Resistances in Series-Parallel

Sets of resistors, all having identical ohmic values, can be connected together in parallel sets of series networks, or in series sets of parallel networks. By doing this, the total power-handling capacity of the resistance can be greatly increased over that of a single resistor.

Sometimes, the total resistance of a series-parallel network is the same as the value of any one of the resistors. This is always true if the components are identical, and are in a network called an n-by-n matrix. That means, when n is a whole number, there are n parallel sets of n resistors in series (following figure A), or else there are n series sets of n resistors in parallel (following figure B). Either arrangement
gives the same practical result.

Series-parallel resistances.
  At A, sets of series resistors are connected in parallel.
  At B, sets of parallel resistances are connected in series. These examples show symmetrical n-by-n matrices with n = 3.

Engineers and technicians sometimes use series-parallel networks to obtain resistances with large power-handling capacity. A series-parallel array of n by n resistors will have n2 times that of a single resistor. Thus, a 3 × 3 series-parallel matrix of 2 W resistors can handle up to 32 × 2 = 9 × 2 = 18 W, for example. A 10 × 10 array of 1-W resistors can dissipate up to 100 W. The total power handling capacity is multiplied by the total number of resistors in the matrix. But this is true only if all the resistors have the same ohmic values, and the same power-dissipation ratings.

It is unwise to build series-parallel arrays from resistors with different ohmic values or power ratings. If the resistors have values and/or ratings that are even a little nonuniform, one of them might be subjected to more current than it can withstand, and it will burn out. Then the current distribution in the network can change so a second component fails, and then a third. It’s hard to predict the current and power distribution in an array when its resistor values are all different.

If you need a resistance with a certain power-handling capacity, you must be sure the network can handle at least that much power. If a 50-W rating is required, and a certain combination will handle 75 W, that’s fine. But it isn’t good enough to build a circuit that will handle only 48 W. Some extra tolerance, say 10 percent over the minimum rating needed, is good, but it’s silly to make a 500-W network using far more resistors than necessary, unless that’s the only convenient combination given the parts available.

Nonsymmetrical series-parallel networks, made up from identical resistors, can increase the power handling capability over that of a single resistor. But in these cases, the total resistance is not the same as the value of the single resistors. The overall power-handling capacity is always multiplied by the total number of resistors, whether the network is symmetrical or not, provided all the ohmic values are identical. In engineering work, cases sometimes arise where nonsymmetrical networks fit the need.

Division of Power

When combinations of resistances are connected to a source of voltage, they draw current. You can figure out how much current they draw by calculating the total resistance of the combination, and then considering the network as a single resistor.

If the resistors in the network all have the same ohmic value, the power from the source is evenly distributed among them, whether they are hooked up in series or in parallel. For example, if there are eight identical resistors in series with a battery, the network consumes a certain amount of power, each resistor bearing 1⁄ 8 of the load. If you rearrange the circuit so that the resistors are in parallel, the circuit will dissipate a certain amount of power (a lot more than when the resistors were in series), but again, each resistor will handle 1⁄ 8 of the total power load.

If the resistances in the network do not all have identical ohmic values, they divide up the power unevenly. Situations like this are discussed in the next section.

Resistances in Parallel

When resistances are placed in parallel, they behave differently than they do in series. One way to look at resistances in parallel is to consider them as conductances instead. In parallel, conductances add up directly, just as resistances add up in series. If you change all the ohmic values to siemens, you can add these figures up and convert the final answer back to ohms.
The symbol for conductance is G. This figure, in siemens, is related to the resistance R, in ohms, by these formulas :

G = 1/R
R = 1/G

Problem 1

Consider five resistors in parallel. Call them R1 through R5, and call the total resistance R as shown in following figure. Let the resistance values be as follows: R1 = 100 Ω, R2 = 200 Ω, R3 = 300 Ω, R4 = 400 Ω, and R5 = 500 Ω. What is the total resistance, R, of this parallel combination?


Five resistors of values R1 through R5, connected in parallel, produce a net resistance R. Illustration for above Problems

Converting the resistances to conductance values, you get: G1 = 1/100 = 0.01 S, G2 = 1/200 = 0.005 S, G3 = 1/300 = 0.00333 S, G4 = 1/400 = 0.0025 S, and G5 = 1/500 = 0.002 S.
Adding these gives G = 0.01 + 0.005 + 0.00333 + 0.0025 + 0.002 = 0.0228 S. The total resistance is therefore R = 1/G = 1/0.0228 = 43.8 Ω.

Problem 2

Suppose you have five resistors, called R1 through R5, connected in parallel as shown in Problem 1 figure. Suppose all the resistances, R1 through R5, are 4.70 kΩ. What is the total resistance, R, of this combination?

When you have two or more resistors connected in parallel and their resistances are all the same, the total resistance is equal to the resistance of any one component divided by the number of components. In this example, convert the resistance of any single resistor to 4700 Ω, and then divide this by 5. Thus, you can see that the total resistance is 4700/5 = 940 Ω.

In a situation like this, where you have a bunch of resistors connected together to operate as a single unit, the total resistance is sometimes called the net resistance. Take note, too, that R is not italicized when it means resistor, but R is italicized when it means resistance!

Resistances in Series

When you place resistances in series, their ohmic values add together to get the total resistance. This is easy to imagine, and it’s easy to remember!


Suppose resistors with the following values are connected in series, as shown in following figure 112 Ω, 470 Ω, and 680 Ω. What is the total resistance of the series combination?
Simply add up the values, getting a total of 112 + 470 + 680 = 1262 Ω. You might round this off to 1260 Ω. It depends on the tolerances of the resistors—how precise their actual values are to the ones specified by the manufacturer.


Three resistors in series. Illustration for mentioned problem . Resistance values are in ohms.

Power Calculations


A circuit for working Ohm’s Law problems

You can calculate the power P, in watts, in a dc circuit such as that shown in above figure, by using the formula P = EI. This formula tells us that the power in watts is the product of the voltage in volts and the current in amperes. If you are not given the voltage directly, you can calculate it if you know the current and the resistance.

Recall the Ohm’s Law formula for obtaining voltage: E = IR. If you know I and R but you don’t know E, you can get the power P this way:
P = EI = (IR)I = I 2R
Suppose you’re given only the voltage and the resistance. Remember the Ohm’s Law formula for obtaining current: I = E/R. Therefore:
P = EI = E(E/R) = E2/R

Problem 1

Suppose that the voltmeter in above figure reads 12 V and the ammeter shows 50 mA. What is the power dissipated by the potentiometer?
Use the formula P = EI. First, convert the current to amperes, getting I = 0.050 A. (Note that the last 0 counts as a significant digit.) Then multiply by 12 V, getting P = EI = 12 × 0.050 = 0.60 W.

Problem 2

If the resistance in the circuit of above figure is 999 Ω and the voltage source delivers 3 V, what is the power dissipated by the potentiometer?
Use the formula P = E2/R = 3 × 3/999 = 9/999 = 0.009 W = 9 mW. You are justified in going to only one significant figure here.

Problem 3

Suppose the resistance in above figure is 47 kΩ and the current is 680 mA. What is the power dissipated by the potentiometer?
Use the formula P = I 2R, after converting to ohms and amperes. Then P = 0.680 × 0.680 × 47,000 = 22,000 W = 22 kW. (This is an unrealistic state of affairs: an ordinary potentiometer, such as the type you would use as the volume control in a radio, dissipating 22 kW, several times more than a typical household!)

Problem 4

How much voltage would be necessary to drive 680 mA through a resistance of 47 kΩ, as is described in the previous problem?
Use Ohm’s Law to find the voltage: E = IR = 0.680 × 47,000 = 32,000 V = 32 kV. That’s the level of voltage you’d expect to find on a major utility power line, or in a high-power tube-type radio broadcast transmitter.

Resistance Calculations

Ohms’ Law can be used to find a resistance between two points in a dc circuit when the voltage and the current are known.


A circuit for working Ohm’s Law problems

Problem 1

If the voltmeter in above figure reads 24 V and the ammeter shows 3.0 A, what is the resistance of the potentiometer?
Use the formula R = E/I, and plug in the values directly, because they are expressed in volts and amperes: R = 24/3.0 = 8.0 Ω. Note that you can specify this value to two significant figures, the 8 and the 0, rather than saying simply 8 Ω. This is because you are given both the voltage and the current to two significant figures. If the ammeter reading had been given as 3 A, you would only be entitled to express the answer as 8 Ω, to one significant digit. The digit 0 can be, and often is, just as important in calculations as any of the other digits 1 through 9.

Problem 2

What is the value of the resistance in above figure if the current is 18 mA and the voltage is 229 mV?
First, convert these values to amperes and volts. This gives I = 0.018 A and E = 0.229 V. Then plug into the equation: R = E/I = 0.229/0.018 = 13 Ω.

Problem 3

Suppose the ammeter in above figure reads 52 μA and the voltmeter indicates 2.33 kV. What is the resistance?
Convert to amperes and volts, getting I = 0.000052 A and E = 2330 V. Then plug into the formula: R = E/I = 2330/0.000052 = 45,000,000 Ω = 45 MΩ.

Voltage Calculations

The second application of Ohm’s Law is to find unknown dc voltages when the current and the resistance are known. Let’s work out some problems of this kind.


A circuit for working Ohm’s Law problems

Problem 1

Suppose the potentiometer in above figure is set to 100 Ω, and the measured current is 10 mA. What is the dc voltage?
Use the formula E = IR. First, convert the current to amperes: 10 mA = 0.01 A. Then multiply: E = 0.01 × 100 = 1.0 V. That’s a little less than the voltage produced by a flashlight cell.

Problem 2

Adjust the potentiometer in above figure to a value of 157 kΩ, and suppose the current reading is 17.0 mA. What is the voltage of the source?
You must convert both the resistance and the current values to their proper units. A resistance of 157 kΩ is 157,000 Ω, and a current of 17.0 mA is 0.0170 A. Then E = IR = 0.017 × 157,000 = 2669 V = 2.669 kV. You should round this off to 2.67 kV. This is a dangerously high voltage.

Problem 3

Suppose you set the potentiometer in above figure so that the meter reads 1.445 A, and you observe that the potentiometer scale shows 99 Ω. What is the voltage?
These units are both in their proper form. Therefore, you can plug them right in and use your calculator: E = IR = 1.445 × 99 = 143.055 V. This can and should be rounded off—but to what extent? This is a good time to state an important rule that should be followed in all technical calculations.

The Rule of Significant Figures

Competent engineers and scientists go by the rule of significant figures, also called the rule of significant digits. After completing a calculation, you should always round the answer off to the least number of digits given in the input data numbers.

If you follow this rule in Problem 3, you must round off the answer to two significant digits, getting 140 V, because the resistance (99 Ω) is only specified to that level of accuracy. If the resistance were given as 99.0 Ω, then you would round off the answer to 143 V. If the resistance were given as 99.00 Ω, then you could state the answer as 143.1 V. However, any further precision in the resistance value would not entitle you to go to any more digits in your answer, unless the current were specified to more than four significant figures.

This rule takes some getting used to if you haven’t known about it or practiced it before. But after a while, it will become a habit.

Current Calculations

The first way to use Ohm’s Law is to determine current in dc circuits. In order to find the current, you must know the voltage and the resistance, or be able to deduce them. Refer to the schematic diagram of following figure. It consists of a dc voltage source, a voltmeter, some wire, an ammeter, and a calibrated, wide-range potentiometer.


A circuit for working Ohm’s Law problems

Problem 4-1

Suppose that the dc generator in above figure produces 10 V and the potentiometer is set to a value of 10 Ω. What is the current?
This is solved by the formula I = E/R. Plug in the values for E and R; they are both 10, because the units are given in volts and ohms. Then I = 10/10 = 1.0 A.

Problem 4-2

Imagine that dc generator in above figure produces 100 V and the potentiometer is set to 10 kΩ. What is the current?
First, convert the resistance to ohms: 10 kΩ = 10,000 Ω. Then plug the values in: I = 100/10,000 = 0.01 A. You might prefer to express this as 10 mA.

Problem 4-3

Suppose that dc generator in above figure is set to provide 88.5 V, and the potentiometer is set to 477 MΩ. What is the current?
This problem involves numbers that aren’t exactly round, and one of them is huge. But you can use a calculator. First, change the resistance value to ohms, so you get 477,000,000 Ω. Then plug into the Ohm’s Law formula: I = E/R = 88.5 / 477,000,000 = 0.000000186 A. It is more reasonable to express this as 0.186 μA or 186 nA.

Ohm’s Law

The interdependence among current, voltage, and resistance in dc circuits is called Ohm’s Law, named after the scientist who supposedly first quantified it. Three formulas denote this law:

E = IR
I = E/R
R = E/I

You need only remember the first of these formulas in order to derive the others. The easiest way to remember it is to learn the abbreviations E for voltage, I for current, and R for resistance, and then remember that they appear in alphabetical order with the equal sign after the E. Sometimes the three symbols are arranged in the so-called Ohm’s Law triangle, shown in following figure. To find the value of a quantity, cover it up and read the positions of the others.


The Ohm’s Law triangle. The voltage is E, the current is I, and the resistance is R. These quantities are expressed in volts, amperes, and ohms, respectively.

Remember that you must use units of volts, amperes, and ohms for the Ohm’s Law formulas to yield a meaningful result! If you use, say, volts and microamperes to calculate a resistance, you cannot be sure of the units you’ll end up with when you derive the final result. If the initial quantities are given in units other than volts, amperes, and ohms, convert to these units, and then calculate. After that, you can convert the calculated current, voltage, or resistance value to whatever size unit you want. For example, if you get 13,500,000 Ω as a calculated resistance, you might prefer to say that it’s 13.5 MΩ.


Most dc circuits can be boiled down to three major components: a voltage source, a set of conductors, and a resistance. This is shown in Following Figure. The voltage or EMF source is E; the current in the conductor is I; the resistance is R.


The basic elements of a dc circuit. The voltage is E, the current is I, and the resistance is R.

You already know that there is a relationship among these three quantities. If one of them changes, then one or both of the others will change. If you make the  resistance smaller, the current will get larger. If you reduce the applied voltage, the current will also decrease. If the current in the circuit increases, the voltage across the resistor will increase. There is a simple arithmetic relationship among these three quantities.

Schematic and Wiring Diagrams

In a schematic diagram, the interconnection of the components is shown, but the actual values of the components are not necessarily indicated. You might see a diagram of a two-transistor audio amplifier, for example, with resistors and capacitors and coils and transistors, but without any data concerning the values or ratings of the components. This is a schematic diagram, but not a true wiring diagram. It gives the scheme for the circuit, but you can’t wire the circuit and make it work, because there isn’t enough information.


Schematic symbols for incandescent lamp
fixed capacitor
fixed inductor with air core
fixed inductor with laminated-iron core
chassis ground
earth ground
signal generator or source of alternating current
pair of terminals
specialized component or device

Suppose you want to build the circuit. You go to an electronics store to get the parts. What values of resistors should you buy? How about capacitors? What type of transistor will work best? Do you need to wind the coils yourself, or can you get ready-made coils? Are there test points or other special terminals that should be installed for the benefit of the technicians who might have to repair the amplifier? How many watts should the potentiometers be able to handle? All these things are indicated in a wiring diagram. You might have seen this kind of diagram in the back of the instruction manual for a hi-fi amplifier, a stereo tuner, or a television set. Wiring diagrams are especially useful when you want to build, modify, or repair an electronic device.

Schematic Symbols

The simplest schematic symbol is the one representing a wire or electrical conductor: a straight, solid line. Sometimes, dashed lines are used to represent conductors, but usually, dashed lines are drawn to partition diagrams into constituent circuits, or to indicate that certain components interact with each other or operate in step with each other. Conductor lines are almost always drawn either horizontally across or vertically up and down the page. This keeps the diagram neat and easy to read.

When two conductor lines cross, they aren’t connected at the crossing point unless a heavy black dot is placed where the two lines meet. The dot should always be clearly visible wherever conductors are to be connected, no matter how many of them meet at the junction. A resistor is indicated by a zigzag. A variable resistor, or potentiometer, is indicated by a zigzag with an arrow through it, or by a zigzag with an arrow pointing at it. These symbols are shown below :


Schematic symbols for a fixed resistor (A), a two-terminal variable resistor (B), and a three-terminal potentiometer (C).

An electrochemical cell (such as a common dime-store battery) is shown by two parallel lines, one longer than the other. The longer line represents the plus terminal. A true battery, which is a combination of two or more cells in series, is indicated by several parallel lines, alternately long and short. It’s not necessary to use more than four lines to represent a battery, although you’ll often see 6, 8, 10, or even 12 lines. Symbols for a cell and a battery are shown in figure :


Schematic symbols for an electrochemical cell (A) and an electrochemical battery (B).

Meters are portrayed as circles. Sometimes the circle has an arrow inside it, and the meter type, such as mA (milliammeter) or V (voltmeter) is written alongside the circle, as shown in (following figure A). Sometimes the meter type is indicated inside the circle, and there is no arrow (following figure B). It doesn’t matter which way you draw them, as long as you’re consistent throughout a schematic diagram.


Meter symbols can have the designator either outside the circle (A) or inside (B). In this case, both symbols represent a milliammeter (mA).

Some other common symbols include the incandescent lamp, the capacitor, the air-core coil, the iron-core coil, the chassis ground, the earth ground, the ac source, the set of terminals, and the black box (general component or device), a rectangle with the designator written inside. These are shown in Next topic