In Lab 4 you used oscilloscope probes (and an oscilloscope) to study RC circuits

Question

In Lab 4 you used oscilloscope probes (and an oscilloscope) to study RC circuits. These probes are actually coaxial cables attached to the oscilloscope input. Much like a voltmeter, you want the oscilloscope to have a high resistance. R_sc so that it doesn't have too much current across it when you take a measurement. A typical value is R_sc = 1 M ohm. The coaxial cable is a cable with an inner conductor and an outer conductor. The inner wire carries the signal and the outer wire is connected to ground. This design is least susceptible to electrical noise. The problem is that coaxial cables have significant capacitance and a typical value for a meter long cable is C_coax = 100 pF. A diagram of the circuit made by the cable and oscilloscope is shown below. In this problem we are going to show some problems that may arise from this set up. What is the effective impedance. Z_off. of this probe configuration from hi to GND at DC (f = 0)? Replace the capacitor with either a short or an open (you have to decide what is appropriate at this frequency) and draw the effective circuit. What is the effective impedance. Z_off. of this probe configuration from hi to GND at the infinite frequency limit (f rightarrow infinity)? Replace the capacitor with either a short or an open (you have to decide what is appropriate at this frequency) and draw the effective circuit. At what frequency f_eq do the cable capacitance and the input resistor have impedances of the same magnitude? For f>>f_eq, make the argument that you can ignore the resistance because the probe acts like a pure capacitor. For R_sc = 1 M ohm and C_coax = 100 pF. what is the effective useful frequency range for this cable configuration where we can think of it acting as a simple 1 M ohm resistor?

Explanation / Answer

This type of circuit is called asingle-time-constant (STC) circuit. A STC circuits are composed of or can be reduced to one reactive component (inductance or capacitance) and one resistance. An STC circuit formed of an inductance L and a resistance R has a time constant = L/R. The time constant of an STC circuit composed of a capacitance C and a resistance R is given by = CR. Using   =2pif

a) At f=0, or = 0 capacitors should be replaced by open circuits(1/jC = )

b) At a = capacitors should replaced with short circuits (1/jC = 0).

c) Zc = (1/C) = R, then frequency is eq = 1/RC

d) for   >>  eq = 1/RC, we are in the limit =

e) eq = 1/RC=104 rad/s

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