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Transcript

Brown University Physics Department Physics 0060 Lab C-260 ELECTROMANETIC PULSE PROPAGATION IN A COAXIAL CABLE The mechanical waves on a stretched string are easily generated and observed but not easily studied in quantitative detail. The propagating waves in an electrical system are however easily generated and easily studied with standard instrumentation. The purpose of this lab is to measure the speed of light using the properties of electromagnetic waves and pulses in a coaxial cable. THEORY: If we simultaneously have two signals(electric circuit is demonstrated in Fig.2): one is input to the oscilloscope directly, the other is reflected from a long cable before input to the oscilloscope, there will be a phase difference between the two signals because the wave inside the cable travels a longer distance. The speed of light is determined by measuring the propagation velocity in the cable. The velocity is given by = v 2* L / ∆t Where L is the length of the cable and ∆t is the time interval of two signals. Note that the factor 2 implies the reflected signal by the cable. Compare the velocity of the signal v with its theoretical value given below: v = c/ ε Where c is the speed of light in vacuum and ε is the dielectric constant of the plastic insulator. You will experimentally determine the dielectric constant from a measurement of the capacitance of the coax cable (See Appendix for further details).From the above equation we can get an electrical measurement of the speed of light. A coaxial cable has a characteristic impedance that relates the voltage to the current in the cable (in our experiment, it’s 50 Ohms). A cable terminated with this impedance will behave equivalently to a pure resistance equal to the characteristic impedance. So if we connect an variable termination resistor (VTR) at the end of the cable and adjust the impedance( in this case the impedance is the same as resistance), the ratio of the amplitudes of reflected and incident signal is Vreflected R − Z 0 = Vincident R + Z 0 140428 1 Brown University Physics Department Physics 0060 Lab C-260 Z 0 is the characteristic impedance. A detailed derivation of the electromagnetic theory is shown in the appendix. Experimental Procedure 1. Measure the radii of the inner copper conductor and inner plastic insulator of the short sample cable. It has its insulation stripped back and conductors exposed. Fig. 1 2. Determine the dielectric constant of the insulation separating the inner and outer conductors by measuring the capacitance of the coiled cable. Use the capacitance meter and solve for the dielectric constant in CGS units (1 Farad = 9 × 1011cm) using equation 9 of the Appendix. 3. Measure velocity of wave in the cable: (a) Be familiar with the TDS2012B oscilloscope and connect it to PC. Use the software Tektronix OpenChoice Desktop to acquire data on PC. (See details on our website: --Lab Manuals--Introduction to the Oscilloscope) (b) Use the SRS DS335 3.1MHz synthesized function generator (please check for availability) to generate a square wave function. Connect the filter circuit directly to the function generator. Set the function generator to ≈ 100 kHz, ≈ 10 V square waves. (To set the frequency, press [FREQ] and then enter “100.00000” and press [Vpp/kHz]. To set the amplitude, press [AMPL] and then enter “10.0” and press [Vpp/kHz].) The filter circuit acts to “differentiate” the square wave, giving short pulses when the amplitude changes. Use a short cable to connect the output of this pulse generator to one side of a Tee connected to the scope input. Connect the coiled cable you used in Part 2 to the other side of the Tee. Important: Do not connect the end of the filter with the 50 Ω resistor to the function generator. (c) Connect SYNC OUT of the function generator to the oscilloscope external trigger input and set the oscilloscope to trigger on this external trigger 140428 2 Brown University Physics Department Physics 0060 Lab C-260 signal to offer a synchronous reference. Note that you have to select the SYNC ON option on the panel of function generator by pressing [SHIFT] and then [.] Fig. 2 (d) Observe the two pulses on the oscilloscope and output the data on PC. You are supposed to see waveforms as is shown below: Fig.3 Measurement of the time interval between incident and reflected pulses. 4. Measure the length of the cable and calculate the velocity of propagation in the cable. 5. Calculate the constant c using the velocity of propagation and the dielectric constant for the cable. Compare your result with the known speed of light. (c = 2.998 × 1010 cm/sec) 6. If you have extra time, you may want to investigate some of the other options. 140428 3 Brown University Physics Department Physics 0060 Lab C-260 Adjust the scope so that you see both the original pulse and a reflected signal from the end of the cable. Connect the variable termination resistor (VTR) to the far end of the cable. Connect an ohmmeter to the other BNC jack on the VTR. (Use the switch on the VTR to toggle between the ohmmeter and the cable.) Notice how the reflected signal changes as you change the termination resistance. Plot the reflected signal amplitude versus termination resistance; be sure to make several measurements in the region where the reflected amplitude is rapidly changing. Use this plot to determine the characteristic impedance of the cable. Compare your results with the predicted impedance for your cable. (1 Ohm = 1.113 × 10-12 cm-1 sec in CGS units) Connect to Ω Meter Switch Fine Adjust Coarse Adjust Connect to far end of cable 140428 Variable Terminator Fig. 4 4 Brown University Physics Department Physics 0060 Lab C-260 APPENDIX Theory of Electromagnetic Wave Propagation in a Coaxial Cable While the effects of signal propagation can usually be neglected for low frequency circuits, propagation effects become very important when the signal changes appreciably during the time it takes the signal to propagate in the circuit. Electromagnetic wave propagation is governed by Maxwell’s equations, and the theory behind electromagnetic wave propagation in a coaxial cable is developed in the next section. The electric and magnetic fields inside the cable must obey Maxwell’s equations: 4πρ ∇⋅E = , (1) ∇ ⋅ B = 0, (2) ε 4π ε ∂E ∇× B = J+ , c c ∂t 1 ∂B ∇× E = − , c ∂t (3) (4) where the dielectric constant ε is needed to take care of the polarization of the material in response to an applied field (see Purcell, Chapter 10, for example). The dominant “mode” for electromagnetic waves in a coaxial cable is one where the electric and magnetic fields are transverse to the axis of the cable. Given this assumption, Eqs. 1 and 3 require that E r ( z, t ) = 2λ ( z , t ) , εr (5) Bφ ( z , t ) = 2 I ( z, t ) , cr (6) where λ (z,t) is the charge density on the inner conductor and I(z,t) is the current carried by the inner conductor. The potential difference between the inner and outer conductors, V, is found by integrating Eq. 5 V ( z, t ) = 140428 2λ ( z , t ) ε ln(b / a), (7) 5 Brown University Physics Department Physics 0060 Lab C-260 where a and b are the radii of the inner and outer conductors, respectively. For a fixed potential difference between conductors with the far end of the cable unconnected, the cable acts like a capacitor with a capacitance of C= εL 2 ln b / a (8) , where L is the length of the cable. Thus, by measuring the capacitance of the cable we can determine the dielectric constant ε ε= 2C ln b / a . L (9) From Eqs. 3 and 4 we have the requirement − ∂Bφ =− ∂z ε ∂E r c ∂t , ∂E r 1 ∂Bφ . =− c ∂t ∂z (10) (11) By differentiating Eq. 10 with respect to t, rearranging the order of differentiation, and substituting Eq. 11 yields a wave equation field: ∂ 2 Er ε ∂ E = 2 2 2r . 2 ∂z c ∂t (12) The solutions of Eq. 12 are travelling waves that propagate in either the +z or –z directions with a velocity given by v= c ε . (13) The magnetic field obeys a similar wave equation and is related to the electric field as follows: Bφ = ε E r . (14) This relation between the electric and magnetic fields requires that there be a fixed ratio between the voltage across the inner and outer conductors and the current on the conductors 140428 6 Brown University Physics Department Physics 0060 Lab C-260 Z= V 2 ln b / a , = I εc (15) where Z is the “characteristic impedance” of the cable. If the end of the cable is “terminated” by connecting a resistor whose value is equal to the characteristic impedance, this relation between the voltage and current is maintained. Such a cable looks like a pure resistance, independent of the length of the cable. A cable terminated by a resistance R ≠ Z will generate a reflected wave so that the ratio between voltage and current is maintained at the characteristic impedance. The reflected wave will propagate back down the cable with the same propagation velocity as the incident wave. At any point in the cable, the voltage and current are determined by superimposing the incident and reflected waves. For an incident wave of amplitude VI , the amplitude of the reflected wave VR can be shown to be VR = R−Z VI . R+Z (16) Such reflections will occur wherever there is a change in impedance seen by signals propagating down the cable. 140428 7