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Subelement E5
ELECTRICAL PRINCIPLES
Section E5C
Coordinate systems and phasors in electronics: Rectangular Coordinates; Polar Coordinates; Phasors
Which of the following represents a capacitive reactance in rectangular notation?
-jX
• +jX
• X
• Omega

Hint:
A negative imaginary (e.g. -jX) number denotes capacitive reactance A positive imaginary (e.g. +jX) number denotes inductive reactance.

When using rectangular coordinates to graph the impedance of a circuit, the horizontal axis represents the resistive component. When using rectangular coordinates to graph the impedance of a circuit, the vertical axis represents the reactive component.

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How are impedances described in polar coordinates?
• By X and R values
• By real and imaginary parts
By phase angle and amplitude
• By Y and G values

Impedances are described by vectors, having an angle (phase angle) and magnitude (amplitude). This is similar to another question in the pool about vectors.

So, the magnitude is the distance from the origin (center of the graph), and the phase angle is the angle (usually from the X-axis). Positive phase angles are impedances with a net inductive reactance. Negative phase angles are reactances with a net capacitive reactance.

-k6yxh

Hint: Polar coordinates deal with angles.

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Which of the following represents an inductive reactance in polar coordinates?
• A positive real part
• A negative real part
A positive phase angle
• A negative phase angle

Phase angles are measure from 0 to 180 for positive (inductive) and 0 to -180 for negative (capacitive)

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Which of the following represents a capacitive reactance in polar coordinates?
• A positive real part
• A negative real part
• A positive phase angle
A negative phase angle

Phase angles displayed in polar coordinates are angles above and below the horizontal line. Capacitive reactances are displayed in the 4th quadrant, the one with a positive resistive element (right side of the graph) and a capacitive reactance - they have a negative phase angle, a point below the horizontal line.

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What is the name of the diagram used to show the phase relationship between impedances at a given frequency?
• Venn diagram
• Near field diagram
Phasor diagram
• Far field diagram

Phase relationship is the key in this question to remember the answer is phasor diagram.

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What does the impedance 50-j25 represent?
• 50 ohms resistance in series with 25 ohms inductive reactance
50 ohms resistance in series with 25 ohms capacitive reactance
• 25 ohms resistance in series with 50 ohms inductive reactance
• 25 ohms resistance in series with 50 ohms capacitive reactance

In any impedance measurement:

• The real part is resistance.
• The imaginary part is reactance seen as either inductive ($+j$ values) or capacitance ($-j$ values)

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What is a vector?
• The value of a quantity that changes over time
A quantity with both magnitude and an angular component
• The inverse of the tangent function
• The inverse of the sine function

A vector is defined as a quantity that includes magnitude and direction. The direction could also be described as "an angular component." This is another example of a question of definition.

Impedances can be described as a vector in cartesian or polar coordinates, i.e., a magnitude (distance from the origin of the graph) and direction (an angle, compared to the horizontal or X-axis of the graph).

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What coordinate system is often used to display the phase angle of a circuit containing resistance, inductive and/or capacitive reactance?
• Elliptical coordinates
Polar coordinates

Polar coordinates are often used to display the phase angle of a circuit containing resistance, inductive and/or capacitive reactance. In a polar-coordinate system, each point on the graph has two values, a magnitude and an angle.

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When using rectangular coordinates to graph the impedance of a circuit, what does the horizontal axis represent?
Resistive component
• Reactive component
• The sum of the reactive and resistive components
• The difference between the resistive and reactive components

On a rectangular coordinate graph, remember that the horizontal axis plots the resistive component and the vertical axis plots the reactive component.

Visual:

When laying horizontal, one is more resistive to get up

When standing vertical, one is more reactive to their surroundings

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When using rectangular coordinates to graph the impedance of a circuit, what does the vertical axis represent?
• Resistive component
Reactive component
• The sum of the reactive and resistive components
• The difference between the resistive and reactive components

When using rectangular coordinates to graph the impedance of a circuit, the vertical axis represents the reactive component and the horizontal axis represents the resistive component.

• whitezw

Fun Hint: Vertical Y axis... getting a rise (up) out of someone is a reaction (Reactance). Horizontal X axis is dragging that someone who resists (Resistance). -KD5JUN

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What do the two numbers that are used to define a point on a graph using rectangular coordinates represent?
• The magnitude and phase of the point
• The sine and cosine values
The coordinate values along the horizontal and vertical axes
• The tangent and cotangent values

This goes back to high school days.. Remember when we had to plot lines on chart paper?

Plotting points on the grid involved determining an X value and a Y value. Another term for this is Rectangular Notation. Remember Rectangles = Rectangular with X (horizontal) and Y (vertical) axis.

-KE0IPR

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If you plot the impedance of a circuit using the rectangular coordinate system and find the impedance point falls on the right side of the graph on the horizontal axis, what do you know about the circuit?
• It has to be a direct current circuit
• It contains resistance and capacitive reactance
• It contains resistance and inductive reactance
It is equivalent to a pure resistance

The horizontal axis represents purely resistive values, so if your answer lies anywhere along this axis, the value is a pure resistance. KD9AGJ

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What coordinate system is often used to display the resistive, inductive, and/or capacitive reactance components of impedance?
• Elliptical coordinates
Rectangular coordinates

A Rectangular coordinate system can be used to represent resistance along the X-axis, and the combination of inductive and capacitive reactance on the Y-axis.

If the sum of the reactance components is inductive, it will be in the 1st quadrant, a positive value of Y.

If the sum of the reactive components is capacitive, it will be in the 4th quadrant, below the X-axis, a negative value of Y.

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Tags: arrl chapter 4 arrl module 4c Which point on Figure E5-2 best represents the impedance of a series circuit consisting of a 400 ohm resistor and a 38 picofarad capacitor at 14 MHz?
• Point 2
Point 4
• Point 5
• Point 6

The impedance of a capacitor, denoted by $X_C$ is: $X_C= \frac{1}{2\pi fC}$ where:

• $f$ is the frequency of interest
• $C$ is the capacitance in Farads.

The impedance of a capacitor has a negative j value. In the example shown we can mostly remove the large scale values by canceling the ($10^{6}$) of the MegaHertz and the ($10^{-12}$) of the picoFarads. So, we are left with ($10^{-6}$) giving an impedance value that is:

\begin{align} \frac{1}{2\pi\cdot14\cdot38\cdot10^{-6}}&\approx 299.2\:Ω\\ &\Rightarrow- j299.2\:Ω \end{align}

In a series circuit with a 400 Ω resistor the total impedance is $400 - j299.2\:Ω$ which is in the lower right quadrant of the figure at about 400 in the +X direction and about 300 in the -Y direction.

vaughanth

Hint: Since this circuit is dealing with a capacitor (no inductor component), the capacitance magnitude will be negative. Only Point 4 is a negative capacitance. The pure resistance is 400, so that puts the point on 400.

-KE0IPR

Memory tip. If the frequency is 21 MHz or more, use the first number of the capacitor for the answer clue. If the frequency is less than 21 MHz, use the first number of the resistor for the answer clue.

-KM4VOW

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Tags: arrl chapter 4 arrl module 4c Which point in Figure E5-2 best represents the impedance of a series circuit consisting of a 300 ohm resistor and an 18 microhenry inductor at 3.505 MHz?
• Point 1
Point 3
• Point 7
• Point 8

The impedance of the inductor $Z_L = 2 \pi f L$, where:

• $f$ is the frequency of interest
• $L$ is the inductance in Henries.

The $j$ value of the impedance of an inductor is positive.

$Z_L = 2 \pi \times 3.505\text{ MHz} \times 18\ \mu\text{H}$

Since the frequency here is in MegaHertz and the Inductance in microhenries, then the Mega ($10^6$) and micro ($10^{-6}$) exponents cancel:

\begin{align} Z_L &= 2 \pi \times 3.505\text{ MHz} \times 18\ \mu\text{H}\\ &= 2 \pi \times 3.505 \times 18\\ &= 396.4\ \Omega \end{align}

In a series circuit with a $300\ \Omega$ resistor, the total impedance is $300 + j396.4\ \Omega$ which is in the upper right quadrant of the figure at about $300$ in the $+x$ direction and about $400$ in the $+y$ direction, corresponding to Point 3 in Figure E5-2.

Test Tip: Because the resistance is $300\ \Omega$, the only correct possibilities will be found at 300 ohms positive from the origin. Because the reactive component is only inductive, any correct possibility will be found relatively far away from the resistance axis in the positive direction. Point 3 is the only option that fits.

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Tags: arrl chapter 4 arrl module 4c Which point on Figure E5-2 best represents the impedance of a series circuit consisting of a 300 ohm resistor and a 19 picofarad capacitor at 21.200 MHz?
Point 1
• Point 3
• Point 7
• Point 8

The impedance of a capacitor, denoted by $X_C$ is: $X_C= \frac{1}{2\pi fC}$ where:

• $f$ is the frequency of interest
• $C$ is the capacitance in Farads.

The impedance of a capacitor has a negative j value. In the example shown we can mostly remove the large scale values by canceling the (106) of the MegaHertz and the (10-12) of the picoFarads. So, we are left with (10-6) giving an impedance value that is: \begin{align} X_C&= \frac{1}{2\pi (21.2\times 10^6 \text{ Hz})(19\times 10^{-12}\text{ F})} \\ &=\frac{1}{2\pi(21.2)(19)\left(10^{-6}\right)}\\ &\approx -j395.1\:\Omega \end{align}

In a series circuit with a 300 Ω resistor the total impedance is $300 - j395.1\:Ω$ which is in the lower right quadrant of the figure at about 300 in the +X direction and about 400 in the -Y direction.

In short, since the problem only specifies a capacitance (and no inductance), only one answer falls on 300 Ω for the resistance (+X) axis and has a negative reactance: Point 1.

Also, seeing that the answer must being in the fourth quadrant, the only choices are Point 1 and Point 4, but only Point 1 is among the answer choices.

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Tags: arrl chapter 4 arrl module 4c Which point on Figure E5-2 best represents the impedance of a series circuit consisting of a 300 ohm resistor, a 0.64-microhenry inductor and an 85-picofarad capacitor at 24.900 MHz?
• Point 1
• Point 3
• Point 5
Point 8

First, we know that the answer is going to be one of the three points that aligns horizontally at 300 ohms (the pure resistive component). Unfortunately, this only eliminates Point 5 as a possible answer, leaving us with Points 1, 3, and 8 as possibilities.

To determine which answer is correct, we need to compute the impedance of both the capacitor and inductor at the frequency given. We'll start with the impedance of the inductor: (Use the $π$ key on your calculator for better accuracy.) \begin{align} X_L &= 2πfL\\ &= 2π\cdot(24.9\cdot10^6\text{ Hz}) \cdot (0.64\cdot10^{-6}\text{ H}) \\ &\approx 100.13 \:\Omega\ \end{align}

Then the impedance of the capacitor:

\begin{align} X_C &= \frac{1}{2πfC} \\ &= \frac{1}{2π\cdot(24.9 \cdot10^6 \text{ Hz}) \cdot (85\cdot10^{-12}\text{ F})} \\&= \frac{1}{0.0133} \\ &\approx 75.20\:\Omega \end{align}

This tells us that we have an inductive component (positive direction on the impedance axis) of approximately 100 $\Omega$ and a capacitive component (negative direction on the impedance axis) of 75 $\Omega$.

The resultant reactive component will be $25\:\Omega$ of inductive impedance at the frequency given ($X_L - X_C\approx 100-75=25\:\Omega$).

This is a point slightly above the resistance axis in the positive direction. Only Point 8 satisfies that condition.

Test Tip: Because the resistance is $300\ \Omega$, the only correct possibilities will be found at 300 ohms positive from the origin. Because the reactive component is relatively balanced, any correct possibility will be found relatively close to the resistance axis in the positive direction. Point 8 is the only option that fits.

Test "cheat": do the math to understand these Figure E5-2 problems, BUT for the test, the one's digit in the frequency names the point (14 MHz is Point 4), EXCEPT if the question involves BOTH inductance and capacitance you double that one's digit (2 x 4 = 8, so Point 8).

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