Inductors in parallel
If there is no mutual inductance among two or more parallel-connected inductors, their
values add up like the values of resistors in parallel. Suppose you have inductances L1,
L2, L3, ..., Ln all connected in parallel. Then you can find the reciprocal of the total inductance,
1/L, using the following formula:
1/L = 1/L1 + 1/L2 + 1/L3 +…+ 1/Ln
The total inductance, L, is found by taking the reciprocal of the number you get for 1/L.
Again, as with inductances in series, it’s important to remember that all the units
have to agree. Don’t mix microhenrys with millihenrys, or henrys with nanohenrys. The
units you use for the individual component values will be the units you get for the final
answer.
Problem 10-3
Suppose there are three inductors, each with a value of 40 μH, connected in parallel
with no mutual inductance, as shown in Fig. 10-4. What is the net inductance of the set?
Call the inductances L1 = 40 μH, L2 = 40 μH, and L3 = 40 μH. Use the formula
above to obtain 1/L= 1/40+1/40+1/40+3/40= 0.075. Then L= 1/0.075= 13.333μH.
This should be rounded off to 13 μH, because the original inductances are specified to
only two significant digits.
Problem 10-4
Imagine that there are four inductors in parallel, with no mutual inductance. Their values
are L1 = 75 mH, L2 = 40 mH, L3 = 333 μH, and L4 = 7.0 H. What is the total net
inductance?
You can use henrys, millihenrys, or microhenrys as the standard units in this problem. Suppose you decide to use henrys. Then L1= 0.075 H, L2= 0.040 H, L3= 0.000333
H, and L4= 7.0 H. Use the above formula to obtain 1/L=13.33+25+3003+0.143=
3041.473. The reciprocal of this is the inductance L=0.00032879 H=328.79 μH. This
can be rounded off to 330 μH because of significant-digits considerations.
This is just about the same as the 333-μH inductor alone. In real life, you could have
only the single 333-μH inductor in this circuit, and the inductance would be essentially
the same as with all four inductors.
If there are several inductors in parallel, and one of them has a value that is far
smaller than the values of all the others, then the total inductance is just a little smaller
than the value of the smallest inductor.
