Measuring RMS Voltage of e(t) at f = 3.22MHz

In summary, to determine the rms value of the voltage at f = 3.22MHz, we can use the voltage divider rule to calculate the voltage at the output of the circuit. With an ideal oscilloscope, the voltage measured would be the same as the voltage at the output. However, with a non-ideal oscilloscope, there will be additional parallel load at the output, resulting in a different voltage measurement. The impedance of the oscilloscope should also be taken into consideration in the calculations. This problem may require converting the given circuit into a simpler schematic and using the parallel combination formula.
  • #1
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Homework Statement



Determine the rms value of the voltage that is measured as a function of frequency at f = 3.22MHz. e(t) = 50sqrt(2)cos(wt). A 100MHz oscilliscope is being used to measure the voltage across nodes AA' as seen in the figure. The input impedence of the oscilliscope is a 1Mohm resistor in parallel with a 13pF capacitor. Also, calculate the rms voltage value that would be measured with an ideal oscilliscope (one with an infinite input imedence).

Homework Equations





The Attempt at a Solution



I do not know where to start with this problem. I need to calculate the rms voltage measured with an ideal oscilliscope and an oscilliscope with the specified characteristics above. I have attached the circuit image. Any help would be appreciated. thanks.
 

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  • #2
1. infinite input impedance:
With ideal scope the schematic will be the same as given. Voltage at the output can be calculated as follows:

Using voltage divider rule (simplify schematic to voltage source with series resistance at 5k and complex impedance):

[tex]V_{out} = e(t)\cdot\frac{Z_{p}}{Z_{p}+R_{s}}[/tex]
where
[tex]Z_{p}[/tex] denotes the parallel impedance (total)
[tex]R_{s}[/tex] denotes series resistance at 5k

2. Finite input impedance
Same procedure, same equation. Non-ideal scope will represent additional parallel load at the output.

[tex]V_{out} = e(t)\cdot\frac{Z_{p}||Z_{scope}}{Z_{p}+R_{s}}[/tex]
where
[tex]Z_{p}[/tex] denotes the parallel impedance (excluding scope load)
[tex]R_{s}[/tex] denotes series resistance at 5k
[tex]Z_{scope}[/tex] denotes scope impedance
|| means parallel combinationRemark: Since scope is a 100Mhz (sampling speed?) no aliasing or folding will occur, but some phase lag will.
P.S Remember that impedance is a function of frequency!
 
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  • #3


To calculate the rms voltage, we can use the formula V_rms = V_max / sqrt(2), where V_max is the maximum voltage measured. In this case, we have e(t) = 50sqrt(2)cos(wt), so the maximum voltage is 50sqrt(2) V. Plugging in the values, we get V_rms = 50sqrt(2) / sqrt(2) = 50 V.

To calculate the rms voltage with an ideal oscilliscope, we can use the same formula but with an infinite input impedance. This means that the input impedance does not affect the measurement and can be ignored. So the rms voltage with an ideal oscilliscope would also be 50 V.

However, with the given oscilliscope, the input impedance does affect the measurement. To calculate the rms voltage with this oscilliscope, we can use the formula V_rms = V_max / sqrt(2) * sqrt(R/(R+1/(jwC))), where R is the resistance of the input impedance (1 Mohm) and C is the capacitance (13 pF). Plugging in the values, we get V_rms = (50sqrt(2) / sqrt(2)) * sqrt(1/(1+(1/(3.22*10^6 * 13*10^-12)))) = 50.009 V.

So the rms voltage measured with the given oscilliscope would be slightly higher than the ideal value due to the input impedance affecting the measurement.
 

1. What does RMS voltage measure?

RMS voltage measures the effective or average voltage of an AC signal over a period of time. It takes into account both the amplitude and frequency of the signal, providing a more accurate representation of the voltage than peak or average measurements.

2. How is RMS voltage calculated?

RMS voltage is calculated by taking the square root of the average of the squared values of the voltage over one complete cycle of the AC signal. It is expressed in units of volts (V).

3. Why is it important to measure RMS voltage at a specific frequency?

Different frequencies can have varying effects on electronic components and systems. Measuring RMS voltage at a specific frequency, such as 3.22MHz, allows for a more precise understanding of how the voltage will behave in a particular circuit or device.

4. What is the significance of 3.22MHz in this measurement?

3.22MHz is the frequency at which the RMS voltage is being measured. It may be a specific frequency that is relevant to the circuit or device being tested, or it may be a standard frequency used for testing and analysis purposes.

5. How can measuring RMS voltage at 3.22MHz be useful in practical applications?

Measuring RMS voltage at 3.22MHz can be useful in a variety of practical applications, such as designing and testing electronic circuits and systems, troubleshooting issues with electrical equipment, and ensuring safe and efficient operation of devices that operate at this frequency. It can also provide valuable data for research and development in the field of electronics.

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