 # Unit 8: Parallel Circuits

After series circuits, which you studied in Unit 7, the next simplest type of circuit is the parallel circuit, which we’ll take up next. Again, we’ll restrict our attention to parallel resistive circuits, which contain only resistors in addition to voltage sources.

##### Parallel Connection
• Recall from Unit 2 that two components are connected in series if they are connected to each other at exactly one point and if no other component is connected to that point.
• Example: In the circuit shown below, R2 and R3 are connected in series, and R3 and R4 are also connected in series. • On the other hand, two components are connected in parallel if they are connected to each other at two points.
• Example: In the circuit shown above, the voltage source and R1 are connected in parallel.
• ##### Parallel-Connected Components Have the Same Voltage
• The most important property of parallel connections is that the voltage is the same across every parallel-connected component.
• Example: In the circuit shown below, the voltage source and R1 are connected in parallel, so we know that the voltage across the source must be the same as the voltage across R1. But R1 and R3 are not connected in parallel, so we cannot assume that the voltage across R1 is equal to the voltage across R3. ##### KCL in Parallel Resistive Circuits
• When applied to a parallel resistive circuit with a single voltage source, KCL says that if you add the currents through all of the resistors, the sum must be equal to the value of the total current leaving the voltage source.
• Here’s why. Consider the parallel circuit shown below, which shows the directions of the currents flowing out of the source and through the resistors. • Looking at the point labeled A, we see that there is one current flowing into that point, namely IT, the circuit’s total current.
• There are two currents leaving point A, namely I1 and I2.
• Since KCL tells us that the sum of the currents entering a point is equal to the sum of the currents leaving that point, we can say that

IT = I1 + I2

• Applying the same kind of reasoning to a parallel circuit with more resistors, we’ll always reach the same conclusion: the sum of the resistor currents is equal to the current flowing out of the voltage source.
• • ##### KCL in Other Circuits
• KCL is a general rule that applies in all circuits, not just parallel circuits and not just circuits containing resistors. In more complicated circuits, it can get tricky to apply KCL correctly, but when applied correctly it is a powerful tool. We’ll see this in later units.

##### Total Parallel Resistance
• Suppose you have n resistors connected in parallel, where n is any number. The total conductance of these resistors is equal to the sum of the n individual conductances. In symbols:

GT = G1 + G2 + … + Gn

• From this we can derive an expression for the total equivalent resistance of n resistors connected in parallel:

RT = 1 ÷ (1÷R1 + 1÷R2 + … + 1÷Rn)

• This formula is often called the reciprocal formula, since it involves taking the reciprocal of the sum of the reciprocals of the resistors. (Remember, the reciprocal of a number just means 1 divided by that number.)
• • An important fact about parallel-connected resistances is that the total equivalent resistance is always less than each of the individual resistances, including the smallest one. For example, looking at the circuit shown below, we can say immediately that the total resistance of the three resistors must be less than 680 Ω. If you get a value greater than 680 Ω when you apply the reciprocal formula to this circuit, you know that you’ve made a mistake. • • Below are three formulas that you can use to find total parallel resistance in certain special cases. However, the reciprocal formula is the general formula that works for all cases of resistors in parallel, including these special cases, so you can always use it if you wish.
##### Special Case #1: Two Parallel Resistors
• In many cases, we wish to figure out the total resistance of two resistors connected in parallel. We could use the reciprocal formula to find this total resistance, or we could use the following special-case formula:

RT = (R1 × R2) ÷ (R1 + R2)

• In words, the total resistance of two parallel resistors is equal to their product divided by their sum.
• For obvious reasons, this rule is often called the product-over-sum rule.
• ##### Special Case #2: Parallel Resistors of Same Value
• Another special case arises when you have two or more resistors in parallel, and all of the resistors have the same individual resistance. (For example, pehaps you have three 100-Ω resistors in parallel with each other.) Again, we could use the reciprocal formula in such cases, or we could use the following special-case rule:
• For n parallel resistors, each having resistance R,

RT = R ÷ n

• In words, if you have several resistors of the same value connected in parallel, the total resistance is equal to the individual resistance value divided by the number of resistors.
• For obvious reasons, this rule is often called the value-over-number rule.
• • We’ve still got one more special case to cover, but this animated lesson summarizes the cases that we’ve covered so far.
• ##### Special Case #3: Resistor in Parallel with a Much Smaller Resistor
• When one resistance is much greater than another one connected in parallel with it, the total resistance of the combination is very nearly equal to the smaller of the two. In symbols:

If R1 >> R2, then R1||R2R2

• Here we have used the symbol || for the parallel combination of two resistors, and we have also used two standard mathematical symbols: >> means "much greater than," and ≈ means "approximately equal to."
• ##### Analyzing Parallel Resistive Circuits
• We noted above that all of the components in a parallel circuit must have the same voltage as each other.
• Of course, once we know the voltage across any resistor, we can use Ohm’s law to find the current through that resistor.
• So we now know enough to be able to find currents and voltage drops in a parallel resistive circuit. There are four basic steps.
1. Recall that in a parallel circuit, every component has the same voltage. Therefore, each resistor’s voltage is equal to the source voltage. In symbols,

VS = V1 = V2 = … = Vn

2. Use Ohm’s law in the form ÷ R to find the current through each resistor. In symbols,

I1 = V1 ÷ R1
I2 = V2 ÷ R2
and so on for each of the resistors.

3. Use the reciprocal formula (or one of the special-case formulas given above) to find the circuit’s total resistance:

RT = 1 ÷ (1÷R1 + 1÷R2 + … + 1÷Rn)

4. Use one of the following methods to find the circuit’s total current:
• Either add together all of the individual resistor currents:

IT = I1 + I2 + … + In

• Or apply Ohm’s law in the form V ÷ R to the entire circuit. In words, the total current produced by the voltage source is equal to the source voltage divided by the total resistance. In symbols,

IT = VS ÷ RT

• • ##### Voltage Sources Connected in Parallel?
• In general, you should not connect different-valued voltage sources in parallel with each other.
• An exception to this is the case of rechargeable batteries. For instance, suppose you’ve got a "dead" car battery whose voltage is close to 0 V. You can recharge the battery by connecting it in parallel with a good car battery or in parallel with a battery charger that produces a voltage of about 12 V.
• Though we generally don’t connect different-valued voltage sources in parallel with each other, we do sometimes connect equal-valued voltage sources in parallel with each other. Why would we want to do this? The following animated lesson explains.
• ##### Current Sources Connected in Parallel
• A current source is a device that supplies the same current to any resistance connected across its terminals.
• The schematic symbol for a current source is shown below. • Current sources can be connected in parallel.
• Current sources connected in parallel can be replaced by a single equivalent current source that produces a current equal to the algebraic sum of the individual sources.

##### Current Divider
• A group of resistors connected in parallel is often called a current divider because the total current entering the group is divided among the various resistors in inverse proportion to the resistance of each one.
• For example, if you have two resistors in parallel and one resistor is twice as large as the other one (for example, suppose that one is 20 kΩ and the other is 10 kΩ), then there will be twice as much current through the smaller resistor as there is through the larger one.
• On the other hand, if one of the parallel resistors is three times as large as the other one (say, 30 kΩ and 10 kΩ), then there will be three times as much current through the smaller resistor as there is through the larger one.
• Remember that, as in these examples, if two resistors of different size are in parallel with each other, the smaller resistor gets more current than the larger resistor.
##### The Current-Divider Rule
• For branches in parallel, the current Ix through any branch equals the ratio of the total parallel resistance RT to the branch’s resistance Rx, multiplied by the total current IT entering the parallel combination. In equation form:

Ix = (RT ÷ Rx) × IT

• Here x is a variable representing the number of the resistor that you’re interested in.
• For instance, if you’re trying to find the current through resistor R1, you would replace x with 1 to get:

I1 = (RT ÷ R1) × IT

• On the other hand, applying the rule to resistor R4 in a parallel circuit gives us:

I4 = (RT ÷ R4) × IT

• Note that RT in this formula means the equivalent resistance (given by the reciprocal formula), not the sum of the resistors.
• The current-divider rule given above applies whenever you have any number of resistors in parallel. There’s another form of the current-divider rule that applies only to cases of two resistors in parallel. However, I’ve found that students usually get confused if they try to remember these special-case formulas in addition to the general formula. Therefore, I recommend that you just remember the general formula and use it for all cases.
• ##### Power in a Parallel Circuit
• To find the power dissipated in a resistor in a parallel circuit, use any of the same formulas that you used for series 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 resistive circuits, there are two ways to compute total power dissipated in a parallel resistive circuit. You’ll get the same answer either way:
1. Either find the power for each resistor, and then add these powers:

PT = P1 + P2 + P3 + … + Pn

2. Or apply any one of the power formulas to the entire circuit:

PT = VS × IT

PT = IT2 × RT

PT = VS2 ÷ RT

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.
• ##### Troubleshooting Parallel Circuits
• Recall from the previous Unit that the two most common types of circuit problems are opens (breaks) and shorts (paths of zero resistance connecting points that should not be connected).
• Recall also that the current through an open is zero, and that the voltage across a short is zero.
• In a parallel circuit, an open resistor has no effect on the current passing through the other resistors. But it does increase the circuit’s total resistance and therefore decreases the circuit’s total current.
• A shorted resistor in a parallel circuit is basically the same thing as connecting a wire directly from the power supply’s positive terminal to its negative terminal. This is a very bad thing to do, and will cause the circuit’s total current to increase to an excessive value.
• If the circuit is properly protected by a fuse or circuit breaker, the fuse will blow or the breaker will trip, cutting off all current to the circuit.
• If the circuit is not properly protected, the excessive current caused by a short can start a fire or damage the circuit’s power supply.
• • ##### Unit 8 Review
• This e-Lesson has covered several important topics, including:
• parallel connections and parallel circuits
• Kirchhoff’s Current Law (KCL)
• total resistance of resistors in parallel
• parallel-connected sources
• current-divider rule
• power in parallel circuits
• shorts and opens in parallel circuits.
• 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.