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Ham Radio: Impedance-4: Coordinate Systems
[last updated: 2020-11-20]
go to: Impedance-1: Why
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- The formula to calculate Impedance can be in two different forms. The form of the expression is associated with the coordinate system used to plot the pictorial representation on a graph. There are two types of graphical coordinate systems that can be used.
- Rectangular (complex) Coordinates:
In this form, resistance R is plotted on the x-axis, and the complex component reactance X is plotted on the y-axis. The hypotenus then is the impedance Z.
- Polar Coordinates:
In this form, resistance R is still plotted on the x-axis, but the impedance Z is plotted as an angle theta from the horizontal x-axis, and the magnitude of the impedance is where the angled impedance line crosses the perpendicular from the end of the resistance vector.
- Rectangular (complex) Coordinates:
- Note that the plot, the graph, looks the same regardless of the coordinate system.
All you're really doing is changing the form of the equation that defines Z.
In polar coordinates: Z = R ∠ θ
In rectangular/complex coordinates: Z = R + jX - In both coordinate systems, the x-axis represents DC Resistance R,
and the y-axis represents Reactance X
------------------------------------------------------------- - The choice of which expression form you use (polar or rectangular) depends on what you need to calculate:
- For example suppose you need to calculate the value of two impedances in parallel.
You use the same formula that you'd use to calculate two resistances in parallel:
"the products over the sum" equation:
The two values multiplied together, divided by the two values added:
Z = (Z1 * Z2) / (Z1 + Z2) - For the numerator, the top of the equation: (Z1 * Z2),
you are multiplying two impedances.
If you multiply impedances, it's way easier to use the polar form. The technique is:-
For: Z1 = R1∠θ1
and: Z2 = R2∠θ2
Z1 * Z2 =
(R1 * R2) ∠ (θ1 + θ2) - For the denominator, you're adding impedances, which is way easier if you use rectangular (complex) form:
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For: Z1 = R1+jX1
and: Z2 = R2+jX2
(R1+jX1) + (R2+jX2) =(R1 + R2) + j (X1 + X2)
- In summary, calculate the numerator (Z1 * Z2) by multiplying the impedances in polar form, and calculate the denominator (Z1 + Z2) by adding the impedances in rectangular/complex form.
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- Convert between polar and rectangular form:
- This requires an understanding of some very basic trigonometry - Sine, Cosine, and Tangent:
basic trig functions
trig functions applied to Impedance- Trig functions are defined based on the angles of a "right triangle."
A Right Triangle is one that has one 90-deg angle in it. - Per the diagram:
Sine: sin θ = (length of opposite side) / (length of hypoteneuse)
Cosine: cos θ = (length of adjacent side) / (length of hypoteneuse)
Tangent: tan θ = (length of opposite side) / (length of adjacent side) - These functions show the relationship between the lengths of the 3 sides and the angles between them.
---------------------------------------------- - Trig tables can be found online or in reference books that show the values of the various trig functions at different angles. For example:
Function 30° 45° 60° Sine 0.500 0.707 0.866 Cosine 0.866 0.707 0.500 Tangent 0.577 1.000 1.732
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---------------------------------------------------------------- - Trig functions are defined based on the angles of a "right triangle."
- Convert rectangular to polar:
If you start with rectangular form (Z = R + jX), then you already know R, and X.
For polar form (Z = R ∠ θ), you need to have R and θ.
You already know R, so you need to find θ.
You can do this 3 different ways:- using Z & R: cos θ = R / Z
- using Z & X: sin θ = X / Z
- using X & R: tan θ = X / R
- For example suppose you need to calculate the value of two impedances in parallel.
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eof