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:
- Add the resistance values to find the circuit's total resistance, RT.
For the circuit shown, this means that
RT = R1 + R2 + R3
- Apply Ohm's law to the entire circuit to find the circuit's
total current:
IT = VS ÷ RT
- 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 = IT
- 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:
Vx = VS×(Rx÷RT)

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, Vab = 0 V. But there
will be 9 V across R3. Also, Vac = 9 V,
and Vbc = 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
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 XC represents a capacitor's
opposition to current.
- Inductive reactance XL 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 ZR when discussing a resistor's
impedance, which is closely related to its resistance R.
- Similarly, we might write ZC when discussing
a capacitor's impedance, which is closely related to its
reactance XC.
- Again, we might write ZL when discussing
an inductor's impedance, which is closely related to its reactance XL.
- Let's look at each of these cases more closely.
Impedance of a Resistor
- A resistor's impedance ZR 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
ZR = R∠0°
- We can easily convert this to rectangular notation, to get
ZR = R + j0
or simply
ZR = R
- Thus, ZR 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 ZC is a complex
quantity whose
magnitude XC 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
ZC = XC ∠−90°
- We can easily convert this to rectangular notation, to get
ZC = 0 − jXC
or simply
ZC = −jXC
- Remember, in each of these equations XC = 1 ÷ (2pfC),
which is also equal to 1 ÷ (ωC).
- Thus, ZC 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 ZL is a complex
quantity whose
magnitude XL 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
ZL = XL ∠90°
- We can easily convert this to rectangular notation, to get
ZL = 0 + jXL
or simply
ZL = jXL
- Remember, in each of these equations XL = 2pfL,
which is also equal to ωL.
- Thus, ZL 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:
- Use complex addition to find the circuit's total impedance, ZT.
- Apply Ohm's law to the entire circuit to find
the circuit's total current.
- Recognize that, since we're dealing with a series circuit, each
component's current is equal to the total current.
- 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
ZT = Z1 + Z2 + … + Zn
- 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 VS and the
total impedance ZT, you can use Ohm's
Law to find the current:
IT = VS ÷ ZT
- 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:
IT = I1 = I2 = … = In

Step 4: Find Voltage Drops
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 Vx across
any impedance Zx in
a series circuit with source voltage VS is
given by:
Vx = (Zx ÷ ZT) × VS
- 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 ÷ XC)
- For an RC lead circuit, the phase angle φ between
the input and output is
φ = tan−1(XC ÷ 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 XC 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(XL ÷ R)
- For an RL lead circuit, the phase angle φ between
the input and output is
φ = tan−1(R ÷ XL)
- Here's a nice memory trick to help you remember those two formulas:
Notice that the order of the R and the XL 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.