Unit 9: Series-Parallel Circuits

In Units 7 and 8 you studied series circuits and parallel circuits. Many real-life circuits are more complicated, with combinations of both series-connected and parallel-connected components. In this unit we'll study such series-parallel circuits. You'll find that the basic rules of the game are the ones that you've already learned for series circuits and parallel circuits, but the job of applying those rules can get pretty tricky.

Unit 8 Review

This unit will build on material that you studied in Unit 8. So let's begin by taking this self-test to review what you learned in that unit.

Identifying Series-Parallel Relationships

A series-parallel circuit is a circuit that contains combinations of series-connected and parallel-connected components, which are in turn connected in series and/or in parallel with other such combinations. Here's an example:
Recall our definitions of series connections and parallel connections from previous units:
- Two components are connected in series if they are connected to each other at exactly one point and no other component is connected to that point.
- Two components are connected in parallel if they are connected to each other at two points.
In the example circuit shown above, R2 is in series with R3, which is in series with R4. So R2, R3, and R4 form a series path. Also, this entire series path, taken as a whole, is in parallel with R1.

Notation for Series-Parallel Relationships

As a short-hand way of describing series-parallel relationships in a circuit, we'll use two symbols:
- we use the symbol + to indicate a series connection between components
- we use the symbol || to indicate a parallel connection
So for the example circuit shown above, we would write
R2 + R3 + R4
to indicate that R2, R3, and R4 form a series path. Then, to show that R1 is in parallel with this series path, we would write
R1 || (R2 + R3 + R4)

Analyzing Series-Parallel Circuits

Units 2 and 3 gave step-by-step rules for analyzing series circuits and parallel circuits. It was possible to do this because one basic approach works on all series circuits, and another basic approach works on all parallel circuits.
But it's not possible to give a step-by-step procedure that will work on all series-parallel circuits. There is too much variety among series-parallel circuits, and an approach that works for one circuit may not work for other circuits.
The key is practice, practice, and more practice. To develop your skills, you need to study as many examples and work out as many problems as you can.
Usually, a good first step is to find the circuit's total equivalent resistance by combining pairs of series-connected or parallel-connected resistors until you've combined all the resistors into a single resistance.
- For example, in the circuit shown below, we would add R2, R3, and R4 together (since they're in series), then use the reciprocal formula to combine that value with R1, since R1 is in parallel with the series path containing R2, R3, and R4. The result would be the circuit's total equivalent resistance, R_T.
After you have the total equivalent resistance, you can use Ohm's law to find the circuit's total current:
I_T = V_S ÷ R_T
After that, you can apply any of the rules and laws from previous units, in any sequence that leads to the solution for a particular voltage or current. The most useful of these rules and laws are:
- Ohm's law
- voltage/current rules for series or parallel combinations
- Kirchhoff's Voltage Law
- Kirchhoff's Current Law
- the voltage-divider rule
- the current-divider rule
Let's review these rules and laws, and discuss their application to series-parallel circuits.

Ohm's Law in Series-Parallel Circuits

Recall that Ohm's law relates resistance, current, and voltage. As you know, the three forms of Ohm's law are:
I = V ÷ R
V = I × R
R = V ÷ I
Ohm's law can be applied to a single resistor, or to an entire resistive circuit, or to any resistive portion of a circuit.
- For instance, in the example circuit that we've been discussing, Ohm's law can be applied to any one of the resistors by itself . Applying it to resistor R3, we can write:
  I₃ = V₃ ÷ R₃
- Or we can apply Ohm's law (in any of its forms) to the entire circuit, letting us write:
  I_T = V_S ÷ R_T
  Remember that I_T means the total current passing through the voltage source.
- Or we can apply Ohm's law to a portion of the circuit. For instace, recall that our example circuit contains a series path made up of R2, R3, and R4. Applying Ohm's law just to this series path, we can say that the current through the series path is equal to the voltage across the series path divided by the resistance of the series path. In symbols:
  I₂₊₃₊₄ = V₂₊₃₊₄ ÷ R₂₊₃₊₄
  To avoid being confused by this notation, remember that here we're using + to stand for series connection, not mathematical addition. So we're using the subscript ₂₊₃₊₄ to denote the series combination of R2, R3, and R4. So R₂₊₃₊₄ is the resistance of this series path, and V₂₊₃₊₄ is the voltage across this series path, and I₂₊₃₊₄ is the current through this series path.
So although Ohm's law is one of the simplest laws in electronics, it is also very powerful and very flexible. Many complicated-looking problems in electronics can be solved using nothing more than Ohm's law.

Voltage/Current Rules for Series or Parallel Combinations

Here are two more simple-but-powerful rules. Recall that:
- Components connected in series must have the same current (but usually they won't have the same voltage).
- Components connected in parallel must have the same voltage (but usually they won't have the same current).
These two rules apply not only to individual components but also to portions of circuits.
- In our example circuit, for instance, resistor R1 is connected in parallel with the series path containing R2, R3, and R4. Therefore, we know that the voltage across R1 must be equal to the voltage across the series path. In symbols:
  V₁ = V₂₊₃₊₄
- Also, in the same circuit, since R2, R3, and R4 form a series path, we know that the three resistors must carry the same current. In symbols:
  I₂ = I₃ = I₄

Kirchhoff's Voltage Law in Series-Parallel Circuits

Recall that KVL says that the sum of the voltage drops around any closed loop in a circuit equals the sum of the voltage rises around that loop.
We originally studied KVL in the context of series circuits, but KVL holds for all circuits.
- In our example circuit, for instance, one closed loop contains just the voltage source and R1. Therefore KVL tells us that the voltage rise across the source must be equal to the voltage drop across R1. In symbols:
  V_S = V₁
  Of course, we could also have seen this from the fact that the source and R1 are connected in parallel. In many cases, there will be more than one way to arrive at the same conclusion.
- Another closed loop in our example circuit contains the voltage source and resistors R2, R3, R4. Therefore KVL tells us that the voltage rise across the source must be equal to the sum of the voltage drops across those three resistors. In symbols:
  V_S = V₂ + V₃ + V₄

Kirchhoff's Current Law in Series-Parallel Circuits

Recall that KCL says that the sum of all currents entering a point is equal to the sum of all currents leaving that point.
We originally studied KCL in the context of parallel circuits, but KCL holds for all circuits.
- In our example circuit, for instance, consider the point where the voltage source meets R1 and R2. KCL tells us that the current flowing into this point from the voltage source must be equal to the sum of the currents flowing out of this point through R1 and R2. In symbols:
  I_T = I₁ + I₂

The Voltage-Divider Rule in Series-Parallel Circuits

The rules listed above (Ohm's law, KVL, and KCL) are your most important tools for analyzing circuits. Most series-parallel circuits can be analyzed using just those rules. In some cases, the voltage-divider rule and the current-divider rule are also useful. However, it's easy to make mistakes when you apply the voltage-divider rule and current-divider rule to series-parallel circuits. Therefore, although we'll now discuss these two rules, I advise you to rely primarily on Ohm's law, KVL, and KCL.
In discussing series circuits, we stated the voltage-divider rule as follows: the voltage across any resistance in a series circuit is equal to the ratio of that resistance to the circuit's total resistance, multiplied by the source voltage. In equation form:
V_x = (R_x ÷ R_T) × V_S
The same basic rule applies to any series path in a series-parallel circuit (but not to an entire series-parallel circuit, and not to any portion of a series-parallel circuit that is not a series path). Since the original wording of the rule applies only to series circuits, we need to change the wording slightly, as follows: the voltage across any resistance in a series path is equal to the ratio of that resistance to the path's total resistance, multiplied by the voltage across the series path. In equation form:
V_x = (R_x ÷ R_path) × V_path
Here I'm using two symbols whose meanings hopefully are clear: R_pathmeans the total resistance of the series path, and V_path means the voltage across the entire series path. As in the original voltage-divider rule, x is a variable representing the number of the resistor that you're interested in.
- In our example circuit, for instance, we have a series path containing resistors R2, R3, R4. If we know the voltage across the entire series path, then the voltage-divider rule lets us find the voltage across any one resistor in that path. For instance, the voltage across R2 is given by:
  V₂ = (R₂ ÷ R_path) × V_path
- It's very important to remember that this rule applies only to resistors in a series path. For instance, since R1 is not part of a series path in this circuit, we cannot use the voltage-divider rule to find its voltage:
  V₁ = (R₁ ÷ R_T) × V_S Incorrect!
Because students often apply the voltage-divider rule incorrectly in series-parallel circuits, I recommend that you rely more heavily on Ohm's law, KVL, and KCL, and try to avoid using the voltage-divider rule.

The Current-Divider Rule in Series-Parallel Circuits

In discussing parallel circuits, we stated the current-divider rule as follows: for branches in parallel, the current I_x through any branch equals the ratio of the total parallel resistance R_T to the branch's resistance R_x, multiplied by the total current I_T entering the parallel combination. In equation form:
I_x = (R_T ÷ R_x) × I_T
The same rule applies to parallel branches in a series-parallel circuit. But you have to be careful to apply it correctly. In particular, note that R_x is the total resistance of the branch whose current you are trying to find. Depending on the circuit you're analyzing, R_x may be a single resistor's resistance, or it may be the combined resistance of several resistors in series and/or parallel. Also, note that in this equation R_T means the total resistance of the parallel branches, which may or may not be the same as the entire circut's total resistance. Similarly, I_T in this equation means the total current entering the parallel branches, which may or may not be the same as the entire circut's total current.
- In our example circuit, for instance, we have two parallel branches. One branch contains just R1, and the other branch is a series path containing R2, R3, R4. If we know the total current entering both branches, then the current-divider rule lets us find the current into either branch. For instance, the current into the series path is given by:
  I_path = (R_T ÷ R_path) × I_T
- But the following would not be a proper application of the current-divider rule, since R2 does not by itself form a complete branch:
  I₂ = (R_T ÷ R₂) × I_T Incorrect!
Because students often apply the current-divider rule incorrectly in series-parallel circuits, I recommend that you rely more heavily on Ohm's law, KVL, and KCL, and try to avoid using the current-divider rule.

Examples

We have now reviewed the use of Ohm's law, Kirchhoff's Laws, and the divider rules in series-parallel circuits. Analysis of any particular series-parallel circuit will involve the use of one or more of these rules and laws.
Usually there will be more than one way to approach a particular problem or circuit. In finding a particular voltage, for instance, one student might use Ohm's law and KVL, while another student uses KVL and the voltage-divider rule, and a third student uses the current-divider rule and Ohm's law. All three approaches will give the same answer, as long as the students apply the rules correctly.
As mentioned earlier, a good first step is usually to find the circuit's total equivalent resistance by combining pairs of series-connected or parallel-connected resistors until you've combined all the resistors into a single resistance. Then find the circuit's total current by using Ohm's Law in the form:
I_T = V_S ÷ R_T

Power in a Series-Parallel Circuit

To find the power dissipated in a resistor in a series-parallel circuit, use any of the same formulas that you used for series circuits and parallel circuits:
P = V × I

P = I² × R

P = V² ÷ R
Recall that in each of these equations, R is the resistor's resistance, V is the voltage across the resistor, and I is the current through the resistor.

Total Circuit Power

Just as with series circuits and parallel circuits, there are two ways to compute total power dissipated in a series-parallel circuit. You'll get the same answer either way:
1. Either find the power for each resistor, and then add these powers:
  P_T = P₁ + P₂ + P₃ + ... + P_n
2. Or apply any one of the power formulas to the entire circuit:
  P_T = V_S × I_T
  
  P_T = I_T² × R_T
  
  P_T = V_S² ÷ R_T
  These are the same power formulas from above, except that now we're applying them to the entire circuit, instead of to a single resistor.

Ground (or Common) Symbol

Recall from Unit 4 that instead of drawing lines to show connections in a schematic diagram, we often use a ground symbol, also called a common symbol, to designate terminals that are connected together. The symbol looks like this:
The main reason for using this symbol is that it let you eliminate lines that clutter a schematic diagram and make it difficult to visualize current flow between components.
Example: Take a look again at this circuit diagram:

Notice that the voltage source, R1, and R4 all come together at a common point. Using the ground symbol to show that these three components are connected together, we could redraw the diagram as shown below.
This is not a very good example, because the redrawn diagram with the ground symbol is no easier to read than the original diagram. But in a complicated schematic diagram with many resistors, the ground symbol can eliminate a lot of lines and therefore make the diagram much easier to read. The important point in this example is to see that the two diagrams are equivalent.

Bubble Symbol for Voltage Sources

In schematic diagrams where the common symbol is used, we often omit that symbol from the side of any voltage source that is connected to common.
The terminal that is not connected to common is shown by a small circle (or "bubble") and is labeled + or - the voltage of the voltage source.
Example: Continuing the same example from above, we could redraw the schematic diagram once more using the bubble symbol to arrive at the following diagram:

Circuit Geometry

The geometry of a circuit (the way the component connections are shown in a schematic diagram) can often make it hard to see voltage and current relations in the circuit.
You should carefully redraw a complicated schematic diagram in as many ways as necessary to make it easier to read.

Unit 9 Review

This e-Lesson has covered several important topics, including:
- identifying series and parallel relationships
- analyzing series-parallel circuits
- power in series-parallel circuits
- ground symbol and bubble symbol.
To finish the e-Lesson, take this self-test to check your understanding of these topics.

Congratulations! You've completed the e-Lesson for this unit.