Parallel resistor-inductor circuits
Let's take the same components for our
series example circuit and connect them in parallel:
Because the power source has the same
frequency as the series example circuit, and the resistor and inductor
both have the same values of resistance and inductance, respectively,
they must also have the same values of impedance. So, we can begin our
analysis table with the same "given" values:
The only difference in our analysis
technique this time is that we will apply the rules of parallel circuits
instead of the rules for series circuits. The approach is fundamentally
the same as for DC. We know that voltage is shared uniformly by all
components in a parallel circuit, so we can transfer the figure of total
voltage (10 volts ∠ 0o) to all components columns:
Now we can apply Ohm's Law (I=E/Z)
vertically to two columns of the table, calculating current through the
resistor and current through the inductor:
Just as with DC circuits, branch currents
in a parallel AC circuit add to form the total current (Kirchhoff's
Current Law still holds true for AC as it did for DC):
Finally, total impedance can be
calculated by using Ohm's Law (Z=E/I) vertically in the "Total" column.
Incidentally, parallel impedance can also be calculated by using a
reciprocal formula identical to that used in calculating parallel
resistances.
The only problem with using this formula
is that it typically involves a lot of calculator keystrokes to carry
out. And if you're determined to run through a formula like this
"longhand," be prepared for a very large amount of work! But, just as
with DC circuits, we often have multiple options in calculating the
quantities in our analysis tables, and this example is no different. No
matter which way you calculate total impedance (Ohm's Law or the
reciprocal formula), you will arrive at the same figure:
- REVIEW:
- Impedances (Z) are managed just like
resistances (R) in parallel circuit analysis: parallel impedances
diminish to form the total impedance, using the reciprocal formula.
Just be sure to perform all calculations in complex (not scalar) form!
ZTotal = 1/(1/Z1 + 1/Z2 + . . . 1/Zn)
- Ohm's Law for AC circuits: E = IZ ; I
= E/Z ; Z = E/I
- When resistors and inductors are mixed
together in parallel circuits (just as in series 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.
- Parallel AC circuits exhibit the same
fundamental properties as parallel DC circuits: voltage is uniform
throughout the circuit, branch currents add to form the total current,
and impedances diminish (through the reciprocal formula) to form the
total impedance.
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