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

[ashkash]

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.

 

[antoker]

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):

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

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

$$V_{out} = e(t)\cdot\frac{Z_{p}||Z_{scope}}{Z_{p}+R_{s}}$$
where
$$Z_{p}$$ denotes the parallel impedance (excluding scope load)
$$R_{s}$$ denotes series resistance at 5k
$$Z_{scope}$$ 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!

 

[piercas]

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.