Impedance

Key Points:

• Impedance is to AC circuits what resistance is to DC circuits.
• Impedance is the total opposition to current. It is a combination of the resistance from resistors, and reactance from capacitors and inductors.
• Impedance is also dependent on the frequency of the AC signal.

Why it Matters:

Using an alternating current (AC) source makes things a little more complicated than using a DC source. In DC circuits, electrical resistance is the only thing that is actively opposing current. In AC circuits, however, current can be opposed by capacitance and inductance as well as resistance. The current-reducing effect of capacitance and inductance in an AC circuit is called reactance. The total opposition to current in AC circuits due to the total effect is called impedance. In this lesson, we’ll learn about reactance and impedance, and dive into the complexities of AC circuits.

Introduction to Impedance

We will see in this lesson that resistors in AC circuits behave the same way as they do in DC circuits. Unfortunately, this is not the case for capacitors and inductors. Capacitors and inductors both cause an opposition to an AC current because there is a time delay in their functionality. In DC circuits, this delay is often not an issue because the system is in steady-state. But in AC circuits, the delay causes the current to lag behind or shift forward, creating an opposition to current flow that is called reactance.

Impedance is the sum of the resistance and the reactance in a circuit:

Impedance = Resistance + Reactance

In order to separately track resistance and reactance, impedance is treated as a combination of two numbers rather than just one. You can think of this as a point on a cartesian plane, with one x-coordinate and one y-coordinate. Similarly, impedance has one resistance value and one reactance value.

Impedance is most often represented using the letter ‘Z’, and reactance using the letter ‘X’. People needed a new way to track resistance versus reactance, so scientists decided to identify reactance using the letter ‘j’. The letter ‘j’ has a mathematical representation which we’ll cover, but the important thing is that it is just used to track reactance.

We can use these abbreviations to re-write impedance using the common representation:

Z = R + jX

Let’s cover impedance in detail, starting with resistors, capacitors, and inductors in AC circuits.

Resistors in AC Circuits

Resistance doesn’t change when we go from DC to AC. As we have seen, electrical resistance is a material property determined by the device itself as well as temperature. Just as in DC circuits, resistance can be found using Ohm’s Law:

R = \frac{V}{I} = \frac{V_{max}sin(\omega t)}{I_{max}sin(\omega t)}= \frac{V_{max}}{I_{max}}

In other words, the resistance doesn’t change because of the sinusoidal waveforms of the voltage and current. This is because the voltage and current are in step with each other. Another way of saying this is the voltage and current are in phase with each other.

For resistors in AC circuits, there is nothing new to learn; resistors behave essentially the same way as they do in DC circuits. The only difference is that the voltage and current through the resistor varies. The resistor treats voltage and current the same, supplying the same resistance as it would if it were connected to a DC circuit.

To understand why this is the case, think about the operation of a resistor. A resistor converts electrical energy into heat. It does so because it is an inefficient conductor, but it otherwise does not alter or change the way that current flows.

This is not the case for capacitors and inductors, however. They both store energy rather than dissipate it as a resistor does. In doing so, they also contribute opposition to current flow in the form of reactance.

Capacitors in AC Circuits

As a reminder, capacitors consist of conductors separated by a dielectric, or insulating, material. Capacitors store energy in an electric field; as current is applied to a capacitor, the molecules in the dielectric become polarized. This is the electrical equivalent to storing energy in a spring; when the current is changed, the molecules in the dielectric ‘spring back’ to their normal (resting) positions, driving current in the direction opposite to the originally applied current.

In an AC circuit, a capacitor will start to store energy as a positive voltage is applied. The applied voltage reaches a peak and then reduces to zero (and then goes negative) following the sine wave. Just as the voltage from the source reaches zero, the voltage across the capacitor is at its’ highest. Then, when the voltage from the supply begins to reverse direction, the capacitor will start to release the energy that was just stored in its dielectric. The capacitor discharges, but the current it produces is not in phase with the source voltage. This is because the peak of the current produced by the capacitor occurs as the voltage across it reaches zero. The current leads the voltage by 90 degrees.

Since the current created by the capacitor’s discharge is no longer in phase with the voltage, it actually works against it. Think about two waves crashing into each other; they cancel out some of each other’s movement. This opposition to current results in capacitive reactance.

It turns out that capacitive reactance depends not only on the capacitor’s value (in farads), but also on the frequency of the AC source. Capacitive reactance is defined by the symbol X_c:

X_c = \frac{1}{\omega C} = \frac{1}{2\pi fC}

In this formula, f is the frequency of the AC source, and C is the capacitance of the capacitor in farads. Note that the greater the capacitance of the capacitor, or the higher the frequency, the lower the reactance. That’s why capacitors are used as low frequency filters, or alternatively, as high-pass filters.

We’ll derive this formula using some ‘light’ calculus and trigonometry in the next lesson.

Inductors in AC Circuits

Similarly to capacitors, inductors also store energy. Instead of storing energy in an electric field, however, inductors store energy in a magnetic field. When the voltage across the inductor is at its’ maximum point (V_max), the current is still increasing because of the delay in the inductor’s response. By the time the current has increased to its’ maximum point, the voltage has dropped to zero. This makes sense because when the current is at its’ peak, it is momentarily unchanging (slope of 0) so there should be no voltage drop across the inductor. By the time the voltage decreases to it’s lowest point (-V_max), the current is at zero. So the current is always 90 degrees behind the voltage, resulting in inductive reactance, X_L:

X_L = \omega L = 2\pi fL

In this formula, f is the frequency of the AC source and L is the inductance of the inductor (in henrys). Note that the greater the inductance of the inductor, or the higher the frequency, the larger the reactance will be. This is why inductors are used as high frequency filters, also known as low-pass filters.

Impedance = Resistance + Reactance

We’ve seen that impedance has two main components: resistance and reactance.

We’ve also learned that reactance has two main components: capacitive reactance and inductive reactance.

So impedance is really the sum of three terms:

Impedance = Resistance + Capacitive Reactance + Inductive Reactance

The attributes of each type of impedance can be seen in the following table:

However, resistance and reactance contribute to impedance in two distinctive ways.

Resistance causes the circuit to lose energy in the form of heat.

Reactance causes the circuit to store and release energy. Energy is not lost, although the current may be momentarily reduced.

Because resistance and reactance differ in how they contribute to impedance, reactance is mathematically computed in a way that distinguishes it from resistance. Reactance is ‘tagged’ with the letter ‘j’, which has a mathematical identity of the square root of -1. This makes impedance a complex number.

We’ll cover this interesting concept in greater detail as we progress through Module 4.

Let’s continue with Lesson 4: Capacitors in AC Circuits.