In an Alternating Current, known commonly as an “AC circuit”, **impedance** is the opposition to current flowing around the circuit. Impedance is a value given in *Ohms* that is the combined effect of the circuits current limiting components within it, such as Resistance (R), Inductance (L), and Capacitance (C).

In a Direct Current, or DC circuit, the opposition to current flow is called Resistance, but in an AC circuit, *impedance* is the result of both the circuits resistive (R), and reactive (X) components. While the amount of electrical resistance present in a DC circuit is denoted by the letter “R“, for an alternating AC circuit, the letter or symbol “Z” is used to represent the opposition to current flow.

Also, just like DC resistance, impedance is expressed in Ohms, and where applicable, multiples and submultiples of the Ohm value are used.

For example microhms (uΩ or 10^{-6}), milliohms (mΩ or 10^{-3}), kilohms (kΩ or 10^{3}), and megohms (MΩ or 10^{6}), etc. In each case, impedance can be described by using Ohm’s law which is:

Z = V/I, I = V/Z, V = I*Z

Where: Z is the impedance given in Ohms, V is in Volts, and I is in Amperes.

## Impedance Form

We said previously that impedance (Z) is the combined effect of the total values of the resistance (R) and the reactance (X) present within an AC circuit. But impedance is also frequency dependant and therefore has a phase angle associated with it.

The phase angle of reactance, either *inductive* or *capacitive*, is always 90^{o} out-of-phase with the resistive component, so the circuits resitive and reactive values cannot be simply added together arithmetically to give the circuits total impedance value. That is R + X does not equal Z.

It is worth noting here that resistors do not change their value with frequency and therefore have no reactance (wirewounds not included), so their resistance is directly equal to their impedance, (R = Z). As a result resistors have no phase angle, so the voltage across them and current flowing through them will always be “in-phase”.

However, reactance in the form of inductive reactance, (X_{L}) or capacitive reactance, (X_{C}) does change with frequency, causing a circuits impedance value to vary as the supply frequency varies. It is for this reason that the expressions of “resistive impedance” (for resistors) and “reactive impedance” (for inductors and capacitors) are sometimes used in AC circuit analysis.

As the circuits resistive and reactive values cannot be added together to find the total impedance (Z), because the two values differ from each other by 90^{o}, that is they are at right-angles to each other, we can therefore plot the values on a two-dimensional graph with the x-axis being the resistive or “real axis”, and the y-axis being the reactive or “imaginary axis”. This is the same method used in the construction of a right-angle triangle.

The following right-angled graphs show how resistance and reactance are combined together to show impedance with the hypotenuse (longest side) of the triangle representing the complex impedance of the circuit.

### Resistance and Inductive Reactance

As we are dealing with what is effectively a three-sided right-angled triangle, we can use Pythagoras’s theorem and associated equations to relate the two sides of the right-angled triangle representing resistance and inductive reactance to the length of the third side being the hypotenuse. Pythagoras’s theorem is defined in terms of impedance, resistance and reactance as being:

Z^{2} = R^{2} + X^{2}

That is:

(Impedance)^{2} = (Resistance)^{2} + (Reactance)^{2}

In this way we can show that the impedance vector (Z) is the resulting vector sum of the resistance vector (R) and the reactance vector (X_{L}) and is a positive slope as shown.

### Impedance of an RL Circuit

The phase angle (φ) defines the angle in degrees between the impedance vector and the resistance vector as shown below.

### Phase Angle of an RL Circuit

As with the previous circuit containing an inductor and inductive reactance, we can also show the complex impedance of an AC circuit containing capacitors and capacitive reactance.

The same right-angled graph can be used to show how resistance and capacitive reactance are combined with the hypotenuse (longest side) of the triangle representing the complex impedance of the circuit.

Remember that for a capacitor the impedance vector (Z) is the vector sum of the resistance vector (R) and the reactance vector (X_{C}). It is drawn in the opposite direction of the previous X_{L} vector as a negative slope. This shows that the effect of capacitive reactance on an AC circuit is opposite to that of inductive reactance.

### Resistance and Capacitive Reactance

Again using Pythagoras’s theorem and equations we can relate the two sides of the right-angled triangle representing resistance and capacitive reactance to the hypotenuse which is the complex impedance. Pythagoras’s theorem is defined in terms of impedance, resistance and reactance as being:

### Impedance of an RC Circuit

The tangent of the phase angle (φ) defines the angle in degrees between the impedance vector and the resistance vector. The phase angle is equal to the reactance divided by the resistance as shown:

### Phase Angle of an RC Circuit

Thus vector diagrams can be used to show how resistance and reactance (inductive and capacitive) are combined together to form impedance. We can also note that we can use the ohmic values of the circuit, either using Z, R or X, to find the *phase angle*, Φ between the supply voltage, V_{S} and the circuit current, I.

## Impedance Example No1

A 53mH inductor and a 15Ω resistor are connected in series. Calculate the total impedance and phase angle at 60Hz.

1. Total Circuit Impedance, Z:

2. Phase Angle, Φ:

## Impedance Example No2

A solenoid coil was found to have a static resistance of 12Ω when measured with a multimeter. If the solenoid coil draws a current of 5 Amperes when connected to a 100 Volt, 1000 Hz supply. Calculate the inductance of the coil and the power factor.

1. The Coil’s Inductance, X_{L}:

2. Power Factor:

We have seen that **Impedance**, (Z) is the combined effect of resistance, (R) and reactance, (X) within an AC circuit and that the purely reactive component, X is 90^{o} out-of-phase with the resistive component, being positive (+90^{o}) for inductance and negative (-90^{o}) for capacitance.

But what if a series AC circuit contained both inductive reactance, X_{L} and capacitive reactance, X_{C}. How would this affect the complex impedance of the circuit.

## Impedance of an RLC Circuit

Reactance is Reactance! while the impedance triangle of an inductor will have a positive slope and the impedance triangle of a capacitor will have a negative slope, the mathematical sum of the two impedances will produce the circuits overall impedance value.

The combined reactance of the series circuit will be the sum of the inductive reactance, X_{L} and the capacitive reactance, X_{C} as shown.

X = X_{L} + (-X_{C}) = X_{L} – X_{C}

Which gives:

As a general rule of thumb, we would subtract the smaller reactance value from the larger value, whether it is X_{L} or X_{C}, it makes no difference. This is because squaring a negative value will always produce a positive result in mathematics. For example -2^{2} is the same result as 2^{2}, which is +4.

So it is correct to use either (X_{L} – X_{C}) or (X_{C} – X_{L}) to find a circuits combined reactance value before adding it to the resistance value.

The resulting impedance triangle would look like:

### RLC Impedance Triangle

With the slope of the impedance being either positive or negative in direction depending on which reactance is greater, Inductive (X_{L} – X_{C}) or Capacitive (X_{C} – X_{L}). Then the circuits impedance in complex form is therefore defined as: Z = R ±jΧ

Clearly then, if an AC circuit contains only Inductance and Capacitance in series, impedance, Z = X_{L} – X_{C}, or vice versa. If the circuit is at resonance, the net reactance becomes zero so Z = 0 as the inductive reactance is equal and opposite in value to the capacitive reactance because X_{L} = X_{C}. This is why circuit current flow is only limited by the dynamic resistance (R) in a series circuit at resonance.

## Impedance Example No3

A non-inductive resistor of 10Ω, a capacitor of 100uF, and an inductor of 0.15H are connected in series to a 240V, 50Hz supply. Calculate the inductive reactance, the capacitive reactance, the circuits complex impedance and the power factor.

R = R = 10Ω

1. Inductive Reactance, X_{L}

2. Capacitive Reactance, X_{C}

3. Complex Impedance, Z

4. Power Factor

We have seen in this tutorial that impedance, symbol Z, is the opposition to current flowing around an AC circuit, and is the combined effect of resistance and reactance. We have also seen that impedance is not equal to the mathematical sum but the vector sum of the resistive and reactive components within the circuit as the reactive component is 90^{o} “out-of-phase” with the resistive component.

Complex impedance in series obey the same Ohms Law rules as for purely resistive circuits.

That is: Z_{T} = Z_{1} + Z_{2} + Z_{3} + Z_{4} + …etc.

But what of parallel connected circuits. How is impedance calculated for them.

## Parallel Impedances

If a single resistance and a single reactance are connected together in parallel, the impedance of each parallel branch must be found. But as there are only two components in parallel, R and X, we can use the standard equation for two resistances in parallel.

It is given as: R_{T} = (R_{1}*R_{2})/(R_{1} + R_{2}).

Where: Z, R and X are all given in Ohms.

Notice also that as we are dealing with AC supplies and frequencies, and so the resistive component is 90^{o} out-of-phase with the reactive component, the product is divided by the vector sum of R and X.

Thus if “n” branches containing complex impedances are connected together in parallel, the total impedance is the vector addition of all the parallel branches. Thus the reciprocal of the total impedance of the circuit is given as:

and this is

### Resistance and Inductance in Parallel

### Resistance and Capacitance in Parallel

### Resistance, Inductance and Capacitance in Parallel

Note here for this RLC parallel circuit that at the resonant frequency, X_{L} = X_{C} which becomes zero, so only resistance (R) is present in the circuit. Therefore at resonance only, the dynamic impedance is defined as being: Z = R.