How to Calculate Impedance in Parallel Circuits

Impedance is a combination of resistance and reactance. Pure resistance as with a resistor is the same value for alternating current (AC) and direct current (DC) circuits. Reactance is analogous to resistance for capacitors and inductors, but the reactance of a device depends on the frequency applied. Reactance will be seen in AC circuits or DC circuits with an AC signal riding on the DC.
In this example we will examine a simple circuit with two branches: a resistor in series with a capacitor branch in parallel with a resistor in series with an inductor.

Difficulty: Easy

Instructions

Things You'll Need:

• Calculator
• Value(s) for resistors (We'll use 2 100-ohm resistors)
• Value(s) for capacitors (We'll use 1 microFarad)
• Values(s) for inductors (We'll use 50 milliHenries)
• Frequency of the signal applied (We'll use 1,000 Hertz)

1. Calculate the reactance of the capacitor using the formula:
   Reactance (capacitive) = 1 / ( 2 X Pi X frequency in Hertz X Capacitance in Farads)
   
   Here this will be 1 / (2 X 3.14159 X 1000 X 0.000001) = 1 / (0.0062832) = 159 ohms

2. Calculate the reactance of the inductor using the formula:
   Reactance (inductive) = 2 X Pi X frequency in Hertz X Inductance in Henries
   
   Here this will be 2 X 3.14159 X 1000 X 0.05 = 314 ohms

3. Add the resistance and reactance in each branch:
   
   Capacitive branch is resistor's resistance plus capacitor's reactance:
   100 ohm + 159 ohm = 259 ohm total branch impedance
   
   Inductive branch is resistor's resistance plus inductor's reactance:
   100 ohm + 314 ohm = 414 ohm total branch impedance

4. Calculate the parallel impedances the same way you would calculate simple parallel resistances:
   
   ( 259 X 414) / (259 + 414) = 107,226 / 673 = 159.3 ohms impedance.

Tips & Warnings

• While resistance does not change with frequency, reactances do. Inductive reactance goes up with frequency and capacitive reactance drops with frequency. For the same components, if we change the frequency from 1000 Hertz to 5000 Hertz, the capacitive reactance becomes 31.8 ohms, the inductive reactance becomes 1,571 ohms and the resulting circuit impedance drops to (132 X 1671) / (132 + 1671) = 122 ohms impedance.

References

• Reactance tutorial
• Impedance tutorial

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