Download MTI TN113: The Series to Parallel Impedance Transformation

 MTI TN113: The Series to Parallel Impedance Transformation
Serial/parallel R-C and R-L networks are ubiquitous in low frequency and RF designs. The necessity to view
these simple circuits in both serial and parallel forms often recurs, especially during analysis.
I was in the lab with a client and we were measuring the characteristics of a lossy passive circuit element using
an LCR meter. I kept alternating between the series and parallel equivalent network readings, to the
confusion of my client, and he wondered which of those numbers was “right”. I explained that they were
both correct, and later dragged out my ancient cheat sheet of the series to parallel impedance transformation,
with the intent of passing it on to my client. My old cheat sheet was hand written, well used (and abused), in bad shape, and generally unclear. Normally, I
would simply scan my cheat sheet and pass it on. It was a loathsome looking document, with numerous
scribbles from past projects.
So, it was well past the time to clean up my cheat sheet. This tech note shows a clean presentation of these
basic circuit networks, and the series ↔ parallel impedance transformation
Getting back to the LCR meter: an LCR meter typically measures the magnitude and phase of an impedance,
and then backs out the numbers into an equivalent parallel and series network. The LCR meter equivalent
parallel and series network numbers are both correct, and will result in the same impedance at that particular
test frequency. Remember that the equivalent parallel and series network values are exact for only one
frequency point.
You can find a summary table of simplified equations for the series ↔ parallel transformation in many
electrical engineering texts. Like many, I prefer to see both the simplified and derived equations in one spot. Thus, the impetus for my very own personalized transformation table.
Recall, from basic electrical engineering, that an impedance vector can be illustrated as:
Where Z is the magnitude
and the tangent of the phase angle is
It is most convenient to separate out the real and imaginary portions of a series or parallel impedance, which
ultimately makes the series ↔ parallel transformation easier to grasp.
Recall, that Q (the “quality factor”), is a measure of how much energy is not lost in a reactive element. The
higher the Q, the less energy is lost. In the circuit world, we model the loss as a simple resistor. Note that
the Q for series (Qs) and parallel (Qp) circuits is numerically the same at any particular frequency, but the
calculation is necessarily different. Also note that Q is equal to the absolute value of the tangent of the
impedance phase angle. You will sometimes see an inverse version of this in capacitor specifications, where instead of the capacitor’s
Q, the
manufacturer will provide a D (dissipation, or dissipation factor) number as tan-delta:
As a series circuit, the ESR (equivalent series resistance) of the capacitor will then be: Note that the angle δ that the capacitor manufacturer talks about is simply:
The derivation of the series ↔ parallel transformation equations simply make use of comparing the real and
imaginary portions, and simplifying by using the appropriate Q calculation.
From the following page, we note that the equivalent series capacitance
So, we are naturally interested in how high a Q is required to make the assumption that the series equivalent
capacitance is equal to the equivalent parallel capacitance. This is shown in the plot below:
In short:
Q=10, Cs =Cp within ~1%.
Q=30, Cs =Cp within ~0.1%.
Q=100, Cs =Cp within ~0.01%.
Don’t have the time to derive formulas for magnitude and phase of common passive circuits? You can refer
to:
Van Valkenburg, Reference Data for Engineers: Radio, Electronics, Computer & Communications. I see that both my
1956 and 1993 copies present the same table, so I presume it is a permanent fixture no matter what edition
you obtain.
or
John R. Brand, Handbook of Electronic Formulas, Symbols, and Definitions Second Edition. New York, NY: Van
Nostrand Reinhold, 1992.
Series and Parallel RC
Impedance Magnitude: Impedance Phase:
Impedance Magnitude: Impedance Phase:
Conversion between the series and parallel RC circuits:
Series and Parallel RL
Impedance Magnitude: Impedance Phase:
Impedance Magnitude: Impedance Phase:
Conversion between the series and parallel RL circuits: