Which equation expresses the reciprocal of the total impedance for two parallel impedances as the sum of reciprocals?

Which equation expresses the reciprocal of the total impedance for two parallel impedances as the sum of reciprocals?

• A. Z_total = Z1 + Z2
• B. 1/Z_total = 1/Z1 + 1/Z2
• C. Z_total = Z1 Z2 / (Z1+Z2)
• D. Z_total = Z1 - Z2

Answer: (B)

In parallel, the two impedances share the same voltage and the currents add. Using I = V/Z for each branch, the total current is I_total = V/Z1 + V/Z2. Since the total current also equals I_total = V/Z_total, you can divide both sides by V to get 1/Z_total = 1/Z1 + 1/Z2. This shows that the reciprocal of the total impedance is the sum of the reciprocals of the individual impedances. If you solve this for Z_total, you’d get Z_total = Z1 Z2 / (Z1 + Z2), which is the actual parallel-impedance formula. The reciprocal form 1/Z_total = 1/Z1 + 1/Z2 is simply that same relationship written in reciprocal terms, which is exactly what the question asks for. The other forms don’t match the asked expression: Z_total = Z1 + Z2 is the series case, and Z_total = Z1 - Z2 isn’t a valid combination for impedances.

In parallel, the two impedances share the same voltage and the currents add. Using I = V/Z for each branch, the total current is I_total = V/Z1 + V/Z2. Since the total current also equals I_total = V/Z_total, you can divide both sides by V to get 1/Z_total = 1/Z1 + 1/Z2. This shows that the reciprocal of the total impedance is the sum of the reciprocals of the individual impedances.

If you solve this for Z_total, you’d get Z_total = Z1 Z2 / (Z1 + Z2), which is the actual parallel-impedance formula. The reciprocal form 1/Z_total = 1/Z1 + 1/Z2 is simply that same relationship written in reciprocal terms, which is exactly what the question asks for.

The other forms don’t match the asked expression: Z_total = Z1 + Z2 is the series case, and Z_total = Z1 - Z2 isn’t a valid combination for impedances.