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Scope Probes and Heartbreak

The questions below are due on Wednesday April 15, 2020; 11:59:00 PM.
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The network shown below models an oscilloscope probe that provides a 10:1 voltage attenuation. Resistor R_1 is a fixed resistor in the probe, resistor R_2 models the input resistance of the oscilloscope, capacitor C_1 is a variable capacitor in the probe, and capacitor C_2 models the combined input capacitance of the oscilloscope and the cable between the probe and the oscilloscope. What relations are required between R_1, R_2, C_1 and C_2 so that for a total signal v_{IN} with sinusoidal-steady-state amplitude V_{in}, the total output signal v_{OUT}(t) = 0.1 v_{IN}(t) for all \omega? Stated another way, what relations are required so that the sinusoidal steady state amplitude of the output V_{out} is related to the input's SSS amplitude as follows: V_{out} = 0.1 V_{in} and \phi = 0 for all \omega? To answer this question generate an expression for C_1 in terms of C_2, R_1, and R_2 (use C2, R1, and R2 as needed).

(Note that the value of C_2 is difficult to guarantee in practice due to variations in cable length and oscilloscope input capacitance, so C_1 is made manually adjustable in the probe.)

Hint: Focus on the requirement that \phi=0 for this first part. Then build on that and the V_{out} = 0.1 V_{in} requirement for the second part.


Building on the relationship you developed above. Come up with an expression for R_1 in terms of the three other circuit components. (use your relationship from above to greatly simplify (substitute your math!))