Review of R, X, and Z
Before we begin to explore the effects of
resistors, inductors, and capacitors connected together in the same AC
circuits, let's briefly review some basic terms and facts.
Resistance is essentially friction
against the motion of electrons. It is present in all conductors to some
extent (except superconductors!), most notably in resistors. When
alternating current goes through a resistance, a voltage drop is
produced that is in-phase with the current. Resistance is mathematically
symbolized by the letter "R" and is measured in the unit of ohms (Ω).
Reactance is essentially inertia against
the motion of electrons. It is present anywhere electric or magnetic
fields are developed in proportion to applied voltage or current,
respectively; but most notably in capacitors and inductors. When
alternating current goes through a pure reactance, a voltage drop is
produced that is 90o out of phase with the current. Reactance
is mathematically symbolized by the letter "X" and is measured in the
unit of ohms (Ω).
Impedance is a comprehensive expression
of any and all forms of opposition to electron flow, including both
resistance and reactance. It is present in all circuits, and in all
components. When alternating current goes through an impedance, a
voltage drop is produced that is somewhere between 0o and 90o
out of phase with the current. Impedance is mathematically symbolized by
the letter "Z" and is measured in the unit of ohms (Ω), in complex form.
Perfect resistors possess resistance, but
not reactance. Perfect inductors and perfect capacitors possess
reactance but no resistance. All components possess impedance, and
because of this universal quality, it makes sense to translate all
component values (resistance, inductance, capacitance) into common terms
of impedance as the first step in analyzing an AC circuit.
The impedance phase angle for any
component is the phase shift between voltage across that component and
current through that component. For a perfect resistor, the voltage drop
and current are always in phase with each other, and so the impedance
angle of a resistor is said to be 0o. For an perfect
inductor, voltage drop always leads current by 90o, and so an
inductor's impedance phase angle is said to be +90o. For a
perfect capacitor, voltage drop always lags current by 90o,
and so a capacitor's impedance phase angle is said to be -90o.
Impedances in AC behave analogously to
resistances in DC circuits: they add in series, and they diminish in
parallel. A revised version of Ohm's Law, based on impedance rather than
resistance, looks like this:
Kirchhoff's
Laws and all network analysis methods and theorems are true for AC
circuits as well, so long as quantities are represented in complex
rather than scalar form. While this qualified equivalence may be
arithmetically challenging, it is conceptually simple and elegant. The
only real difference between DC and AC circuit calculations is in regard
to power. Because reactance doesn't dissipate power as resistance does,
the concept of power in AC circuits is radically different from that of
DC circuits. More on this subject in a later chapter!
Series R, L, and C
Let's take the following example circuit
and analyze it:
The first step is to determine the
reactances (in ohms) for the inductor and the capacitor.
The next step is to express all
resistances and reactances in a mathematically common form: impedance.
Remember that an inductive reactance translates into a positive
imaginary impedance (or an impedance at +90o), while a
capacitive reactance translates into a negative imaginary impedance
(impedance at -90o). Resistance, of course, is still regarded
as a purely "real" impedance (polar angle of 0o):
Now, with all quantities of opposition to
electric current expressed in a common, complex number format (as
impedances, and not as resistances or reactances), they can be handled
in the same way as plain resistances in a DC circuit. This is an ideal
time to draw up an analysis table for this circuit and insert all the
"given" figures (total voltage, and the impedances of the resistor,
inductor, and capacitor).
Unless otherwise specified, the source
voltage will be our reference for phase shift, and so will be written at
an angle of 0o. Remember that there is no such thing as an
"absolute" angle of phase shift for a voltage or current, since it's
always a quantity relative to another waveform. Phase angles for
impedance, however (like those of the resistor, inductor, and
capacitor), are known absolutely, because the phase relationships
between voltage and current at each component are absolutely defined.
Notice that I'm assuming a perfectly
reactive inductor and capacitor, with impedance phase angles of exactly
+90 and -90o, respectively. Although real components won't be
perfect in this regard, they should be fairly close. For simplicity,
I'll assume perfectly reactive inductors and capacitors from now on in
my example calculations except where noted otherwise.
Since the above example circuit is a
series circuit, we know that the total circuit impedance is equal to the
sum of the individuals, so:
Inserting this figure for total impedance
into our table:
We can now apply Ohm's Law (I=E/R)
vertically in the "Total" column to find total current for this series
circuit:
Being a series circuit, current must be
equal through all components. Thus, we can take the figure obtained for
total current and distribute it to each of the other columns:
Now we're prepared to apply Ohm's Law (E=IZ)
to each of the individual component columns in the table, to determine
voltage drops:
Notice something strange here: although
our supply voltage is only 120 volts, the voltage across the capacitor
is 137.46 volts! How can this be? The answer lies in the interaction
between the inductive and capacitive reactances. Expressed as
impedances, we can see that the inductor opposes current in a manner
precisely opposite that of the capacitor. Expressed in rectangular form,
the inductor's impedance has a positive imaginary term and the capacitor
has a negative imaginary term. When these two contrary impedances are
added (in series), they tend to cancel each other out! Although they're
still added together to produce a sum, that sum is actually less than
either of the individual (capacitive or inductive) impedances alone. It
is analogous to adding together a positive and a negative (scalar)
number: the sum is a quantity less than either one's individual absolute
value.
If the total impedance in a series
circuit with both inductive and capacitive elements is less than the
impedance of either element separately, then the total current in that
circuit must be greater than what it would be with only the inductive or
only the capacitive elements there. With this abnormally high current
through each of the components, voltages greater than the source voltage
may be obtained across some of the individual components! Further
consequences of inductors' and capacitors' opposite reactances in the
same circuit will be explored in the next chapter.
Once you've mastered the technique of
reducing all component values to impedances (Z), analyzing any AC
circuit is only about as difficult as analyzing any DC circuit, except
that the quantities dealt with are vector instead of scalar. With the
exception of equations dealing with power (P), equations in AC circuits
are the same as those in DC circuits, using impedances (Z) instead of
resistances (R). Ohm's Law (E=IZ) still holds true, and so do
Kirchhoff's Voltage and Current Laws.
To demonstrate Kirchhoff's Voltage Law in
an AC circuit, we can look at the answers we derived for component
voltage drops in the last circuit. KVL tells us that the algebraic sum
of the voltage drops across the resistor, inductor, and capacitor should
equal the applied voltage from the source. Even though this may not look
like it is true at first sight, a bit of complex number addition proves
otherwise:
Aside from a bit of rounding error, the
sum of these voltage drops does equal 120 volts. Performed on a
calculator (preserving all digits), the answer you will receive should
be exactly 120 + j0 volts.
We can also use SPICE to verify our
figures for this circuit:
ac r-l-c circuit
v1 1 0 ac 120 sin
r1 1 2 250
l1 2 3 650m
c1 3 0 1.5u
.ac lin 1 60 60
.print ac v(1,2) v(2,3) v(3,0) i(v1)
.print ac vp(1,2) vp(2,3) vp(3,0) ip(v1)
.end
freq v(1,2) v(2,3) v(3) i(v1)
6.000E+01 1.943E+01 1.905E+01 1.375E+02 7.773E-02
freq vp(1,2) vp(2,3) vp(3) ip(v1)
6.000E+01 8.068E+01 1.707E+02 -9.320E+00 -9.932E+01
The SPICE simulation shows our
hand-calculated results to be accurate.
As you can see, there is little
difference between AC circuit analysis and DC circuit analysis, except
that all quantities of voltage, current, and resistance (actually,
impedance) must be handled in complex rather than scalar form so as to
account for phase angle. This is good, since it means all you've learned
about DC electric circuits applies to what you're learning here. The
only exception to this consistency is the calculation of power, which is
so unique that it deserves a chapter devoted to that subject alone.
- REVIEW:
- Impedances of any kind add in series:
ZTotal = Z1 + Z2 + . . . Zn
- Although impedances add in series, the
total impedance for a circuit containing both inductance and
capacitance may be less than one or more of the individual impedances,
because series inductive and capacitive impedances tend to cancel each
other out. This may lead to voltage drops across components exceeding
the supply voltage!
- All rules and laws of DC circuits
apply to AC circuits, so long as values are expressed in complex form
rather than scalar. The only exception to this principle is the
calculation of power, which is very different for AC.
Parallel R, L, and C
We can take the same components from the
series circuit and rearrange them into a parallel configuration for an
easy example circuit:
The fact that these components are
connected in parallel instead of series now has absolutely no effect on
their individual impedances. So long as the power supply is the same
frequency as before, the inductive and capacitive reactances will not
have changed at all:
With all component values expressed as
impedances (Z), we can set up an analysis table and proceed as in the
last example problem, except this time following the rules of parallel
circuits instead of series:
Knowing that voltage is shared equally by
all components in a parallel circuit, we can transfer the figure for
total voltage to all component columns in the table:
Now, we can apply Ohm's Law (I=E/Z)
vertically in each column to determine current through each component:
There are two strategies for calculating
total current and total impedance. First, we could calculate total
impedance from all the individual impedances in parallel (ZTotal
= 1/(1/ZR + 1/ZL + 1/ZC), and then
calculate total current by dividing source voltage by total impedance
(I=E/Z). However, working through the parallel impedance equation with
complex numbers is no easy task, with all the reciprocations (1/Z). This
is especially true if you're unfortunate enough not to have a calculator
that handles complex numbers and are forced to do it all by hand
(reciprocate the individual impedances in polar form, then convert them
all to rectangular form for addition, then convert back to polar form
for the final inversion, then invert). The second way to calculate total
current and total impedance is to add up all the branch currents to
arrive at total current (total current in a parallel circuit -- AC or DC
-- is equal to the sum of the branch currents), then use Ohm's Law to
determine total impedance from total voltage and total current (Z=E/I).
Either method, performed properly, will
provide the correct answers. Let's try analyzing this circuit with SPICE
and see what happens:
ac r-l-c circuit
v1 1 0 ac 120 sin
vi 1 2 ac 0
vir 2 3 ac 0
vil 2 4 ac 0
rbogus 4 5 1e-12
vic 2 6 ac 0
r1 3 0 250
l1 5 0 650m
c1 6 0 1.5u
.ac lin 1 60 60
.print ac i(vi) i(vir) i(vil) i(vic)
.print ac ip(vi) ip(vir) ip(vil) ip(vic)
.end
freq i(vi) i(vir) i(vil) i(vic)
6.000E+01 6.390E-01 4.800E-01 4.897E-01 6.786E-02
freq ip(vi) ip(vir) ip(vil) ip(vic)
6.000E+01 -4.131E+01 0.000E+00 -9.000E+01 9.000E+01
It took a little bit of trickery to get
SPICE working as we would like on this circuit (installing "dummy"
voltage sources in each branch to obtain current figures and installing
the "dummy" resistor in the inductor branch to prevent a direct
inductor-to-voltage source loop, which SPICE cannot tolerate), but we
did get the proper readings. Even more than that, by installing the
dummy voltage sources (current meters) in the proper directions, we were
able to avoid that idiosyncrasy of SPICE of printing current figures 180o
out of phase. This way, our current phase readings came out to exactly
match our hand calculations.
Series-parallel R, L, and C
Now that we've seen how series and
parallel AC circuit analysis is not fundamentally different than DC
circuit analysis, it should come as no surprise that series-parallel
analysis would be the same as well, just using complex numbers instead
of scalar to represent voltage, current, and impedance.
Take this series-parallel circuit for
example:
The first order of business, as usual, is
to determine values of impedance (Z) for all components based on the
frequency of the AC power source. To do this, we need to first determine
values of reactance (X) for all inductors and capacitors, then convert
reactance (X) and resistance (R) figures into proper impedance (Z) form:
Now we can set up the initial values in
our table:
Being a series-parallel combination
circuit, we must reduce it to a total impedance in more than one step.
The first step is to combine L and C2 as a series combination
of impedances, by adding their impedances together. Then, that impedance
will be combined in parallel with the impedance of the resistor, to
arrive at another combination of impedances. Finally, that quantity will
be added to the impedance of C1 to arrive at the total
impedance.
In order that our table may follow all
these steps, it will be necessary to add additional columns to it so
that each step may be represented. Adding more columns horizontally to
the table shown above would be impractical for formatting reasons, so I
will place a new row of columns underneath, each column designated by
its respective component combination:
Calculating these new (combination)
impedances will require complex addition for series combinations, and
the "reciprocal" formula for complex impedances in parallel. This time,
there is no avoidance of the reciprocal formula: the required figures
can be arrived at no other way!
Seeing as how our second table contains a
column for "Total," we can safely discard that column from the first
table. This gives us one table with four columns and another table with
three columns.
Now that we know the total impedance
(818.34 Ω ∠ -58.371o) and the total voltage (120 volts ∠ 0o),
we can apply Ohm's Law (I=E/Z) vertically in the "Total" column to
arrive at a figure for total current:
At this point we ask ourselves the
question: are there any components or component combinations which share
either the total voltage or the total current? In this case, both C1
and the parallel combination R//(L--C2) share the same
(total) current, since the total impedance is composed of the two sets
of impedances in series. Thus, we can transfer the figure for total
current into both columns:
Now, we can calculate voltage drops
across C1 and the series-parallel combination of R//(L--C2)
using Ohm's Law (E=IZ) vertically in those table columns:
A quick double-check of our work at this
point would be to see whether or not the voltage drops across C1
and the series-parallel combination of R//(L--C2) indeed add
up to the total. According to Kirchhoff's Voltage Law, they should!
That last step was merely a precaution.
In a problem with as many steps as this one has, there is much
opportunity for error. Occasional cross-checks like that one can save a
person a lot of work and unnecessary frustration by identifying problems
prior to the final step of the problem.
After having solved for voltage drops
across C1 and the combination R//(L--C2), we again
ask ourselves the question: what other components share the same voltage
or current? In this case, the resistor (R) and the combination of the
inductor and the second capacitor (L--C2) share the same
voltage, because those sets of impedances are in parallel with each
other. Therefore, we can transfer the voltage figure just solved for
into the columns for R and L--C2:
Now we're all set for calculating current
through the resistor and through the series combination L--C2.
All we need to do is apply Ohm's Law (I=E/Z) vertically in both of those
columns:
Another quick double-check of our work at
this point would be to see if the current figures for L--C2
and R add up to the total current. According to Kirchhoff's Current Law,
they should:
Since the L and C2 are
connected in series, and since we know the current through their series
combination impedance, we can distribute that current figure to the L
and C2 columns following the rule of series circuits whereby
series components share the same current:
With one last step (actually, two
calculations), we can complete our analysis table for this circuit. With
impedance and current figures in place for L and C2, all we
have to do is apply Ohm's Law (E=IZ) vertically in those two columns to
calculate voltage drops.
Now, let's turn to SPICE for a computer
verification of our work:
ac series-parallel r-l-c circuit
v1 1 0 ac 120 sin
vit 1 2 ac 0
vilc 3 4 ac 0
vir 3 6 ac 0
c1 2 3 4.7u
l 4 5 650m
c2 5 0 1.5u
r 6 0 470
.ac lin 1 60 60
.print ac v(2,3) vp(2,3) i(vit) ip(vit)
.print ac v(4,5) vp(4,5) i(vilc) ip(vilc)
.print ac v(5,0) vp(5,0) i(vilc) ip(vilc)
.print ac v(6,0) vp(6,0) i(vir) ip(vir)
.end
freq v(2,3) vp(2,3) i(vit) ip(vit) C1
6.000E+01 8.276E+01 -3.163E+01 1.466E-01 5.837E+01
freq v(4,5) vp(4,5) i(vilc) ip(vilc) L
6.000E+01 1.059E+01 -1.388E+02 4.323E-02 1.312E+02
freq v(5) vp(5) i(vilc) ip(vilc) C2
6.000E+01 7.645E+01 4.122E+01 4.323E-02 1.312E+02
freq v(6) vp(6) i(vir) ip(vir) R
6.000E+01 6.586E+01 4.122E+01 1.401E-01 4.122E+01
Each line of the SPICE output listing
gives the voltage, voltage phase angle, current, and current phase angle
for C1, L, C2, and R, in that order. As you can
see, these figures do concur with our hand-calculated figures in the
circuit analysis table.
As daunting a task as series-parallel AC
circuit analysis may appear, it must be emphasized that there is nothing
really new going on here besides the use of complex numbers. Ohm's Law
(in its new form of E=IZ) still holds true, as do the voltage and
current Laws of Kirchhoff. While there is more potential for human error
in carrying out the necessary complex number calculations, the basic
principles and techniques of series-parallel circuit reduction are
exactly the same.
- REVIEW:
- Analysis of series-parallel AC
circuits is much the same as series-parallel DC circuits. The only
substantive difference is that all figures and calculations are in
complex (not scalar) form.
- It is important to remember that
before series-parallel reduction (simplification) can begin, you must
determine the impedance (Z) of every resistor, inductor, and
capacitor. That way, all component values will be expressed in common
terms (Z) instead of an incompatible mix of resistance (R), inductance
(L), and capacitance (C).
Susceptance and
Admittance
In the study of DC circuits, the student
of electricity comes across a term meaning the opposite of resistance:
conductance. It is a useful term when exploring the mathematical formula
for parallel resistances: Rparallel = 1 / (1/R1 +
1/R2 + . . . 1/Rn). Unlike resistance, which
diminishes as more parallel components are included in the circuit,
conductance simply adds. Mathematically, conductance is the reciprocal
of resistance, and each 1/R term in the "parallel resistance formula" is
actually a conductance.
Whereas the term "resistance" denotes the
amount of opposition to flowing electrons in a circuit, "conductance"
represents the ease of which electrons may flow. Resistance is the
measure of how much a circuit resists current, while conductance is the
measure of how much a circuit conducts current. Conductance used to be
measured in the unit of mhos, or "ohms" spelled backward. Now, the
proper unit of measurement is Siemens. When symbolized in a mathematical
formula, the proper letter to use for conductance is "G".
Reactive components such as inductors and
capacitors oppose the flow of electrons with respect to time, rather
than with a constant, unchanging friction as resistors do. We call this
time-based opposition, reactance, and like resistance we also measure it
in the unit of ohms.
As conductance is the complement of
resistance, there is also a complementary expression of reactance,
called susceptance. Mathematically, it is equal to 1/X, the reciprocal
of reactance. Like conductance, it used to be measured in the unit of
mhos, but now is measured in Siemens. Its mathematical symbol is "B",
unfortunately the same symbol used to represent magnetic flux density.
The terms "reactance" and "susceptance"
have a certain linguistic logic to them, just like resistance and
conductance. While reactance is the measure of how much a circuit reacts
against change in current over time, susceptance is the measure of how
much a circuit is susceptible to conducting a changing current.
If one were tasked with determining the
total effect of several parallel-connected, pure reactances, one could
convert each reactance (X) to a susceptance (B), then add susceptances
rather than diminish reactances: Xparallel = 1/(1/X1
+ 1/X2 + . . . 1/Xn). Like conductances (G),
susceptances (B) add in parallel and diminish in series. Also like
conductance, susceptance is a scalar quantity.
When resistive and reactive components
are interconnected, their combined effects can no longer be analyzed
with scalar quantities of resistance (R) and reactance (X). Likewise,
figures of conductance (G) and susceptance (B) are most useful in
circuits where the two types of opposition are not mixed, i.e. either a
purely resistive (conductive) circuit, or a purely reactive (susceptive)
circuit. In order to express and quantify the effects of mixed resistive
and reactive components, we had to have a new term: impedance, measured
in ohms and symbolized by the letter "Z".
To be consistent, we need a complementary
measure representing the reciprocal of impedance. The name for this
measure is admittance. Admittance is measured in (guess what?) the unit
of Siemens, and its symbol is "Y". Like impedance, admittance is a
complex quantity rather than scalar. Again, we see a certain logic to
the naming of this new term: while impedance is a measure of how much
alternating current is impeded in a circuit, admittance is a measure of
how much current is admitted.
Given a scientific calculator capable of
handling complex number arithmetic in both polar and rectangular forms,
you may never have to work with figures of susceptance (B) or admittance
(Y). Be aware, though, of their existence and their meanings.
Summary
With the notable exception of
calculations for power (P), all AC circuit calculations are based on the
same general principles as calculations for DC circuits. The only
significant difference is that fact that AC calculations use complex
quantities while DC calculations use scalar quantities. Ohm's Law,
Kirchhoff's Laws, and even the network theorems learned in DC still hold
true for AC when voltage, current, and impedance are all expressed with
complex numbers. The same troubleshooting strategies applied toward DC
circuits also hold for AC, although AC can certainly be more difficult
to work with due to phase angles which aren't registered by a handheld
multimeter.
Power is another subject altogether, and
will be covered in its own chapter in this book. Because power in a
reactive circuit is both absorbed and released -- not just dissipated as
it is with resistors -- its mathematical handling requires a more direct
application of trigonometry to solve.
When faced with analyzing an AC circuit,
the first step in analysis is to convert all resistor, inductor, and
capacitor component values into impedances (Z), based on the frequency
of the power source. After that, proceed with the same steps and
strategies learned for analyzing DC circuits, using the "new" form of
Ohm's Law: E=IZ ; I=E/Z ; and Z=E/I
Remember that only the calculated figures
expressed in polar form apply directly to empirical measurements of
voltage and current. Rectangular notation is merely a useful tool for us
to add and subtract complex quantities together. Polar notation, where
the magnitude (length of vector) directly relates to the magnitude of
the voltage or current measured, and the angle directly relates to the
phase shift in degrees, is the most practical way to express complex
quantities for circuit analysis.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first. See Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document
formatting, which led to a much better-looking second edition.
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