# circuit theory 4

### SERIES AND PARALLEL CIRCUITS

## What are “series” and “parallel” circuits?

Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.

There are two basic ways in which to connect more than two circuit components: *series* and *parallel*. First, an example of a series circuit:

Here, we have three resistors (labeled R_{1}, R_{2}, and R_{3}), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling — those little numbers to the lower-right of the letter “R” — are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The defining characteristic of a series circuit is that there is only one path for electrons to flow. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4.

Now, let’s look at the other type of circuit, a parallel configuration:

Again, we have three resistors, but this time they form more than one continuous path for electrons to flow. There’s one path from 8 to 7 to 2 to 1 and back to 8 again. There’s another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there’s a third path from 8 to 7 to 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R_{1}, R_{2}, and R_{3}) is called a *branch*.

The defining characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points.

And, of course, the complexity doesn’t stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too:

In this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R_{1} (from point 2 to point 1). In this configuration, we’d say that R_{2} and R_{3} are in parallel with each other, while R_{1} is in series with the parallel combination of R_{2} and Rsub>3.

This is just a preview of things to come. Don’t worry! We’ll explore all these circuit configurations in detail, one at a time!

The basic idea of a “series” connection is that components are connected end-to-end in a line to form a single path for electrons to flow:

The basic idea of a “parallel” connection, on the other hand, is that all components are connected across each other’s leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to flow, but only one voltage across all components:

Series and parallel resistor configurations have very different electrical properties. We’ll explore the properties of each configuration in the sections to come.

**REVIEW:**- In a series circuit, all components are connected end-to-end, forming a single path for electrons to flow.
- In a parallel circuit, all components are connected across each other, forming exactly two sets of electrically common points.
- A “branch” in a parallel circuit is a path for electric current formed by one of the load components (such as a resistor).

## Simple series circuits

Let’s start with a series circuit consisting of three resistors and a single battery:

The first principle to understand about series circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path for electrons to flow in a series circuit, and because free electrons flow through conductors like marbles in a tube, the rate of flow (marble speed) at any point in the circuit (tube) at any specific point in time must be equal.

From the way that the 9 volt battery is arranged, we can tell that the electrons in this circuit will flow in a counter-clockwise direction, from point 4 to 3 to 2 to 1 and back to 4. However, we have one source of voltage and three resistances. How do we use Ohm’s Law here?

An important caveat to Ohm’s Law is that all quantities (voltage, current, resistance, and power) must relate to each other in terms of the same two points in a circuit. For instance, with a single-battery, single-resistor circuit, we could easily calculate any quantity because they all applied to the same two points in the circuit:

Since points 1 and 2 are connected together with wire of negligible resistance, as are points 3 and 4, we can say that point 1 is electrically common to point 2, and that point 3 is electrically common to point 4. Since we know we have 9 volts of electromotive force between points 1 and 4 (directly across the battery), and since point 2 is common to point 1 and point 3 common to point 4, we must also have 9 volts between points 2 and 3 (directly across the resistor). Therefore, we can apply Ohm’s Law (I = E/R) to the current through the resistor, because we know the voltage (E) across the resistor and the resistance (R) of that resistor. All terms (E, I, R) apply to the same two points in the circuit, to that same resistor, so we can use the Ohm’s Law formula with no reservation.

However, in circuits containing more than one resistor, we must be careful in how we apply Ohm’s Law. In the three-resistor example circuit below, we know that we have 9 volts between points 1 and 4, which is the amount of electromotive force trying to push electrons through the series combination of R_{1}, R_{2}, and R_{3}. However, we cannot take the value of 9 volts and divide it by 3k, 10k or 5k Ω to try to find a current value, because we don’t know how much voltage is across any one of those resistors, individually.

The fgure of 9 volts is a *total* quantity for the whole circuit, whereas the figures of 3k, 10k, and 5k Ω are *individual* quantities for individual resistors. If we were to plug a figure for total voltage into an Ohm’s Law equation with a figure for individual resistance, the result would not relate accurately to any quantity in the real circuit.

For R_{1}, Ohm’s Law will relate the amount of voltage across R_{1} with the current through R_{1}, given R_{1}‘s resistance, 3kΩ:

But, since we don’t know the voltage across R_{1} (only the total voltage supplied by the battery across the three-resistor series combination) and we don’t know the current through R_{1}, we can’t do any calculations with either formula. The same goes for R_{2} and R_{3}: we can apply the Ohm’s Law equations if and only if all terms are representative of their respective quantities between the same two points in the circuit.

So what can we do? We know the voltage of the source (9 volts) applied across the series combination of R_{1}, R_{2}, and R_{3}, and we know the resistances of each resistor, but since those quantities aren’t in the same context, we can’t use Ohm’s Law to determine the circuit current. If only we knew what the *total* resistance was for the circuit: then we could calculate *total* current with our figure for *total* voltage (I=E/R).

This brings us to the second principle of series circuits: the total resistance of any series circuit is equal to the sum of the individual resistances. This should make intuitive sense: the more resistors in series that the electrons must flow through, the more difficult it will be for those electrons to flow. In the example problem, we had a 3 kΩ, 10 kΩ, and 5 kΩ resistor in series, giving us a total resistance of 18 kΩ:

In essence, we’ve calculated the equivalent resistance of R_{1}, R_{2}, and R_{3} combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of R_{1}, R_{2}, and R_{3}:

Now we have all the necessary information to calculate circuit current, because we have the voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 kΩ):

Knowing that current is equal through all components of a series circuit (and we just determined the current through the battery), we can go back to our original circuit schematic and note the current through each component:

Now that we know the amount of current through each resistor, we can use Ohm’s Law to determine the voltage drop across each one (applying Ohm’s Law in its proper context):

Notice the voltage drops across each resistor, and how the sum of the voltage drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts. This is the third principle of series circuits: that the supply voltage is equal to the sum of the individual voltage drops.

However, the method we just used to analyze this simple series circuit can be streamlined for better understanding. By using a table to list all voltages, currents, and resistances in the circuit, it becomes very easy to see which of those quantities can be properly related in any Ohm’s Law equation:

The rule with such a table is to apply Ohm’s Law only to the values within each vertical column. For instance, E_{R1} only with I_{R1} and R_{1}; E_{R2} only with I_{R2} and R_{2}; etc. You begin your analysis by filling in those elements of the table that are given to you from the beginning:

As you can see from the arrangement of the data, we can’t apply the 9 volts of E_{T} (total voltage) to any of the resistances (R_{1}, R_{2}, or R_{3}) in any Ohm’s Law formula because they’re in different columns. The 9 volts of battery voltage is *not* applied directly across R_{1}, R_{2}, or R_{3}. However, we can use our “rules” of series circuits to fill in blank spots on a horizontal row. In this case, we can use the series rule of resistances to determine a total resistance from the *sum* of individual resistances:

Now, with a value for total resistance inserted into the rightmost (“Total”) column, we can apply Ohm’s Law of I=E/R to total voltage and total resistance to arrive at a total current of 500 µA:

Then, knowing that the current is shared equally by all components of a series circuit (another “rule” of series circuits), we can fill in the currents for each resistor from the current figure just calculated:

Finally, we can use Ohm’s Law to determine the voltage drop across each resistor, one column at a time:

Just for fun, we can use a computer to analyze this very same circuit automatically. It will be a good way to verify our calculations and also become more familiar with computer analysis. First, we have to describe the circuit to the computer in a format recognizable by the software. The SPICE program we’ll be using requires that all electrically unique points in a circuit be numbered, and component placement is understood by which of those numbered points, or “nodes,” they share. For clarity, I numbered the four corners of our example circuit 1 through 4. SPICE, however, demands that there be a node zero somewhere in the circuit, so I’ll re-draw the circuit, changing the numbering scheme slightly:

All I’ve done here is re-numbered the lower-left corner of the circuit 0 instead of 4. Now, I can enter several lines of text into a computer file describing the circuit in terms SPICE will understand, complete with a couple of extra lines of code directing the program to display voltage and current data for our viewing pleasure. This computer file is known as the *netlist* in SPICE terminology:

series circuit

v1 1 0

r1 1 2 3k

r2 2 3 10k

r3 3 0 5k

.dc v1 9 9 1

.print dc v(1,2) v(2,3) v(3,0)

.end

Now, all I have to do is run the SPICE program to process the netlist and output the results:

v1 v(1,2) v(2,3) v(3) i(v1)

9.000E+00 1.500E+00 5.000E+00 2.500E+00 -5.000E-04

This printout is telling us the battery voltage is 9 volts, and the voltage drops across R_{1}, R_{2}, and R_{3} are 1.5 volts, 5 volts, and 2.5 volts, respectively. Voltage drops across any component in SPICE are referenced by the node numbers the component lies between, so v(1,2) is referencing the voltage between nodes 1 and 2 in the circuit, which are the points between which R_{1} is located. The order of node numbers is important: when SPICE outputs a figure for v(1,2), it regards the polarity the same way as if we were holding a voltmeter with the red test lead on node 1 and the black test lead on node 2.

We also have a display showing current (albeit with a negative value) at 0.5 milliamps, or 500 microamps. So our mathematical analysis has been vindicated by the computer. This figure appears as a negative number in the SPICE analysis, due to a quirk in the way SPICE handles current calculations.

In summary, a series circuit is defined as having only one path for electrons to flow. From this definition, three rules of series circuits follow: all components share the same current; resistances add to equal a larger, total resistance; and voltage rops add to equal a larger, total voltage. All of these rules find root in the definition of a series circuit. If you understand that definition fully, then the rules are nothing more than footnotes to the definition.

**REVIEW:**- Components in a series circuit share the same current: I
_{Total}= I_{1}= I_{2}= . . . I_{n} - Total resistance in a series circuit is equal to the sum of the individual resistances: R
_{Total}= R_{1}+ R_{2}+ . . . R_{n} - Total voltage in a series circuit is equal to the sum of the individual voltage drops: E
_{Total}= E_{1}+ E_{2}+ . . . E_{n}

## Simple parallel circuits

Let’s start with a parallel circuit consisting of three resistors and a single battery:

The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across R_{1} is equal to the voltage across R_{2} which is equal to the voltage across R_{3} which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values:

Just as in the case of series circuits, the same caveat for Ohm’s Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm’s Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:

At this point we still don’t know what the total current or total resistance for this parallel circuit is, so we can’t apply Ohm’s Law to the rightmost (“Total”) column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor (“branch”) currents:

As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through R_{1}, some more splits off at point 6 to go up through R_{2}, and the remainder goes up through R_{3}. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through R_{1}, R_{2}, and R_{3} join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through R_{1}, R_{2}, and R_{3}.

This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the I_{T} spot on our table with the sum of I_{R1}, I_{R2}, and I_{R3}:

Finally, applying Ohm’s Law to the rightmost (“Total”) column, we can calculate the total circuit resistance:

Please note something very important here. The total circuit resistance is only 625 Ω: *less* than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be *greater* than any one of the resistors individually. Here in the parallel circuit, hoever, the opposite is true: we say that the individual resistances *diminish* rather than *add* to make the total. This principle completes our triad of “rules” for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:

The same basic form of equation works for *any* number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.

Just as with the series circuit, we can use computer analysis to double-check our calculations. First, of course, we have to describe our example circuit to the computer in terms it can understand. I’ll start by re-drawing the circuit:

Once again we find that the original numbering scheme used to identify points in the circuit will have to be altered for the benefit of SPICE. In SPICE, all electrically common points must share identical node numbers. This is how SPICE knows what’s connected to what, and how. In a simple parallel circuit, all points are electrically common in one of two sets of points. For our example circuit, the wire connecting the tops of all the components will have one node number and the wire connecting the bottoms of the components will have the other. Staying true to the convention of including zero as a node number, I choose the numbers 0 and 1:

An example like this makes the rationale of node numbers in SPICE fairly clear to understand. By having all components share common sets of numbers, the computer “knows” they’re all connected in parallel with each other.

In order to display branch currents in SPICE, we need to insert zero-voltage sources in line (in series) with each resistor, and then reference our current measurements to those sources. For whatever reason, the creators of the SPICE program made it so that current could only be calculated *through* a voltage source. This is a somewhat annoying demand of the SPICE simulation program. With each of these “dummy” voltage sources added, some new node numbers must be created to connect them to their respective branch resistors:

The dummy voltage sources are all set at 0 volts so as to have no impact on the operation of the circuit. The circuit description file, or *netlist*, looks like this:

Parallel circuit

v1 1 0

r1 2 0 10k

r2 3 0 2k

r3 4 0 1k

vr1 1 2 dc 0

vr2 1 3 dc 0

vr3 1 4 dc 0

.dc v1 9 9 1

.print dc v(2,0) v(3,0) v(4,0)

.print dc i(vr1) i(vr2) i(vr3)

.end

Running the computer analysis, we get these results (I’ve annotated the printout with descriptive labels):

v1 v(2) v(3) v(4)

9.000E+00 9.000E+00 9.000E+00 9.000E+00

battery R1 voltage R2 voltage R3 voltage

voltage

v1 i(vr1) i(vr2) i(vr3)

9.000E+00 9.000E-04 4.500E-03 9.000E-03

battery R1 current R2 current R3 current

voltage

These values do indeed match those calculated through Ohm’s Law earlier: 0.9 mA for I_{R1}, 4.5 mA for I_{R2}, and 9 mA for I_{R3}. Being connected in parallel, of course, all resistors have the same voltage dropped across them (9 volts, same as the battery).

In summary, a parallel circuit is defined as one where all components are connected between the same set of electrically common points. Another way of saying this is that all components are connected across each other’s terminals. From this definition, three rules of parallel circuits follow: all components share the same voltage; resistances diminish to equal a smaller, total resistance; and branch currents add to equal a larger, total curret. Just as in the case of series circuits, all of these rules find root in the definition of a parallel circuit. If you understand that definition fully, then the rules are nothing more than footnotes to the definition.

**REVIEW:**- Components in a parallel circuit share the same voltage: E
_{Total}= E_{1}= E_{2}= . . . E_{n} - Total resistance in a parallel circuit is
*less*than any of the individual resistances: R_{Total}= 1 / (1/R_{1}+ 1/R_{2}+ . . . 1/R_{n}) - Total current in a parallel circuit is equal to the sum of the individual branch currents: I
_{Total}= I_{1}+ I_{2}+ . . . I_{n}.