 # Unit 7: Parallel AC Circuits

In Unit 6 we saw that analyzing series AC circuits involves the same steps as analyzing series DC circuits, but that at each step you must use complex numbers instead of real numbers. We'll see in this unit that, as you may have guessed, parallel AC circuits are a lot like parallel DC circuits, except that again you need to use complex numbers throughout the analysis.

Also, the same rules that are useful in analyzing parallel DC circuits (such as Kirchhoff's Current Law and the Current-Divider Rule) are also useful for parallel AC circuits. Again, though, the math is more complicated because you have to use complex numbers.

We'll also look at quantities called admittance and susceptance, which are similar to the quantity called conductance that you may recall from your studies of parallel DC circuits. When dealing with parallel circuits, some people find it more convenient to use conductance, susceptance, and admittance in place of resistance, reactance, and impedance. I'm not one of those people, but I do think that you should at least understand what these terms mean and understand how they relate to each other.

##### Review of Parallel DC Circuits
• In EET 1150 you learned how to analyze parallel DC circuits like the one shown below. Let's do a quick review of what you learned there. • You should recall that the basic steps in analyzing a circuit like this one are:
1. Recognize that in a parallel circuit, every component has the same voltage. Therefore, each resistor's voltage is equal to the source voltage. For the circuit shown, this means that

VS = V1 = V2 = V3

2. Use Ohm's law in the form ÷ R to find the current through each resistor. For the circuit shown,

I1 = V1 ÷ R1      and       I2 = V2 ÷ R2     and      I3 = V3 ÷ R3

3. Use the reciprocal formula to find the circuit's total resistance. For the circuit shown,

RT = 1 ÷ (1÷R1 + 1÷R2 + 1÷R3)

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 + I3

• 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

• ##### Review: Kirchhoff's Current Law in Parallel DC Circuits
• In EET 1150 you also learned that Kirchhoff's Current Law (KCL) says that the sum of all currents entering a point is equal to the sum of all currents leaving that point.
• In terms of a simple parallel DC circuit like the one you just analyzed, this means 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.
• ##### Review: Current Divider Rule in Parallel DC Circuits
• In EET 1150 you also learned that the current-divider rule is one way to find how a current coming into a node in a circuit will split up between the different parallel branches attached to that node.
• The rule says that for branches in parallel, the current through any branch equals the ratio of the total parallel resistance to the branch's resistance, multiplied by the total current entering the parallel combination.
• In equation form, this rule is expressed as:

Ix = (RT ÷ Rx) × I

where Ix is the current through the branch you're interested in, RT is the total resistance of all the parallel branches, Rx is the resistance of the branch you're interested in, and I is the total current entering the parallel combination.
• ##### Review: Troubleshooting Parallel DC Circuits
• Recall 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 DC 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 DC 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.
• • That ends our quick review of parallel DC circuits. If you'd like a more thorough review, go to Unit 8 of EET 1150. Now let's get back to AC circuits.

##### Analyzing Parallel AC Circuits • Follow these steps to analyze any parallel AC circuit, such as the parallel RLC circuit shown above:
1. Recognize that, since we're dealing with a parallel circuit, each component's voltage is equal to the source voltage.
2. Apply Ohm's law separately to each component to find each component's current.
3. Use the reciprocal formula to find the circuit's total impedance, ZT.
4. Find the circuit's total current either by adding all of the components' currents, or by applying Ohm's law to the entire circuit.
• These are very similar to the steps that you followed in EET 1150 to analyze a simple parallel DC circuit containing resistors. The big difference is that throughout this procedure, we must now use complex numbers instead of real numbers.
• Let's look at each step in more detail.
##### Step 2: Find Individual Currents
• Now that you know the voltage across each component, use Ohm's Law to find the current through each component:

I1 = V1 ÷ Z1     and     I2 = V2 ÷ Z2     and     I3 = V3 ÷ Z3     and  ...

• Here Z1 is the first component's impedance, which will have an angle of 0° if the first component is a resistor, or an angle of −90° if the first component is a capacitor, or an angle of 90° if the first component is an inductor. Similarly for Z2, Z3, and so on for however many components the circuit contains. So this step will require you first to find the reactances of any capacitors or inductors in the circuit.
• Remember: we're using complex numbers here, not real numbers.
• ##### Step 3: Find Total Impedance
• For impedances in parallel, total impedance is given by the reciprocal formula:

ZT = 1 ÷ (1÷Z1 + 1÷Z2 + … + 1÷Zn)

• Each Z on the right-hand side of this equation may be the impedance of a resistor, an inductor, or a capacitor.
• Remember to treat these quantities as complex numbers, not as real numbers.
• ##### Step 4: Find Total Current
• Use one of the following methods to find the circuit's total current:
• Either add together all of the individual component currents:

IT = I1 + I2 + … + In

• Or apply Ohm's law to the entire circuit:

IT = VS ÷ ZT

• Whichever method you use, remember to treat all quantities as complex numbers, not as real numbers.
• ##### Kirchhoff's Current Law
• As in DC circuits, Kirchhoff's Current Law (KCL) says that the sum of all currents entering a point is equal to the sum of all currents leaving that point.
• We made use of KCL in Step 4 above when we wrote IT = I1 + I2 + … + In.
• Whenever you apply KCL to an AC circuit, you must use complex numbers, not real numbers. If you just add the magnitudes of the currents, instead of adding the magnitudes along with their angles, you won't get good results. For example, in a parallel AC circuit it's usually not true that IT = I1 + I2 + … + In.
• In this unit our main focus is parallel circuits, but KCL applies to all circuits, whether series, parallel, or series-parallel.
• For example, in the series-parallel circuit shown below, the capacitor is in series with the voltage source, so all current leaving the source will pass through the capacitor. But to the right of the capacitor, this current splits in two, with some going through the resistor and some passing through the inductor. Therefore, Kirchhoff's Current Law tells us that the capacitor's current is equal to the sum of the resistor's current plus the inductor's current. • ##### Current-Divider Rule
• As in DC circuits, the current-divider rule tells you how a current coming into a node will split up between the different parallel branches attached to that node.
• The current-divider rule says that the current Ix through a branch with impedance Zx in parallel with other branches is given by:

Ix = (ZT ÷ Zx) I

where I is the incoming current and ZT is the total impedance of all the parallel branches.
• The current-divider rule holds for any combination of components in parallel, regardless of whether the entire circuit is a parallel circuit or a series-parallel circuit.
• For instance, in the series-parallel circuit shown below, the branch containing R1 is in parallel with the branch containing L1. Suppose that you have found the current coming into the node at which C1 meets these two parallel branches. Then the current-divider rule provides one way to find out how this incoming current will split up between the two parallel branches. • Again, use complex numbers, not real numbers.
• ##### Troubleshooting Parallel AC Circuits
• Above we reviewed the basics of troubleshooting parallel DC circuits. Almost all of these same points apply to parallel AC circuits. (But there's one important difference noted below.) In particular:
• A short has zero resistance and zero voltage.
• A short in a parallel circuit causes excessive current to flow, resulting in blown fuses, tripped circuit breakers, or damage to the circuit's power supply.
• No current flows through an open.
• If a component in a parallel circuit becomes open, this will not have any effect on the current flowing through the circuit's other components.
• So far, everything we've said about shorts and opens in parallel AC circuits is the same as what we said earlier about shorts and opens in parallel DC circuits. But here's a difference:
• An open in a parallel DC circuit will always increase the circuit's total resistance, decreasing the circuit's total current.
• But in a parallel AC circuit, an open could either increase or decrease the circuit's total impedance, and therefore could either decrease or increase the total current.
• Why this difference between opens in DC circuits and opens in AC circuits? It's because of the difference between real numbers and complex numbers.
• To find a parallel DC circuit's total resistance, you use the reciprocal formula on two or more real numbers. If one of the circuit's resistors becomes open, then you remove one of these numbers from the formula, and this will increase the total.
• For example, suppose a parallel circuit contains a 100 Ω resistor, a 150 Ω resistor, and a 200 Ω resistor. Then the total resistance is 46.2 Ω. But if the 200 Ω resistor is open, then you remove its resistance from the reciprocal formula, and so the total resistance increases to 60 Ω. This increase in total resistance will decrease the circuit's total current.
• But to find a parallel AC circuit's total impedance, you apply the reciprocal formula to two or more complex numbers. If one of the circuit's components becomes open, then you remove one of these complex numbers from the formula. This could either decrease or increase the total.
• For example, suppose a parallel AC circuit contains a 100∠0° Ω resistive impedance, a 150∠90° Ω inductive impedance, and a 200∠−90° Ω capacitive impedance. Then the total impedance is 98.6 ∠9.46° Ω. If the capacitor becomes open, then you remove its impedance from the reciprocal formula, and so the total impedance decreases to 83.2 ∠33.7° Ω. This decrease in total impedance will increase the circuit's total current.
• On the other hand, suppose that in the same circuit the capacitor is okay but the resistor becomes open. Then the circuit's total impedance is 600 ∠90° Ω, which is an increase from the original total impedance. This increase in total impedance will decrease the circuit's total current.
• So an open in a parallel AC circuit may either increase or decrease the total impedance.
• ##### Conductance
• Recall from EET 1150 that a resistor's conductance, which is abbreviated G, is the reciprocal of its resistance:

G = 1 ÷ R

• Recall also that conductance is measured in units called siemens (abbreviated S).
• For example, a 1 kΩ resistor has a conductance of 1 mS.
• Why is this concept of conductance useful? Primarily because in analyzing parallel circuits, the expression 1 ÷ R appears frequently, and so it's useful to have a shorthand way of referring to the reciprocal of resistance.
• For example, you're familiar with the reciprocal formula for finding total resistance of resistors in parallel, which can be written as:

RT = 1 ÷ (1÷R1 + 1÷R2 + ... + 1÷Rn)

Using the definition of conductance, we can rewrite this in the following simpler form:

GT = G1 + G2 + ... + Gn

• People who prefer to work with conductance instead of resistance also sometimes rewrite Ohm's law in terms of conductance rather than resistance. For example, you know that one form of Ohm's law says that a resistor's current is equal to its voltage divided by its resistance (in symbols, ÷ R). Another way of saying the same thing is that a resistor's current is equal to its voltage times its conductance (in symbols, I = V×G).
• I find it easier to use resistance instead of conductance--this saves me from having to remember a bunch of new formulas. But if you find it easier to work with conductance, feel free to do so. And you should at least know understand what conductance is and how it's related to resistance.
• ##### Susceptance
• Recall that in AC circuits, capacitors and inductors oppose the flow of current, and that we use the term "reactance" for this opposition to current. Recall also that if we know the frequency of the AC current and the size of the capacitor or inductor, we can compute the reactance as follows:
• A capacitor's reactance XC is equal to 1 ÷ (2pfC).
• An inductor's reactance XL is equal to 2pfL.
• Just as it is sometimes useful to have a shorthand way of referring to the reciprocal of a resistor's resistance, it's also sometimes useful to have a shorthand way of referring to the reciprocal of a capacitor's or inductor's reactance. The reciprocal of reactance is called susceptance.
• The unit of susceptance is the siemens.
• The abbreviation for susceptance is B.
• To find a capacitor's susceptance:

BC = 1 ÷ XC = 2pfC

• To find an inductor's susceptance:

BL = 1 ÷ XL = 1 ÷ (2pfL)

• • You know from Unit 6 that impedance can be thought of as a generalization of the concepts of resistance and reactance. Impedance is a complex quantity, expressible in either rectangular form or polar form.
• Just as it's useful to have a shorthand way of referring to the reciprocal of resistance and reactance, it's also useful to have a shorthand way of referring to the reciprocal of impedance. The reciprocal of impedance is called admittance.
• The unit of admittance is the siemens.
• Like impedance, admittance is a complex quantity, expressible in either rectangular form or polar form. The abbreviation for an admittance is Y. Therefore we can write

Y = 1 ÷ Z

• At times we may be interested in the magnitude of the admittance, which we denote by Y.
• When you take the reciprocal of a complex number with a negative angle, you'll get a complex number with a positive angle, and vice versa. Therefore, if an impedance has a positive angle, the corresponding admittance will have a negative angle, and vice versa.
• For example, if a circuit's total impedance has an angle of +36.4°, then the circuit's total admittance must have an angle of −36.4°.
• If you prefer to work with admittances instead of impedances, you can use modified forms of Ohm's law: for example, instead of writing I = V ÷ Z (which says that current equals voltage divided by impedance) you could instead write I = V × Y (current equals voltage times admittance). But I usually find it easier to stick with impedance rather than using admittance.
• ##### Summary of New Terms
• Let's quickly summarize these new terms and abbreviations. Each row in the table below shows a pair of quantities that are reciprocals of each other. The quantities in the left column are all measured in ohms, while those in the right column are all measured in siemens.
 Quantity measured in ohms Reciprocal quantity measured in siemens resistance R conductance G reactance X susceptance B impedance Z admittance Y
• As a memory aid, notice that the terms in the left-hand column come from English words that imply opposing or working against something: to resist or impede someone means to try to stop him, and to react against something means to work against it.
• Conversely, the terms in the right-hand column come from English words that imply helping or working with something: to conduct or admit someone means to help her progress, and to be susceptible to something means that you go along with it.
• These similarities to English words make sense, because all of these terms describe how much (or how little) a component or circuit opposes the flow of current.
• If you increase a circuit's resistance, reactance, or impedance, then you're increasing the circuit's opposition to current, and therefore less current will flow.
• Conversely, if you increase a circuit's conductance, susceptance, or admittance, then you're decreasing the circuit's opposition to current, and therefore more current will flow.
• ##### Admittance of a Resistor
• From Unit 6 you know that in AC circuits a resistor's impedance ZR is a complex quantity whose magnitude in ohms is R and whose angle is 0°.
• Therefore, the resistor's admittance is a complex quantity whose magnitude in siemens is 1/R (which is just the conductance G) and whose angle is 0°.
• So in polar notation, a resistor's admittance is

YR = G∠0°

• We can easily convert this to rectangular notation, to get

YR = G + j0

or simply

YR = G

• Thus, YR for any resistor has a real part but no imaginary part.
• For example, consider a 2 kΩ resistor. Its conductance G is 500 µS, so we can write its admittance (in polar form) as

YR = 500 ∠0° µS

• ##### Admittance of a Capacitor
• From Unit 6 you know that in AC circuits a capacitor's impedance ZR is a complex quantity whose magnitude in ohms is 1/(2pfC) and whose angle is −90°.
• Therefore, the same capacitor's admittance is a complex quantity whose magnitude in siemens is 2pfC and whose angle is +90°.
• So in polar notation, a capacitor's admittance is

YC = 2pfC∠90°

• We can easily convert this to rectangular notation, to get

YC = 0 + j2pfC

or simply

YC = j2pfC

• Thus, YC for any capacitor has a positive imaginary part but no real part.
• For example, consider a 1 µF capacitor in a circuit whose frequency is 500 Hz. The capacitor's reactance X is equal to 1/(2pfC), which works out to 318.3 Ω. Therefore its susceptance B is 3.14 mS, so we can write its admittance (in polar form) as

YC = 3.14 ∠90° mS

##### Admittance of an Inductor
• From Unit 6 you know that in AC circuits an indutor's impedance ZL is a complex quantity whose magnitude in ohms is 2pfL and whose angle is 90°.
• Therefore, the indutor's admittance is a complex quantity whose magnitude in siemens is 1/(2pfL) and whose angle is −90°.
• So in polar notation, an indutor's admittance is

YL = 1/(2pfL)∠−90°

• We can easily convert this to rectangular notation, to get

YL = 0 − j/(2pfL)

or simply

YL = − j/(2pfL)

• Thus, YL for any indutor has a negative imaginary part but no real part.
• For example, consider a 10 mH inductor in a circuit whose frequency is 200 Hz. The inductor's reactance X is equal to 2pfL, which works out to 12.6 Ω. Therefore its susceptance B is 79.6 mS, so we can write its admittance (in polar form) as

YL = 79.6 ∠−90° µS

• ##### Unit 7 Review
• This e-Lesson has covered several important topics, including:
• parallel AC circuits
• Kirchhoff's Current Law
• current-divider rule
• troubleshooting parallel AC circuits
• conductance, susceptance, and admittance.
• 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.