# Impedances in Parallel

Given two impedances Z1 and Z2 connected in parallel, what is their impedance expressed in real and imaginary parts?

For impedances in parallel, the formula is:

$1 Z = 1 Z 1 + 1 Z 2$

Expressing each impedance Z1 and Z2 individually as:

MathML formulaFormula ImageStraight text formula
$Z 1 = a + b j$ Z1 = a + bj
$Z 2 = c + d j$ Z2 = c + dj

Here is the online calculator tool:

Real Part Imaginary Part
Z1
Z2
ZT

(Note: Polar notation using angle symbol Ð to be added later)

(Remember that the imaginary part is positive for inductive reactance, and negative for capacitive reactance)

## Formula Derivation

We can now derive an expression for the combined impedance Z:

$1 Z = Z 1 + Z 2 Z 1 Z 2$ $⇒ Z = Z 1 Z 2 Z 1 + Z 2$ $⇒ Z = a + b j c + d j a + b j + c + d j$

Re-arranging the denominator gives:

$⇒ Z = a + b j c + d j a + c + b + d j$

Now, multiplying above and below by:

$a + c - b + d j$

Gives Us:

$Z = a + b j c + d j a + c - b + d j a + c + b + d j a + c - b + d j$

Now, recognising that:

$j 2 = -1$

We can expand to give:

$Z = a c + a d j + b c j - b d a + c - b j - d j a + c 2 + b + d 2$ $⇒ Z = a 2 c + a c 2 - a b c j - a c d j + a 2 d j + a c d j - a b d j 2 - a d 2 j 2 + a b c j + b c 2 j - b 2 c j 2 - b c d j 2 - a b d - b c d + b 2 d j + b d 2 j a + c 2 + b + d 2$ $⇒ Z = a 2 c + a c 2 - a b c j - a c d j + a 2 d j + a c d j + a b d + a d 2 + a b c j + b c 2 j + b 2 c + b c d - a b d - b c d + b 2 d j + b d 2 j a + c 2 + b + d 2$ $⇒ Z = a 2 c + a c 2 + a b d + a d 2 + b 2 c + b c d - a b d - b c d - a b c j - a c d j + a 2 d j + a c d j + a b c j + b c 2 j + b 2 d j + b d 2 j a + c 2 + b + d 2$ $⇒ Z = a 2 c + a c 2 + a d 2 + b 2 c + a 2 d + b c 2 + b 2 d + b d 2 j a + c 2 + b + d 2$

So, the solution is:

MathML formula
$⇒ Z = a 2 c + a c 2 + a d 2 + b 2 c a + c 2 + b + d 2 + a 2 d + b c 2 + b 2 d + b d 2 a + c 2 + b + d 2 j$
Formula Image
Straight text formula
Z = [ [a2c + ac2 + ad2 + b2c] / [(a + c)2 + (b + d)2] ] + [ [a2d + bc2 + b2d + bd2] / [(a + c)2 + (b + d)2] ] j