Series resistor-inductor circuits
In the previous section, we explored what
would happen in simple resistor-only and inductor-only AC circuits. Now
we will mix the two components together in series form and investigate
the effects.
Take this circuit as an example to work
with:
The resistor will offer 5 Ω of resistance
to AC current regardless of frequency, while the inductor will offer
3.7699 Ω of reactance to AC current at 60 Hz. Because the resistor's
resistance is a real number (5 Ω ∠ 0o, or 5 + j0 Ω), and the
inductor's reactance is an imaginary number (3.7699 Ω ∠ 90o,
or 0 + j3.7699 Ω), the combined effect of the two components will be an
opposition to current equal to the complex sum of the two numbers. This
combined opposition will be a vector combination of resistance and
reactance. In order to express this opposition succinctly, we need a
more comprehensive term for opposition to current than either resistance
or reactance alone. This term is called impedance, its symbol is
Z, and it is also expressed in the unit of ohms, just like resistance
and reactance. In the above example, the total circuit impedance is:
Impedance is related to voltage and
current just as you might expect, in a manner similar to resistance in
Ohm's Law:
In fact, this is a far more comprehensive
form of Ohm's Law than what was taught in DC electronics (E=IR), just as
impedance is a far more comprehensive expression of opposition to the
flow of electrons than resistance is. Any resistance and any
reactance, separately or in combination (series/parallel), can be and
should be represented as a single impedance in an AC circuit.
To calculate current in the above
circuit, we first need to give a phase angle reference for the voltage
source, which is generally assumed to be zero. (The phase angles of
resistive and inductive impedance are always 0o and
+90o, respectively, regardless of the given phase angles for
voltage or current).
As with the purely inductive circuit, the
current wave lags behind the voltage wave (of the source), although this
time the lag is not as great: only 37.016o as opposed to a
full 90o as was the case in the purely inductive circuit.
For the resistor and the inductor, the
phase relationships between voltage and current haven't changed. Across
voltage across the resistor is in phase (0o shift) with the
current through it; and the voltage across the inductor is +90o
out of phase with the current going through it. We can verify this
mathematically:
The voltage across the resistor has the
exact same phase angle as the current through it, telling us that E and
I are in phase (for the resistor only).
The voltage across the inductor has a
phase angle of 52.984o, while the current through the
inductor has a phase angle of -37.016o, a difference of
exactly 90o between the two. This tells us that E and I are
still 90o out of phase (for the inductor only).
We can also mathematically prove that
these complex values add together to make the total voltage, just as
Kirchhoff's Voltage Law would predict:
Let's check the validity of our
calculations with SPICE:
ac r-l circuit
v1 1 0 ac 10 sin
r1 1 2 5
l1 2 0 10m
.ac lin 1 60 60
.print ac v(1,2) v(2,0) i(v1)
.print ac vp(1,2) vp(2,0) ip(v1)
.end
freq v(1,2) v(2) i(v1)
6.000E+01 7.985E+00 6.020E+00 1.597E+00
freq vp(1,2) vp(2) ip(v1)
6.000E+01 -3.702E+01 5.298E+01 1.430E+02
Note that just as with DC circuits, SPICE
outputs current figures as though they were negative (180o
out of phase) with the supply voltage. Instead of a phase angle of
-37.016o, we get a current phase angle of 143o
(-37o + 180o). This is merely an idiosyncrasy of
SPICE and does not represent anything significant in the circuit
simulation itself. Note how both the resistor and inductor voltage phase
readings match our calculations (-37.02o and 52.98o,
respectively), just as we expected them to.
With all these figures to keep track of
for even such a simple circuit as this, it would be beneficial for us to
use the "table" method. Applying a table to this simple series
resistor-inductor circuit would proceed as such. First, draw up a table
for E/I/Z figures and insert all component values in these terms (in
other words, don't insert actual resistance or inductance values in Ohms
and Henrys, respectively, into the table; rather, convert them into
complex figures of impedance and write those in):
Although it isn't necessary, I find it
helpful to write both the rectangular and polar forms of each
quantity in the table. If you are using a calculator that has the
ability to perform complex arithmetic without the need for conversion
between rectangular and polar forms, then this extra documentation is
completely unnecessary. However, if you are forced to perform complex
arithmetic "longhand" (addition and subtraction in rectangular form, and
multiplication and division in polar form), writing each quantity in
both forms will be useful indeed.
Now that our "given" figures are inserted
into their respective locations in the table, we can proceed just as
with DC: determine the total impedance from the individual impedances.
Since this is a series circuit, we know that opposition to electron flow
(resistance or impedance) adds to form the total opposition:
Now that we know total voltage and total
impedance, we can apply Ohm's Law (I=E/Z) to determine total current:
Just as with DC, the total current in a
series AC circuit is shared equally by all components. This is still
true because in a series circuit there is only a single path for
electrons to flow, therefore the rate of their flow must uniform
throughout. Consequently, we can transfer the figures for current into
the columns for the resistor and inductor alike:
Now all that's left to figure is the
voltage drop across the resistor and inductor, respectively. This is
done through the use of Ohm's Law (E=IZ), applied vertically in each
column of the table:
And with that, our table is complete. The
exact same rules we applied in the analysis of DC circuits apply to AC
circuits as well, with the caveat that all quantities must be
represented and calculated in complex rather than scalar form. So long
as phase shift is properly represented in our calculations, there is no
fundamental difference in how we approach basic AC circuit analysis
versus DC.
Now is a good time to review the
relationship between these calculated figures and readings given by
actual instrument measurements of voltage and current. The figures here
that directly relate to real-life measurements are those in polar
notation, not rectangular! In other words, if you were to connect a
voltmeter across the resistor in this circuit, it would indicate
7.9847 volts, not 6.3756 (real rectangular) or 4.8071 (imaginary
rectangular) volts. To describe this in graphical terms, measurement
instruments simply tell you how long the vector is for that particular
quantity (voltage or current).
Rectangular notation, while convenient
for arithmetical addition and subtraction, is a more abstract form of
notation than polar in relation to real-world measurements. As I stated
before, I will indicate both polar and rectangular forms of each
quantity in my AC circuit tables simply for convenience of mathematical
calculation. This is not absolutely necessary, but may be helpful for
those following along without the benefit of an advanced calculator. If
we were restrict ourselves to the use of only one form of notation, the
best choice would be polar, because it is the only one that can be
directly correlated to real measurements.
- REVIEW:
- Impedance
is the total measure of opposition to electric current and is the
complex (vector) sum of ("real") resistance and ("imaginary")
reactance. It is symbolized by the letter "Z" and measured in ohms,
just like resistance (R) and reactance (X).
- Impedances (Z) are managed just like
resistances (R) in series circuit analysis: series impedances add to
form the total impedance. Just be sure to perform all calculations in
complex (not scalar) form! ZTotal = Z1 + Z2
+ . . . Zn
- A purely resistive impedance will
always have a phase angle of exactly 0o (ZR = R
Ω ∠ 0o).
- A purely inductive impedance will
always have a phase angle of exactly +90o (ZL =
XL Ω ∠ 90o).
- Ohm's Law for AC circuits: E = IZ ; I
= E/Z ; Z = E/I
- When resistors and inductors are mixed
together in circuits, the total impedance will have a phase angle
somewhere between 0o and +90o. The circuit
current will have a phase angle somewhere between 0o and
-90o.
- Series AC circuits exhibit the same
fundamental properties as series DC circuits: current is uniform
throughout the circuit, voltage drops add to form the total voltage,
and impedances add to form the total impedance.
|
