Impedance is the opposition of a circuit to the passage of alternating current and is measured in ohms. To calculate impedance, you need to know the value of all resistors and the impedance of inductors and capacitors, which offer varying amounts of opposition to current depending on how the current changes. To calculate the impedance you can use a simple mathematical formula.
Notes on the formula
- Impedance Z = R or XL or XC (if only one is present)
- Impedance serial only Z = √ (R2 + X2) if only R and one type of X are present)
- Impedance serial only Z = √ (R2 + (| XL - XC|)2) (if R, XL, and XC they are all present)
- Impedance in any circuit = R + jX (j is the imaginary number √ (-1))
- Resistance R = I / ΔV
- Inductive reactance XL = 2πƒL = ωL
- Capacitive reactance XC = 1 / 2πƒC = 1 / ωC
Part 1 of 2: Calculate Resistance and Reactance
Step 1. Define the impedance
Impedance is represented by the symbol Z and is measured in ohms (Ω). You can measure the impedance of any electrical circuit or component. The result will tell you how much the circuit resists the flow of electrons (current). There are two different effects that slow down the current, both of which contribute to the impedance:
- The resistance (R) is the deceleration of the current due to effects of the material and the shape of the component. This effect is greatest for resistors, but all components have at least some resistance.
- Reactance (X) is the deceleration of current due to electric and magnetic fields opposing changes in current or voltage. This is most important for capacitors and inductors.
Step 2. Check the resistance
Resistance is a fundamental concept in the study of electricity. You'll see it most often in Ohm's law: ΔV = I * R. This equation allows you to calculate any of these values if you know the other two. For example, to calculate resistance, write the formula as R = I / ΔV. You can also easily measure resistance with a multimeter.
- ΔV is the voltage measured in volts (V). It is also called a potential difference.
- I is the current measured in amps (A).
- R is the resistance measured in ohms (Ω).
Step 3. Identify what type of reactance to calculate
Reactance is only generated in AC (alternating current) circuits. Like resistance, it is measured in ohms (Ω). There are two types of reactance, which can occur in different electrical components:
- Inductive reactance XL It is produced by inductors, also called coils. These components create a magnetic field that opposes directional changes in an AC circuit. The faster the direction changes, the greater the inductive reactance.
- Capacitive reactance XC It is produced by capacitors, which hold an electrical charge. Just as the flow of current in an AC circuit changes direction, the capacitor charges and discharges repeatedly. The longer the capacitor has to charge, the more it opposes the current. Because of this, the faster the direction changes, the lower the capacitive reactance.
Step 4. Calculate the inductive reactance
As described above, inductive reactance increases with the rate of change in the direction of the current, or the frequency of the circuit. This frequency is represented by the symbol ƒ and is measured in Hertz (Hz). The complete formula for calculating inductive reactance is XL = 2πƒL, where L is the inductance measured in henries (H).
- The inductance L depends on the characteristics of the inductor, such as the number of its coils. It is also possible to measure inductance directly.
- If you are familiar with goniometric circumference, imagine an alternating current represented by this circumference, with a complete rotation of 2π radians representing a circle. If you multiply it by ƒ measured in hertz (units per second), you will get a result in radians per second. This is the angular velocity of the circuit and can be written in a lowercase omega ω. You could see the formula for inductive reactance written as XL= ωL.
Step 5. Calculate the capacitive reactance
This formula is similar to inductive reactance, except that capacitive reactance is inversely proportional to frequency. Capacitive reactance is XC = 1 / 2πƒC. C is the capacitance of the capacitor measured in farads (F).
- You can measure capacitance with a multimeter and some basic calculations.
- As explained above, this can be written as 1 / ωL.
Part 2 of 2: Calculate Total Impedance
Step 1. Add up the resistors in the same circuit
Total impedance is simple if the circuit has multiple resistors, but no inductors or capacitors. First, measure the resistance across each resistor (or any component with resistance) or refer to the circuit diagram for the resistance marked in ohms (Ω). Combine them based on how the components are connected:
- Resistors in series (connected end-to-end along the wire) can be added together. The total resistance would be R = R1 + R2 + R3…
- Resistors in parallel (each on a different wire that connects to the same circuit) add up as their reciprocals. The total resistance would be R = 1 / R1 + 1 / R2 + 1 / R3 …
Step 2. Add the similar reactances in the same circuit
If there are only inductors in the circuit, or only capacitors, the total impedance equals the total reactance. Do the calculation as follows:
- Inductors in series: Xtotal = XL1 + XL2 + …
- Capacitors in series: Ctotal = XC1 + XC2 + …
- Inductors in Parallel: Xtotal = 1 / (1 / XL1 + 1 / XL2 …)
- Capacitors in parallel: Ctotal = 1 / (1 / XC1 + 1 / XC2 …)
Step 3. Subtract the inductive and capacitive reactance to get the total reactance
Since one of these effects increases while the other decreases, they have to cancel each other out. To find the total effect, subtract the smallest from the largest.
- You will get the same result from formula Xtotal = | XC - XL|
Step 4. Calculate the impedance of the resistance and the series reactance
You can't just add both, because the two values are "out of date." This means that both values change over time as part of the AC cycle, but peak at different times. Fortunately, if all the components are in series (that is, if there is only one wire), we can use the simple formula Z = √ (R2 + X2).
The math behind this formula involves "phasors"; however, it might also look familiar from geometry. It seems that we can represent the two components R and X as the legs of a right triangle, with the impedance Z as the hypotenuse
Step 5. Calculate the impedance from the resistance and reactance in parallel
This is actually a general way of expressing impedance, but it requires an understanding of complex numbers. This is the only way to calculate the total impedance of a parallel circuit that includes both resistance and reactance.
- Z = R + jX, where j is the imaginary component: √ (-1). Use j instead of i to avoid confusion with I for current.
- You cannot combine the two numbers. For example, an impedance could be expressed as 60 Ω + j120 Ω.
- If you have two circuits like this in series, you can add the real and imaginary components together separately. For example, if Z1 = 60 Ω + j120 Ω and is in series with a resistor with Z2 = 20 Ω, so Ztotal = 80 Ω + j120 Ω.