Example series-parallel R, L, and C circuit.
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:
determine values of impedance image 1
Now we can set up the initial values in our table:
initial values in table 1
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:
initial values in table 2
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 arrive at no other way!
initial values in table 3
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 ∠ -58.371°) and the total
voltage (120 volts ∠ 0°), we can apply Ohm’s Law (I=E/Z) vertically in
the “Total” column to arrive at a figure for total current:
initial values in table 4
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:
initial values in table 5
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:
initial values in table 7
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!
series-parallel kirchhoffs voltage law
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.
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 into the columns for R and L—C2:
initial values in table 9
rule of series circuits table 3
Now we’re all set for calculating the 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:
initial values in table 10
initial values in table 11
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:
total current kirchhoffs current law
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:
rule of series circuits table 1
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.
rule of series circuits table 2
Now, let’s turn to SPICE for a computer verification of our work:
series-parallel rlc spice circuit
Example series-parallel R, L, C SPICE circuit.
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: