Attenuator Coaxial-Cable Oscilloscope Interface

这是一篇关于Attenuator Coaxial-Cable Oscilloscope Interface 的物理实验报告，通过实验并计算实验数据、画图等得出此报告 1. ABSTRACT
  This experiment report shows how an attenuator used to reduce the amplitude of a given input without distorting its waveform. The set-up of this experiment is a simple voltage divider circuit connected to an oscilloscope. The effect of the capacitance between the two coaxial-cables of oscilloscope cannot be ignored, and must be calculated first. The following step is to determine the value of two resistors RA and RB which make up the voltage divider circuit to acquire the specific gains. It was concluded after building and testing the circuit that the attenuator design meets all the constraints given through the oscilloscope readings.

2. THE EXPERIMENT
2.1 INTRODUCTION
  The objective of this experiment was to assemble and test an attenuator to meet all the design specifications. The initial plan was putting a 1MΩ resistor in the voltage divider. However, it provided a too slow time constant. The reason of this failure is the capacitance of the coaxial cable was ignored. By taking this capacitance into consideration and calculation, the appropriate values for two resistor RA and RB were determined. The new circuit was set up based on the these resistors and capacitors, through observing whose oscilloscope, we found that the design specification of converting a 11V 50HZ square wave signal to a 1V 50HZ square signal was met this time. The theory and calculation procedure are presented below to support the results and conclusions.

2.2 THEORY and PROCEDURE
2.2.1 The capacitance of coaxial cable
Assemble the test circuit per Figure 2.1. 
 
Figure 2.1 Model Circuit for measuring Capacitance of Coaxial Cable
Define all the symbols with the physical meaning as listed below:
Rτ = Total equivalent resistance of circuit
           Rs = Resistance of function generator
       Rc = Resistance of capacitor model
       RSC = Resistance of scope model 
       REXT = External resistance chosen based on magnitude of Rc 
                             Rc < 50Ω , REXT = 50Ω
                             Rc < 100Ω, REXT = Rc
                                 Rc > 10 kΩ, REXT = 10 kΩ
         Cτ = Equivalent capacitance of the circuit
         CSC = Capacitance of scope model
         C = Capacitance of Capacitor model
         τ = Time constant of R-C circuit

Then follow the steps listed below to measure the capacitance of coaxial cable.
Step 1:  the total equivalent Resistance of the R-C circuit was calculated as
      Rτ = (RS + REXT) || RC || RSC 
Step2:  Time constant of R-C circuit is given as τ= Rτ * Cτ, thus
      Cτ = τ / Rτ 
Step3:  Capacitance of capacitor model is obtained by the formula below as capacitor model and scope are connected in parallel
      C = Cτ - CSC
 2.2.2 Voltage divider circuit resistors
According to the voltage divider resistance relationship demonstrated below, we can determine the values of RA and RB.
Gain(µ) = RB / (RA + RB )
2.2.3 Time constant measurement
After having the values for RA and RB, we set up the circuit per Figure 2.2. The time constant value can be measured through the oscilloscope display.
 
Figure 2.2 Model Circuit for Attenuator
2.2.4 Verification
By comparing the experiment obtained values and the design specifications, we can verify this experiment and come to the conclusion. 

2.3 Calculation and Results
2.3.1 The capacitance of coaxial cable
All the given information of the RC circuit is: 
RC = 1 MΩ ( measured directly using Digital Multimeter)
REXT = 1 kΩ
RSC = 10 MΩ
CSC = 2pF = 2e-12 F
Referring to the theory part in 2.2.1
 Rτ = (RS + REXT) || (RC) || (RSC)
              = (50Ω + 1kΩ) || 1 MΩ || 10 MΩ=1.05 kΩ
From oscilloscope reading,  τ = 1.84 ns
Thus, Cτ = τ / Rτ
              = (1.84e-9) / 1050
              = 175.4 pF
C = Cτ - CSC
      = 175.4-2
      = 173.4 pF
2.3.2 Voltage divider circuit resistors
The demanded input and output is 11V and 1V respectively. Then we can obtain the gain as 1/11. Referring to the theory part in 2.2.2:
  1/11= RB / (RA +RB+50)
In this experiment we choose RA to be 2.2kΩ, then we can calculate that RB is 225Ω. Using the design constraints, we can check the feasibility of these values:
Upper bound:
Then we apply the time constant constraint:
τ=[(RA + 50) || RB || 1 MΩ]*(CSC + C)=35.7ns < 250ns
Lower bound:
Another constraint RA + R¬B =2225Ω> 2000Ω is also satisfied.
2.3.3 Procedure for building a 0.4 attenuator
Following a similar procedure with 2.3.2:
Step1: choose RA value(1kΩ).
Step2: Using 0.7= RB / (RA +RB+50), calculate the corresponding RB.
Step3: Check the upper and lower bound.
Step4: Increase RA value if upper bound of RB is less than 0 and go back to step2.
After this kind of procedure we can calculate the results as:

RA=1000Ω, RB=2450Ω with τ=130ns

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