Unit 6: Series AC Circuits

In this unit we'll pull together a lot of things from earlier units and use them to analyze series circuits containing AC voltage sources. You'll need to remember what you've learned about AC fundamentals, capacitors, inductors, complex numbers, and phasors. Once you take all of that into account, though, you'll find that to analyze a series AC circuit, you follow the same steps that you follow to analyze a series DC circuit.

Also, the same rules that hold for series DC circuits (such as Ohm's law, Kirchhoff's Voltage Law, and the Voltage-Divider Rule) also hold for series AC circuits. But the math is a little more complicated, because each step involves complex numbers instead of real numbers.

Units 4 and 5 Review

• This unit will build on material that you studied in Unit 4 and Unit 5 . So let's begin by taking these two self-tests to review what you learned in those units.
• 

Review of Series DC Circuits

• In EET 1150 you learned how to analyze series 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. Add the resistance values to find the circuit's total resistance, R_T. For the circuit shown, this means that

     R_T = R₁ + R₂ + R₃

  2. Apply Ohm's law to the entire circuit to find the circuit's total current:

     I_T = V_S ÷ R_T

  3. Recognize that, since we're dealing with a series circuit, each resistor's current is equal to the total current. For the circuit shown:

     I₁ = I₂ = I₃ = I_T

  4. Apply Ohm's law to each resistor to find the voltage drops:

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

• 

Review: Kirchhoff's Voltage Law in Series DC Circuits

• In EET 1150 you also learned that Kirchhoff's Voltage Law (KVL) says that the sum of the voltage drops around any closed loop equals the sum of the voltage rises around that loop.
• In terms of a simple series DC circuit like the one you just analyzed, this means that the sum of all the resistor voltage drops must equal the source voltage.
• 

Review: Voltage Divider Rule in Series DC Circuits

• In EET 1150 you also learned that the voltage-divider rule is a shortcut rule that you can use to find the voltage drop across a resistor in a series circuit.
• The rule says that 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, this rule is expressed as:

  V_x = V_S×(R_x÷R_T)

• 

Review: Troubleshooting Series DC Circuits

• Troubleshooting a non-working circuit means finding the problem that is preventing the circuit from working correctly.
• The two most common types of problems are open circuits and short circuits.
• An open circuit, or "open," is a break in a circuit path.
• The most important thing to remember about opens is that no current can flow through an open.
• Therefore, no current can flow anywhere in a series circuit containing an open.
• Since no current flows through an open, you can think of the open as having infinite resistance (R = ∞).
• Usually, an open will not have a voltage drop of 0 V. In fact, in a series DC circuit that contains an open, the entire source voltage will appear across the open, and no voltage will appear across any of the other resistors.
• So if you measure the voltage between any two points in a series circuit containing an open, you'll measure 0 V if the two points are on the same side of the open, but you'll measure the entire source voltage if the points are on opposite sides of the open.
  • For example, suppose R3 is open in the circuit shown below. Then there will be 0 V across R1, across R2, and across R4. Also, V_ab = 0 V. But there will be 9 V across R3. Also, V_ac = 9 V, and V_bc = 9 V.
    
• 
• A short circuit, or "short," is a path of zero resistance connecting two points in a circuit that are not supposed to be connected.
• Since a short has zero resistance, the voltage across it must be zero. This follows from Ohm's law, V = I × R.
• A component is said to be short-circuited, or "shorted out," when there is a short circuit connected in parallel with it. No current flows through a short-circuited component. Instead, current is diverted through the short itself.
  • For example, suppose that in the circuit shown below there is a short between points a and b, perhaps caused by a loose wire clipping that connects these two points. Then R2 is short-circuited. No current will flow through R2; instead, current will follow the path of zero resistance through the short itself (the wire clipping).
    
• A short in a series DC circuit reduces the circuit's total resistance, causing more current to flow out of the voltage source.
  • For example, in the circuit shown above, if R2 is short-circuited by a wire clipping that connects points a and b, then R2's resistance disappears from the circuit, and the circuit's total resistance is equal to R₁ + R₃ + R₄.
• 
• That ends our quick review of series DC circuits. If you'd like a more thorough review, go to Unit 7 of EET 1150. Now let's get back to AC circuits.

Sinusoidal Response of AC Circuits

• Here's an important point that we've mentioned a couple of times before and that is worth repeating: When a sinusoidal voltage is applied to any circuit containing resistors, capacitors, and inductors, all of the circuit's current waveforms and voltage waveforms are sinusoids and have the same frequency as the source voltage.
• So, for example, if you're given the circuit shown below, and if you're told that the source voltage is a sinusoid having a frequency of 5 kHz, then you can say immediately that the current through every component is a 5-kHz sinusoidal current, and the voltage drop across every component is a 5-kHz sinusoidal voltage.
  
• Where things get a bit tricky is figuring out the peak values and phase shifts of these current and voltage waveforms. But we'll be able to do it, thanks to complex numbers.

Boldface Notation for Phasors

• Up to now we have used italicized, non-boldface letters to represent voltage and current. In particular:
  • V represents voltage.
  • I represents current.
• From this point onward, we will usually treat voltage and current as phasors, which means we'll treat them as complex numbers with both a magnitude and an angle. We'll use the italic letters listed above to denote the magnitude of the phasor, and we'll use boldface letters to represent the total phasor quantity (which includes both the magnitude and the angle). In particular:
  • V represents a phasor voltage, which has both a magnitude (V) and an angle.
  • I represents a phasor current, which has both a magnitude (I) and an angle.

Impedance

• In AC circuits, every component or combination of components has a quantity called an impedance, which we denote with the boldface letter Z.
• If we know a component's voltage V and its current I, we can use the following equation to find the component's impedance:

  Z = V ÷ I

• According to this equation, impedance Z is equal to the ratio of two complex numbers, V and I. Therefore impedance itself is a complex number. (That's why we denote it with a boldface letter.)
• Like any complex quantity, impedance can be expressed in either polar form or rectangular form. When we express it in polar form, we specify its magnitude and its angle.
• Because impedance is the ratio of a voltage to a current, impedance is measured in ohms.
• For example, suppose that dividing a component's voltage by its current gives the result

  V ÷ I = 736 ∠26.5° Ω

  Then the component's impedance Z has a magnitude of 736 Ω and an angle of 26.5°. At times we might be interested in talking just about this impedance's magnitude, in which case we would write

  Z = 736 Ω

  Note that in this last equation, Z is not written in boldface, because it does not denote a complex number. Rather it just denotes a complex number's magnitude.

Ohm's Law for AC Circuits

• Look again at the equation

  Z = V ÷ I

• This equation, which applies to components in AC circuits, is similar to Ohm's law, which you know from your study of DC circuits:

  R = V ÷ I

• In fact, we'll refer to the equation Z = V ÷ I as Ohm's law for AC circuits. In AC circuits, impedance plays a role very similar to the role played by resistance in DC circuits. But you must keep in mind that impedance is a complex quantity, which has both a magnitude and an angle.
• Of course, you can also rearrange this equation to solve for voltage if you know current and impedance, or to solve for current if you know voltage and impedance:

  V = I × Z

  I = V ÷ Z

• Remember, in each case, all quantities are complex numbers, not real numbers.
• 

How is Impedance Related to Resistance and Reactance?

• Recall from previous Units that:
  • Resistance R represents a resistor's opposition to current.
  • Capacitive reactance X_C represents a capacitor's opposition to current.
  • Inductive reactance X_L represents an inductor's opposition to current.
• Recall also that each of these quantities is measured in ohms.
• Impedance Z, which is also measured in ohms, can be thought of as a generalization of the concepts of resistance and reactance. It represents any component's (or combination of components) opposition to AC current.
• Sometimes we attach a subscript to Z to indicate the type of component whose impedance we're talking about:
  • We might write Z_R when discussing a resistor's impedance, which is closely related to its resistance R.
  • Similarly, we might write Z_C when discussing a capacitor's impedance, which is closely related to its reactance X_C.
  • Again, we might write Z_L when discussing an inductor's impedance, which is closely related to its reactance X_L.
• Let's look at each of these cases more closely.

Impedance of a Resistor

• A resistor's impedance Z_R is a complex quantity whose magnitude R is the resistance in ohms and whose angle is 0°.
  • We use 0° because voltage and current are in phase in resistors. (In other words, there is a 0° phase angle between a resistor's current and its voltage.)
• So in polar notation, a resistor's impedance is

  Z_R = R∠0°

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

  Z_R = R + j0

  or simply

  Z_R = R

• Thus, Z_R for any resistor has a real part but no imaginary part. In the complex plane, a resistor's impedance lies along the positive real axis, as shown in the following diagram representing a resistor whose resistance is 50 Ω:
  
• 

Impedance of a Capacitor

• A capacitor's impedance Z_C is a complex quantity whose magnitude X_C is 1 ÷ (2pfC), and whose angle is −90°.
  • We use −90° because voltage lags current by 90° in a capacitor.
• So in polar notation, a capacitor's impedance is

  Z_C = X_C ∠−90°

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

Z_C = 0 − jX_C

or simply

Z_C = −jX_C

• Remember, in each of these equations X_C = 1 ÷ (2pfC), which is also equal to 1 ÷ (ωC).
• Thus, Z_C for any capacitor has a negative imaginary part but no real part. In the complex plane, a capacitor's impedance lies along the negative imaginary axis, as shown in the following diagram representing a capacitor whose reactance is 50 Ω:
  .
• 

Impedance of an Inductor

• An inductor's impedance Z_L is a complex quantity whose magnitude X_L is 2pfL and whose angle is +90°.
  • We use +90° because voltage leads current by 90° in an inductor.
• So in polar notation, an inductor's impedance is

  Z_L =   X_L ∠90°

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

Z_L = 0 + jX_L

or simply

Z_L = jX_L

• Remember, in each of these equations X_L = 2pfL, which is also equal to ωL.
• Thus, Z_L for any inductor has a positive imaginary part but no real part. In the complex plane, an inductor's impedance lies along the positive imaginary axis, as shown in the following diagram representing an inductor whose reactance is 50 Ω:
  .
• 

Putting It All Together

• We've seen that Ohm's law for AC circuits can be written in any of the following forms:

  I = V ÷ Z

  V = I × Z

  Z = V ÷ I

• We've also seen how to calculate a resistor's impedance, or a capacitor's impedance, or an inductor's impedance.
• Let's look at some problems that require us to combine these pieces.
• 

Series Impedance

• Suppose we have a resistance in series with a reactance. We'd like to find the total impedance of these two components, but we can't simply add resistances and reactances as real numbers.  For example, a 1 kΩ resistance in series with a 2 kΩ capacitive reactance does not add up to 3 kΩ.
• Instead, we must add them as complex numbers. So, to find the total impedance of a 1 kΩ resistance in series with a 2 kΩ capacitive reactance, we must add 1∠0° kΩ plus 2∠−90° kΩ, which gives us a total of 2.24∠−63.4° kΩ.
• It may seem strange that you can combine a 1 kΩ resistance with a 2 kΩ reactance and come up with a total of only 2.24 kΩ, but that's how it works.

Analyzing Series AC Circuits

• Now that you know how to treat resistances and reactances as complex quantities, and how to use the phasor form of Ohm's law, and how to use complex numbers to find total impedance, you're ready to analyze any series AC circuit, such as the series RLC circuit shown below.

• Here are the steps to follow:
  1. Use complex addition to find the circuit's total impedance, Z_T.
  2. Apply Ohm's law to the entire circuit to find the circuit's total current.
  3. Recognize that, since we're dealing with a series circuit, each component's current is equal to the total current.
  4. Apply Ohm's law to each component to find the voltage drops.
• These are very similar to the steps that you followed in EET 1150 to analyze a simple series 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 1: Find Total Impedance

• For n impedances in series, total impedance is given by

  Z_T = Z₁ + Z₂ + … + Z_n

• Each Z on the right-hand side of this equation may be the impedance of a resistor, an inductor, or a capacitor. So this step will require you first to find the reactances of any capacitors or inductors in the circuit.
• Remember: we're adding complex numbers here, not real numbers.
• 

Step 2: Find Total Current

• Knowing the source voltage V_S and the total impedance Z_T, you can use Ohm's Law to find the current:

  I_T = V_S ÷ Z_T

• Again, remember that we're dividing complex numbers, not real numbers.
• 

Step 3: Find Individual Currents

• This step is the easiest. It simply requires you to remember that in any series circuit (DC or AC), every component's current is equal to the total current:

  I_T = I₁ = I₂ = … = I_n

• 

Step 4: Find Voltage Drops

• Now that you know the current through each component, use Ohm's Law to find the voltage drop across each component:

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

• Remember, 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.
• Again, remember that we're multiplying complex numbers, not real numbers.
• 

Practice Problems

• Want more practice analyzing series AC circuits? Here are a couple of lessons that will generate as many practice problems as you want, and then let you check your answers against the correct answers.
• The first one covers series RC circuits:
• 
• And the second one covers series RL circuits:
• 

Phasor Diagrams

• After you've analyzed a circuit by finding currents and voltage drops, you can draw a phasor diagram that shows in graphical form how these quantities relate to each other. A phasor diagram simply shows each voltage and current as a vector in the complex plane, drawn with the appropriate angle and magnitude.
• Here is a simple example showing the current and voltage phasors for a single component:
  
• Here's another example showing the phasors for all voltages and currents in a particular series RC circuit.
  
• From the diagram you can quickly see the relationship between the circuit's current and voltages.
• In every phasor diagram for a series circuit, you should find that the current and the resistor's voltage drop have the same angle, since current and voltage in a resistor are always in phase with each other.
• Also, you should find that inductor voltage and capacitor voltage are always at a 90° angle to the current, which should make sense. (Remember ELI the ICEman from Unit 4?)

Kirchhoff's Voltage Law

• As in DC circuits, Kirchhoff's Voltage Law (KVL) says that the sum of the voltage drops around any closed loop equals the sum of the voltage rises around that loop.
• As we'll see in later units of this course, KVL applies to all circuits, whether series, parallel, or series-parallel.
• In this unit we're restricting our attention to series circuits containing a single voltage source. In these circuits, KVL simplifies to the following form: the source voltage in a series circuit is equal to the sum of the voltage drops across the circuit's resistors, capacitors, and inductors.
• Whenever you apply KVL to an AC circuit, you must use complex numbers, not real numbers. If you just add the magnitudes of the voltages, instead of adding the magnitudes along with their angles, you won't get good results.
• 

Voltage-Divider Rule

• As in DC circuits, this is a shortcut rule for finding voltage drops in a series circuit.
• The voltage-divider rule says that the voltage V_x across any impedance Z_x in a series circuit with source voltage V_S is given by:

  V_x = (Z_x ÷ Z_T) × V_S

• Again, use complex numbers, not real numbers.
• 

Troubleshooting Series AC Circuits

• Above we reviewed the basics of troubleshooting series DC circuits. Almost all of these same points apply to series AC circuits. (But there's one important difference noted below.) In particular:
• No current flows through an open.
• Therefore, no current flows anywhere in a series circuit containing an open.
• Since no current flows through an open, you can think of the open as having infinite resistance (R = ∞).
• Usually, an open will not have a voltage drop of 0 V. In fact, in a series circuit that contains an open, the entire source voltage will appear across the open, and no voltage will appear across any of the other resistors, capacitors, or inductors.
• So if you measure the voltage between any two points in a series circuit containing an open, you'll measure 0 V if the two points are on the same side of the open, but you'll measure the entire source voltage if the points are on opposite sides of the open.
• A short has zero resistance and zero voltage.
  • A component is said to be short-circuited, or "shorted out," when there is a short connected in parallel with it. No current flows through a short-circuited component. Instead, current is diverted through the short itself.
• So far, everything we've said about opens and shorts in series AC circuits is the same as what we said earlier about opens and shorts in series DC circuits. But here's a difference:
  • A short in a series DC circuit will always reduce the circuit's total resistance, increasing the circuit's total current.
  • But in a series AC circuit, a short could either increase or decrease the circuit's total impedance, and therefore could either decrease or increase the total current.
• Why this difference between shorts in DC circuits and shorts in AC circuits? It's because of the difference between real numbers and complex numbers.
  • To find a series DC circuit's total resistance, you add two or more real numbers. If one of the circuit's resistors is shorted out, then you replace one of these numbers with zero, and this will decrease the total.
    • For example, suppose a series circuit contains a 100 Ω resistor, a 150 Ω resistor, and a 200 Ω resistor. Then the total resistance is 450 Ω. But if the 200 Ω resistor is shorted out, then you replace its resistance with zero, and so the total resistance decreases to 250 Ω. This decrease in total resistance will increase the circuit's total current.
  • But to find a series AC circuit's total impedance, you add two or more complex numbers. If one of the circuit's components is shorted out, then you replace one of these complex numbers with zero. When you're adding complex numbers, replacing one of them by zero could either decrease or increase the total.
    • For example, suppose a series 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 112∠−26.6° Ω. If the capacitor is shorted out, then you replace its impedance with zero, and so the total impedance increases to 180∠56.3° Ω. This increase in total impedance will decrease the circuit's total current.
    • On the other hand, suppose that in the same circuit the capacitor is okay but the resistor is shorted out. Then the circuit's total impedance is 50∠−90° Ω, which is a decrease from the original total impedance. This decrease in total impedance will increase the circuit's total current.
    • So a short in a series AC circuit may either increase or decrease the total impedance.
• 

"Systems View" of a Circuit

• In your electronics courses up to now, you've considered a circuit as a collection of individual components such as resistors, capacitors, and inductors. Most of the circuits you've studied have been very small, with just a few components.
• But real-world circuits are usually much more complex, with hundreds or even thousands of components. To study and analyze such circuits, you have to shift your way of looking at circuits. Instead of concentrating on individual components, it's more useful to think of the circuit as a system made up of parts that perform certain functions. The "parts" that I'm referring to here are not individual components. Rather, they are sub-circuits that contain many components connected together to perform some function.
• For example, in your later courses you'll study amplifier circuits. These circuits contain transistors as well as resistors, capacitors, and inductors, so we're not ready to understand the details now. But there are a number of standard designs for amplifier circuits, and rather than focusing on the details you might just want to consider the entire amplifier circuit as a "box." This box has two input terminals to which you can connect an input voltage, and two output terminals at which the amplifier's output voltage will appear. Here's a diagram:
  
• The amplifier circuit contains many components (resistors, capacitors, transistors). But in this diagram we're not showing those details.
• A more complete circuit might consist of an oscillator connected to an amplifier connected to a power amplifier, as shown here:
  
• In this diagram we have three boxes representing three complicated sub-circuits. We're not showing the details of these sub-circuits, but we are showing how the sub-circuits are connected to each other. (For example, the two lines between the oscillator and the amplifier show that the oscillator's output voltage is also the amplifier's input voltage.)
• Next we'll look at a few simple series RC and RL circuits that we can think of as "boxes" that perform a certain function.

Lag Circuits and Lead Circuits

• In some applications, a designer needs to shift a voltage's phase angle by a certain amount. In such cases the designer uses a circuit that introduces a phase shift between the circuit's output voltage and its input voltage.
• There are two basic possibilities here. Either:
• The circuit is designed so that its output voltage lags its input voltage, in which case we're dealing with a lag circuit. Or:
• The circuit is designed so that its output voltage leads its input voltage, in which case we're dealing with a lead circuit.
• 

RC Lag Circuits and Lead Circuits

• A simple series RC circuit can serve as either a lag circuit or a lead circuit, depending on whether you take the output voltage across the resistor or across the capacitor. In particular:
  • In any series RC circuit, the capacitor's voltage lags the source voltage, so you'll have a lag circuit if you take the output voltage across the capacitor, as shown here:
    
  • Also, in any series RC circuit, the resistor's voltage leads the source voltage, so you'll have a lead circuit if you take the output voltage across the resistor, as shown here:
    
• If you remember ELI the ICEman, you'll be able to quickly identify circuits like the ones above as either lead circuits or lag circuits.
  • ICE reminds you that a capacitor's voltage tends to lag everything else in the circuit, so you've got a lag circuit if you're taking the output voltage across the capacitor.
• Whenever you encounter circuits like these, you can always use the general techniques you learned above to analyze the circuit and figure out how far the output voltage is shifted from the input voltage. Or you can remember the following formulas.
  • For an RC lag circuit, the phase angle φ between the input and output is

    φ = −tan⁻¹(R ÷ X_C)

  • For an RC lead circuit, the phase angle φ between the input and output is

    φ = tan⁻¹(X_C ÷ R)

• Using these formulas, and by choosing appropriate values of R and C, you can design a lead circuit or lag circuit to shift the voltage by any desired angle.
• Here's a nice memory trick to help you remember those two formulas: Notice that the order of the R and the X_C in each formula is the same as the order of the resistor and the capacitor in the corresponding schematic diagram.
• 

RL Lag Circuits and Lead Circuits

• A simple series RL circuit can also serve as either a lag circuit or a lead circuit, depending on whether you take the output voltage across the resistor or across the inductor. In particular:
  • In any series RL circuit, the resistor's voltage lags the source voltage, so you'll have a lag circuit if you take the output voltage across the resistor, as shown here:
    
  • Also, in any series RL circuit, the inductor's voltage leads the source voltage, so you'll have a lead circuit if you take the output voltage across the inductor, as shown here:
    
• If you remember ELI the ICEman, you'll be able to quickly identify circuits like the ones above as either lead circuits or lag circuits.
  • ELI reminds you that an inductor's voltage tends to lead everything else in the circuit, so you've got a lead circuit if you're taking the output voltage across the inductor.
• Whenever you encounter circuits like these, you can always use the general techniques you learned above to analyze the circuit and figure out how far the output voltage is shifted from the input voltage. Or you can remember the following formulas.
  • For an RL lag circuit, the phase angle φ between the input and output is

    φ = −tan⁻¹(X_L ÷ R)

  • For an RL lead circuit, the phase angle φ between the input and output is

    φ = tan⁻¹(R ÷ X_L)

• Here's a nice memory trick to help you remember those two formulas: Notice that the order of the R and the X_L in each formula is the same as the order of the resistor and the inductor in the corresponding schematic diagram.
• 

Unit 6 Review

• This e-Lesson has covered several important topics, including:
  • series AC circuits
  • phasor diagrams
  • Kirchhoff's Voltage Law
  • voltage-divider rule
  • troubleshooting series AC circuits
  • lag circuits and lead 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.

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