A Quick Explanation of Impedance

terminology

#1

This post covers the concept of impedance in general terms. Seen from the definition on the hyperlink, it is the combination of resistance and reactance. In simpler terms, impedance can be thought of how much passive elements in an alternating current circuit reduce or impede the 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.) Some DC waveforms that can be manipulated from a stable DC input. These include square waves, sawtooth waves, triangle waves, and other pulsating modes.

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. Here are the equations for voltage and current over a capacitor:

I_{Cap}=C*\frac{dV_{Cap}}{dt}
V_{Cap}=\frac{1}{C}*\int_{0}^{t}I_{Cap} dt + V_{0}

Here are the equations for voltage and current over an inductor:

V_{Ind}=L*\frac{dI_{Ind}}{dt}
I_{Ind}=\frac{1}{L}*\int_{0}^{t}V_{Ind}dt+I_{0}

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 allow 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.
NOTE: Modern devices have higher operating frequencies with the MLCC designs, however, there are still many parts that fall under the 1 to 3 Megahertz range. 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}}=-j*\frac{1}{\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 milliOhms 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.

Note 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 except it is in terms of impedance.

Determining impedance for the special applications such as pulsating DC signals is more complex than the scope of my knowledge, but in general, the same ideas still apply. Modern day devices run from anywhere to several Megahertz to several Gigahertz on these types of signals. There still needs to be the consideration of impedance because these levels of frequency cause major design problems for a mixture of different components. Design gets as specific as selecting the right cable, making sure PCB traces aren’t too close together, the right value of capacitors, inductors, and resistors along with their operating frequency must be considered, the ground plane has to have a special design, shielding must be applied with specific materials to reduce EMI emission, and many more requirements beyond these.


#2

Impedance is not the equivalent of resistance, since:

  1. They are two different terms and have different meanings. Resistance is a constant value regardless of frequency, and impedance changes with regard to frequency
  2. Resistance (resistors) disspiate energy in the irreversable form of heat, where reactors (inductors, capacitors) do not dissipate energy, they store and return it.
  3. REsistance is a portion of complex impedance, but not vice versa, resistance is a material property (resistivity)

It is already hard enough solving equations with these Calculus expressions in a DC circuit."

There is no such thing as calculus expression for DC as DC does not vary in time. Dx C = 0 which is meaningless.

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. "

Not true. With modern SMT components, resistors are essentially 'pure components" into many 10s or 100s of MHz.

“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).”

Not true. Complex numbers do not apply to DC.


#3

@daveca You are correct about impedance not equaling resistance, I will make corrections to that statement. It can be easier to imagine the term if you think of a component that reduces the overall signal as a resistor would.

However, there are very applicable calculus expressions for DC over capacitors and inductors, specifically. I will put those equations in the post for Voltage and Current for the Capacitors and Inductors. DC applications do change over time if there is a function generation by manipulating a pure DC signal into an oscillating signal. This can include square waves, sawtooth waves, and triangular waves.

Also, every single component (even modern MLCC parts) have their frequency limitations. Most components have a lower natural frequency in general, but you are also correct in saying modern devices have higher ratings, I will be more specific.

For the last point, the same concept applies when designers have to consider the possibility of different pulsated DC waves. There is more work done because the waves are not as simple as sinusoidal waves. Impedance measurements in traces, cabling, active components, and passive components very much affect the whole design of a board that contains signals of 1MHz to several GHz of a square wave signal.

Thank you for pointing out things I can change though, I will be more specific to make this clear.