 Series:

The values of the decade 100-1000 for the E96 series are:

100, 102, 105, 107, 110, 113, 115, 118, 121, 124, 127, 130, 133, 137, 140, 143, 147, 150, 154, 158, 162, 165, 169, 174, 178, 182, 187, 191, 196, 200, 205, 210, 215, 221, 226, 232, 237, 243, 249, 255, 261, 267, 274, 280, 287, 294, 301, 309, 316, 324, 332, 340, 348, 357, 365, 374, 383, 392, 402, 412, 422, 432, 442, 453, 464, 475, 487, 491, 511, 523, 536, 549, 562, 576, 590, 604, 619, 634, 649, 665, 681, 698, 715, 732, 750, 768, 787, 806, 825, 845, 866, 887, 909, 931, 959, 976,

Multiply or divide the values by 10 for other decades.

E192 Series:

The values of the decade 100-1000 for the E192 series are:

100, 101, 102, 104, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 120, 121, 123, 124, 126, 127, 129, 130, 132, 133, 135, 137, 138, 140, 142, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 162, 164, 165, 167, 169, 172, 174, 176, 178, 180, 182, 184, 187, 189, 191, 193, 196, 198, 200, 203, 205, 208, 210, 213, 215, 218, 221, 223, 226, 229, 232, 234, 237, 240, 243, 246, 249, 252, 255, 258, 261, 264, 267, 271, 274, 277, 280, 284, 287, 291, 294, 298, 301, 305, 309, 312, 316, 320, 324, 328, 332, 336, 340, 344, 348, 352, 357, 361, 365, 370, 374, 379, 383, 388, 392, 397, 402, 407, 412, 417, 422, 427, 432, 437, 442, 448, 453, 459, 464, 470, 475, 481, 487, 493, 499, 505, 511, 517, 523, 530, 536, 542, 549, 556, 562, 569, 576, 583, 590, 597, 604, 612, 619, 626, 634, 642, 649, 657, 665, 673, 681, 690, 698, 706, 715, 723, 732, 741, 750, 759, 768, 777, 787, 796, 806, 816, 825, 835, 845, 856, 866, 876, 887, 898, 909, 920, 931, 942, 953, 965, 976, 988

Multiply or divide the values by 10 for other decades.

Resistor Colour Code

3 Band Resistor Codes:

Colour	Band 1	Band 2	Multiplier
Black	0	0	1
Brown	1	1	10
Red	2	2	100
Orange	3	3	1000
Yellow	4	4	10000
Green	5	5	100000
Blue	6	6	1000000
Violet	7	7
Grey	8	8
White	9	9

4 Band Resistor Codes:

Colour	Band 1	Band 2	Band 3	Multiplier
Black	0	0	0	1
Brown	1	1	1	10
Red	2	2	2	100
Orange	3	3	3	1000
Yellow	4	4	4	10000
Green	5	5	5	100000
Blue	6	6	6	1000000
Violet	7	7	7
Grey	8	8	8
White	9	9	9

Tolerance:

Colour:	None	Silver	Gold	Red	Brown
Tolerance:	20%	10%	5%	2%	1%

Potentiometers

The most common form of rotary control on electronic equipment is the potentiometer. They are used for applications such as volume or tone controls on audio equipment and are available in panel and PCB mounting versions. A potentiometer has three terminals and is basically a variable potential divider.

Operation:

A potentiometer has a resistive track (normally made of carbon), the ends of which are connected to two of the pins (T1 and T2). The vaolue marked on a potentiomer refers to the resistance of this track. The third pin is the the 'wiper' connection (W); this touches the track at an adjustable point set by the control shaft. The internal construction of a potentiometer is shown below:

Comparing this to a potential divider circuit  it can be seen that the two parts of the track either side of the wiper form R1 and R2, and the wiper provides the output from the divider.

Symbol and Connections:

The diagram below shows the standard symbol used for a potentiometer and typical connections when used as a volume control or similar. See below for details of using a potentiometer as a variable resistor:

E.G. for a linear potentiometer, if the input voltage (V_in) is 1V and the shaft is mid-way the output voltage (V_out) will be 0.5V.

Potentiometer Types:

Linear potentiometers have a constant resistance all along the track, so the output voltage is proportional to the position of the control shaft. For audio applications logarithmic types are more suitable; these have a non-linear track (supposedly logarithmic, but normally an approximation) to give a smooth response with logarithmic audio signals. Using a linear potentiometer as a volume control will give a large variation in level at one end of the rotation with little variation otherwise. Linear types are often marked A, with logarithmic types being marked B.
 

Other Potentiometer Configurations:

A potentiometer can be used as a variable resistor by only connecting one track connection and the wiper. In this case the resistance will be 0 at one end of the shaft's rotation, and the resistance quoted on the component at the other. Often in this configuration one end of the track is connected to the wiper instead of being left open-circuit (see diagram above).

Potential Divider

A Potential (or voltage) Divider is made up of two resistors. The output voltage from a potential divider will be a proportion of the input voltage and is determined by the resistor values.

Operation:

Consider a standard resistor connected across a voltage supply. If you were able to open it up and measure the voltage at any point along it you would find the voltage varied linearly along it (assuming the resistance was constant along it's length). For example if the resistor was connected between 10V and 0V the voltage half-way along it would be 5V. Similarly the voltage 10% from the 0V end would be 1V. The diagram below illustrates this:

A potential divider works in the same way. Obviously a resistor is normally enclosed so you can't tap the voltage off at any point along it, so two resistors are used; the tap point being between these resistors. The diagram below shows a potential divider circuit with the standard component notation used:

V₁ is the voltage in and V₂ is the voltage out.

Equation:

The voltage out of the divider is determined by the resistor values using the equation:
V₂ = (R₂ / (R₁ + R₂)) * V₁



By rearranging this and specifying the total resistance (R₁ + R₂) the resistor values can be found from the desired output voltage V₂.

Phasor Diagrams

Phasor diagrams can be used to represent voltages and currents in A.C. circuits. They show not only the magnitude (size) of the voltage or current, but also the phase angle of it.



See the diagrams below for some examples. Diagrams for components in series are drawn using the current as the reference phasor (since it is the same for all the components in series), those for parallel circuits use the voltage as the reference phasor.




The length of the line for each component represents the magnitude of the voltage or current, and the angle between it and the reference phasor is the phase angle.

Note that often phasor diagrams are only sketched, rather than drawn to scale!

Example:

Phase Angle in R, L & C

Inductors and Capacitors, or a combination of these and resistors in A.C. circuits have a phase angle - the current is out of phase with the voltage:

Resistance:

For a resistor the current and the voltage is in phase.

Inductance:

For an inductance Voltage leads Current by 90 degrees (p / 2 radians).

Capacitance:

For a capacitance Voltage lags Current by 90 degrees (p / 2 radians).

OHM's Law

This simple formula describes the relationship between voltage, current and resistance:

Voltage = Current X Resistance or V = IR
Current = Voltage / Resistance or I = V / R
Resistance = Voltage / Current or R = V / I



Ohms law can easily be remembered using the triangle below. Cover Up the value you want to find, and treat letters next to each other  as multiplications, and letters above each other as divisions. Try it!

Logic Gates

Logic gates give an output depending on the condition of the inputs. Below are the symbols and truth tables for the common types of logic gates. Note that due to font limitations the Boolean symbol for XOR expressed here as (+) is actually a + sign inside a circle.

AND:

	A	B	Output
	0	0	0
	0	1	0
	1	0	0
	1	1	1
	Expression: Output = A.B

OR:

A	B	Output
0	0	0
0	1	1
1	0	1
1	1	1
Expression: Output = A+B

XOR:

A	B	Output
0	0	0
0	1	1
1	0	1
1	1	0
Expression: Output = A(+)B

NOT:

Input	Output
0	1
1	0
Expression: Output = Input

NAND:

A	B	Output
0	0	1
0	1	1
1	0	1
1	1	0
Expression: Output = A.B

A NOT gate can be formed from a NAND gate by connecting A and B together.

NOR:

A	B	Output
0	0	1
0	1	0
1	0	0
1	1	0
Expression: Output = A+B

A NOT gate can be formed from a NOR gate by connecting A and B together.

XNOR:

A	B	Output
0	0	1
0	1	0
1	0	0
1	1	1
Expression: Output = A(+)B

A NOT gate can be formed from an XNOR gate by connecting one input to ground.

LED Current Limiting Resistor

LEDs (Light Emitting Diodes) must always be operated with a resistor in series to limit the current to a safe level. Most LEDs can only tolerate a current of about 30mA maximum. To calculate the resistor value required the forward voltage (V_f) and current (I_f)of the LED must be known. These can be found from the suppliers catalogue. For standard LEDs V_f is about 2V and I_f is about 20mA.
The resistor value can be calculated as follows:



R = (V_s - V_f)  / I_f

= (Supply Voltage - Forward Voltage) / Forward Current

Note that I_f is in Amps, NOT Milliamps! (20mA = 0.02A)

Kirchoff's Laws

Kirchoff's voltage and current laws basically state that voltage or current in a circuit must be accounted for - it cannot just 'disappear'. The laws are as follows:



Kirchoff's Current Law:

Kirchoff's Current Law states that 'the algebraic sum of the current meeting at any point in a circuit is zero'.

For example:

I3 = I1 + I2

or I1 + I2 - I3 = 0 (currents towards point designated as positive, those away from point negative).
Kirchoff's Current Law

In other words the sum of all currents entering a junction must equal the sum of those leaving it.



Kirchoff's Voltage Law:

Kirchoff's Voltage Law states that 'in travelling round any closed mesh (section) of a network (circuit) , the algebraic sum of the emfs (voltages) acting in the mesh is equal to the algebraic sum of the IR voltage drops for the individual resistance in the mesh.'

For example:

Working anticlockwise:

IR1 + IR2 = E1 - E2

Working clockwise:

-IR1 - IR2 = E2 - E1
Kirchoff's Voltage Law

In other words the sum of all voltage sources must equal the sum of all voltages dropped across resistances in the circuit, or part of circuit.

IP Ratings

IP numbers are often quoted on enclosures, connectors etc to indicate protection from solids, liquids and impact. The table below gives a description of these ratings. Note that the third number, protection against impacts, is often omitted.

IP #	First Number - Protection against solids	Second Number - Protection against liquids	Third Number - Protection against mechanical impacts (often omitted)
0	No Protection	No Protection	No Protection
1	Protected against solid objects over 50mm	Protected against vertically falling drops of water	Protected against impact of 0.225 joules
2	Protected against solid objects over 12mm	Protected against direct sprays up to 15 deg. from vertical	Protected against impact of 0.375 joules
3	Protected against solid objects over 2.5mm	Protected against direct sprays up to 60 deg. from vertical	Protected against impact of 0.5 joules
4	Protected against solid objects over 1mm	Protected against direct sprays from all directions - limited ingress permitted	Protected against impact of 2.0 joules
5	Protected against dust - limited ingress permitted	Protected against low pressure jets from all directions - limited ingress permitted	Protected against impact of 6.0 joules
6	Totally protected against dust	Protected against strong jets from all directions - limited ingress permitted	Protected against impact of 20.0 joules
7		Protected against effects of immersion from 15cm - 1m
8		Protected against long periods of immersion under pressure

DC Power

Power in DC circuits is calculated in the same way as Apparent Power in AC circuits:

Power = V I

= Voltage * Current

Units are Watts (W)

This equation can be combined with Ohms Law to give an expression for the power dissipated in a resistor as:




Power = I2 R

= (Current through Resistor)2 * Resistance

Units are Watts (W)

This is derived as follows:

Power = VI

V = I R (from Ohms Law)

Combining these gives Power = I I R

or Power = I2R

Conductance

Conductance is the reciprocal of resistance,

i.e. Conductance G = (1 / Resistance)

Units of Conductance are Siemens (S)




Calculating Resistors in Parallel Using Conductance:

G = G1 + G2 + G3 .....

(Total conductance is greater with more resistors in parallel so resistance is less.)

Calculating Resistors in Series Using Conductance:

(1/G) = (1/G1) + (1/G2) + (1/G3) .....

(Total conductance is less, because resistance is greater.)