Most often when we plot values on a graph, we use the rectangular, or Cartesian, coordinate system. The two numbers that are used to define a point on a graph using rectangular coordinates are the coordinate values along the horizontal and vertical axes. In the graph above, point P is at x,y.
Rectangular coordinates is the coordinate system often used to display the resistive, inductive, and/or capacitive reactance components of an impedance. When using rectangular coordinates to graph the impedance of a circuit, the X axis, or horizontal axis, represents the resistive component, and the Y axis, or vertical axis, represents the reactive component.
QUESTION: What coordinate system is often used to display the resistive, inductive, and/or capacitive reactance components of impedance? (E5C04)
ANSWER: Rectangular coordinates
QUESTION: When using rectangular coordinates to graph the impedance of a circuit, what do the axes represent? (E5C09)
ANSWER: The X axis represents the resistive component and the Y axis represents the reactive component
QUESTION: Where is the impedance of a pure resistance plotted on rectangular coordinates? (E5C07)
ANSWER: On the horizontal axis
In rectangular notation, we’d represent an impedance as R +/- jX, where X is the value of the reactance. When X is inductive reactance, the reactance is a positive value. When X is negative, the reactance is capacitive.
QUESTION: Which of the following represents capacitive reactance in rectangular notation? (E5C01)
ANSWER: -jX
QUESTION: What does the impedance 50-j25 represent? (E5C06)
ANSWER: 50 ohms resistance in series with 25 ohms capacitive reactance
If the impedance was 50 + j25, then the circuit would have 50 ohms resistance in series with 25 ohm of inductive reactance because +jX represents an inductive reactance.
To figure out the impedance of a circuit, you first plot the inductive reactance on the positive y-axis and the capacitive reactance on the negative y-axis. The net reactance, X, will be the sum of the two reactances. After you’ve computed the net reactance, you plot the resistance on the x-axis and compute the magnitude of the impedance, shown by r in the graph above. If you consider that r is the third side of a right triangle made up of the sides r, x, and y, r is equal to the square root of x2 + y2.
When thinking about how capacitive reactances, inductive reactances, and resistances combine in a circuit containing resistors, capacitors, and inductors, it’s useful to think in terms of polar coordinates. Polar coordinates show you both the magnitude of an impedance (shown by r in the figure above) and the phase angle of an impedance (shown by θ in the figure above).
QUESTION: What coordinate system is often used to display the phase angle of a circuit containing resistance, inductive and/or capacitive reactance? (E5C08)
ANSWER: Polar coordinates
QUESTION: How are impedances described in polar coordinates? (E5C02)
ANSWER: By phase angle and magnitude
In polar coordinates, a positive phase angle represents an inductive reactance, and a a negative phase angle represents a capacitive reactance. Phasor diagram is the name of the diagram used to show the phase relationship between impedances and resistances at a given frequency.
QUESTION: Which of the following represents an inductive reactance in polar coordinates? (E5C03)
ANSWER: A positive phase angle
QUESTION: What is the name of the diagram used to show the phase relationship between impedances at a given frequency? (E5C05)
ANSWER: Phasor diagram
Now, let’s take a look at some actual circuits.
QUESTION: Which point on Figure E5-1 best represents the impedance of a series circuit consisting of a 400-ohm resistor and a 38-picofarad capacitor at 14 MHz? (E5C10)
ANSWER: Point 4
Here’s how to figure that out. Right off the bat, we know that the only choices are really Points 2, 4, and 6 because the resistance is 400 ohms. Next, we calculate the capacitive reactance: XC = 1/2πfC = 1/(2 × 3.14 × 14×106 × 38×10-12) ≈ 300 ohms. Because the reactance is capacitive, it’s as a negative value. Point 4 is the only point that has a resistance of 400 ohms a reactance of -300 ohms.
QUESTION: Which point in Figure E5-1 best represents the impedance of a series circuit consisting of a 300-ohm resistor and an 18-microhenry inductor at 3.505 MHz? (E5C11)
ANSWER: Point 3
The resistance for this question is specified to be 300 ohms and the reactance is: XL = 2πfL = 2 × 3.14 × 3.505×106 × 18×10-6) ≈ 400 ohms. Because the reactance is inductive, we know it is a positive value. Point 3 is the only point with a resistance of 300 ohms and a reactance of +400 ohms.
QUESTION: Which point on Figure E5-1 best represents the impedance of a series circuit consisting of a 300-ohm resistor and a 19-picofarad capacitor at 21.200 MHz? (E5C12)
ANSWER: Point 1
The resistance is 300 ohms, and the reactance is: XC = 1/2πfC = 1/(2 × 3.14 × 21.2×106 × 19×10-12) ≈ 400 ohms. Because the reactance is capacitive, it’s -400 ohms. Point 1 is the only point with a resistance of 300 ohms and a reactance of -400 ohms.
Jack Vaughan says
Good stuff. Thanks for sharing this material on imedance, mismatch, and related.
David Ryeburn VE7EZM and AF7BZ says
Change “r is equal to the square root of x^2 and y^2” to “r is equal to the square root of x^2 + y^2”.
In 1/(2 × 3.14 × 14×106 × 38×10-12) the exponents 6 and -12 got lowered again.
The same thing happened to the exponents 6 and -6 in 2 × 3.14 × 3.505×106 × 18×10-6).
And once again for the exponents 6 and -12 in 1/(2 × 3.14 × 21.2×106 × 19×10-12).
This seems almost always to happen until you fix it. I’d suggest you double-check anything that has an exponent in it.
Dan KB6NU says
Thanks. It’s hard to copyedit your own work, but you’re right. I should check the exponents specifically.