A Quick Explanation of Impedance

terminology

#1

This post covers the concept of impedance in general terms. Seen from the definition on the hyper-link, it is the equivalent of resistance in alternating current. The same terminology is applicable when talking about high frequency radio applications or high frequency digital applications because all these applications have something in common. They all have some form of changing voltage in any periodic waveform. (NOTE: This is not limited to sinusoidal waves explicitly.)

Why not just use resistance if impedance is basically the same thing? There is more background on why a different term is used. The main difference between impedance and resistance is the frequency/frequencies at which the circuit operates. In a direct current application, there tends to be no frequency on the inputs and/or outputs (disregard clock generation and other oscillation designs for now). For those who have taken circuits, knows the general equations for voltage, current, and power across resistors, capacitors, and inductors. It is already hard enough solving equations with these Calculus expressions in a DC circuit. It becomes harder to solve in alternating current using the same equations because the input voltage is now changing over time (so does current). Fortunately, a time saving shortcut was discovered after the Fourier Transform [Khan Academy is a very useful resource when it comes to complex math topics in general, I recommend watching the playlist if you are interested] was developed. This method turns the complex equations for inductors and capacitors into imaginary (complex) numbers which allows circuits to be solved using the same techniques for basic DC analysis (Ohm’s Law and other methods in DC analysis). Here are the applicable equations derived from converting to the frequency domain:

1. Resistors

Z_{R}=R_{N}

Where:
Rn is equal to the resistance of one Resistor N in Ohms.
At lower frequencies (typically lower than 1 to 3 Megahertz is considered low frequency), the impedance of a resistor (Zr) is simply equal to to the resistance value. The same is true for the following parts at lower frequencies where Capacitance doesn’t change and Inductance doesn’t change.

2. Capacitors

Z_{C}=\frac{1}{-j*\omega*C_{N}}

Where:

j=\sqrt{-1}=i
\omega=2*\pi*f ; f=frequency(Hz)

Cn is Capacitance of one Capacitor N in Farads
The letter “j” is used as a convention in electrical analysis because current is typically noted with the letter “i” so this avoids confusion. Also, another convention in analysis is to use radians and angular frequency rather than using linear frequency and degrees.

3. Inductors

Z_{L}=j*\omega*L_{N}

Where:

j=\sqrt{-1}=i
\omega=2*\pi*f; f=frequency(Hz)

Ln is equal to the inductance of one Inductor N in Henry’s

Each item in an AC circuit must be converted before any analysis takes place. Impedance is also measured in Ohms and typically the “j”/complex part is omitted when values are spoken or written for measurement purposes. Here is an example of impedance calculation:

For an inductor that is 50 micro Henry’s (50µH) and the frequency of the voltage/current source is 1000 Hertz (1kHz), the impedance is calculated as:

Z_{L}=j*2*\pi*1000*50*10^{-6}=0. 314*j(\Omega's)=314m\Omega

The Ohms measurement is typically written after the “j”. The j is often removed when spoken or written in a report to make it less confusing. So 314 milli Ohms of impedance is the answer for this specific inductor at 1kHz. The “j” is only important when using it in circuit analysis as the imaginary part determines the phase shift of the periodic wave. Topics on analysis can be discussed further upon request.

One last note is that most data sheets that list impedance will talk about overall input/output impedance rather than list all the impedance values in the circuit or design. This is the same concept as total or effective resistance in a DC circuit. Please ask questions if you need clarification or ask about unmentioned details.