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).
|
