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?
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Point 2
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Correct AnswerPoint 4
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Point 5
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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.
Hint: Since the real part (resistance) is 400, we are limited to the three points at X=400. Since this circuit is dealing with a capacitor (no inductor component), the imaginary (Y) magnitude will be negative. Of those three points at X=400, only Point 4 is a negative reactance (capacitance). No calculations required.
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.
Mnemonic: One rider is over two pedals (pi), two feet frequency), and two legs (L) or calves (C).
Last edited by mcdanlj. Register to edit
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?
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Point 1
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Correct AnswerPoint 3
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Point 7
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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.
Memorization Tip: 3.505 mhz starts with the number 3, therefore, "Point 3" is the answer.
Last edited by knguyen553. Register to edit
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?
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Correct AnswerPoint 1
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Point 3
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Point 7
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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 be in the fourth quadrant, the only choices are Point 1 and Point 4, but only Point 1 is among the answer choices.
Last edited by marvsherman419. Register to edit