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- 1. Exercise 1: The resistivity measurement system that used was based on the four point probe Schlumberger configuration and it is shown in figure 1 Figure 1 :The Measurement System The circuit diagram of a constant current source is shown in figure 2.If we apply input voltage Vin ,this circuit produces the same output current irrespective the changes in load resistor Rload .Since the output current Io equals the current through R on the resistor R,hence the the circuit shown here has a nearly infinite load resistance. For Vin=0.6V and R=3000 ohms ……{given} current Io can be calculated as follows Io= Vin/R Io= 0.6/3000 = 0.0002A or 0.2 mA Actually by changing the load resistance Rload the opamp output changes its voltage to keep the current Io remain constant. Vin = 0.6V Vload R=3KOhm Figure 2 : Constant current generator
- 2. Below is the table showing the values of Vload for different values of Rload for the resistivity measurement system used i.e,Vin=0.6 V. Rload(Ohms) V load (Volts) Io load(A) 1000 0.2 0.0002 2000 0.4 0.0002 3000 0.6 0.0002 4000 0.8 0.0002 5000 1 0.0002 6000 1.2 0.0002 Table showing results of Vload at Different Rload Plotting graph between Vload and Rload we have got Vload Vs Rload 1.4 1.2 y = 0.0002x 1 Vload (Vlots) 0.8 Series1 0.6 Linear (Series1) 0.4 0.2 0 0 5000 10000 Rload (ohms) Graph between Vload Vs Rload From the above graph between Vload and Rload the equation of line is given by Y=0.0002x Where Gradient=Vload/Rload=Io Therefore Io=0.0002 A Results: From the graph we can see that the gradient remains the same for all values of Rload. Hence we can say that in this constant current source the output current Io is independent of the changes in Rload and therefore the circuit has nearly infinite load resistance within the compliance range.
- 3. . Exercise 2:Measurement Strategies Figure 1:Resistivity Measurement strategy used in archaeology site The measuring system used in a archaeology site (as shown by figure 1) for measuring electrical resistivity or conductivity of any material was based on the four point probe method.This is a simple and low cost method which is first proposed by Thomson in 1861 and first used by Schlumberger in 1920 to measure the resistivity of Earth. Principle The basic principle of four point probe method is illustrated in the figure below Ground Four Point Probe Method “By applying constant current to the outer electrodes ,Potential Difference can be measured by the inner electrodes at different positions on the surface of the material .The current remains constant in the circuit irrespective to the
- 4. changes in resistances and the output voltage is totally dependent on the resistance changes.” The four point method offers advantages over Direct or two probes method which offer very poor performance due to polarization and electrode contact issues. By using constant current source the current flowing through the material will largely independent to the probe/contact impedances as current remains constant as resistance changes.For verification below is the experimental results showing that the current remains same for different values of load resistances and only the output voltage will be changed with resistance.this experiment was performed with 1 mA current source with different values of resistances.and the output voltage of 15 V as opamp maximum i/p was 15 V and the opamp saturated at 13 volts approximately. R Vout I ohms (volts) (amperes) 1000 1.02 0.00102 2000 2 0.001 3000 3.05 0.001017 4000 4.07 0.001018 5000 5.08 0.001016 6000 6.1 0.001017 7000 7.12 0.001017 8000 8.15 0.001019 9000 9.165 0.001018 10000 10.18 0.001018 11000 11.18 0.001016 12000 12.19 0.001016 13000 13.21 0.001016 Results from Laborator experiment for verification of constant current source The graph between different Voltage and resistances values is shown below .
- 5. V vs R(Linear Range) 14 y = 0.0010x - 0.0050 12 10 8 Vo (V) Linear 6 Linear (Linear) 4 2 0 0 5000 10000 15000 R(ohm) Also to ensure that virtually no current will flow through the voltage measuring electrodes an instrumentational amplifier is used as shown in the figure. In designing the voltage measuring electrodes should be as near to resistance as possible to minimize contact resistance and loading effects. The current follows the path according to the distribution of resistivity within the region.Thereore the potential difference between the two inner probes contains the information about the distribution of electrical resistivity. For measuring resistivity of the ground currents are injected into the ground and the resulting potential differences are measured at the surface.The in situ measurement of resistivity is impractical as it is the resistance measured between the faces of a unit cube of material.Therefore the common method is to measure the resistance on the surface of a material and then calculate the apparent resistivity values using the current and voltage measurements and the electrode configuration used Mostly low frequency alternating sources are applied to the ground for avoiding the polarization effect. In practice the frequency of the alternating current required to penetrate in the ground decreases with the increase in depth. F a.c. α 1/depth In general a frequency of 100 Hz is required to penetrate in 10m depth whereas 10 Hz for 100m. The resistivity for a conducting cylinder of resistance ∆R,length ∆L and cross sectional area ∆A is given by: =∆R ∆A/∆L ………eq(1)
- 6. Current Flow in the ground: Consider a homogeneous material,when current I will flow through it it will follow the radial pattern from the electrode and create hemispherical equipotential surfaces. From Ohms’s Law and substituting values in eq 1 we know that ∆V/∆L = I/∆A = - i …………………..eq(2) The potential gradient ∆V/∆L is dependent on the current density I and the resistivity of the material. If the electrode is at distance r the surface area is 2πr² and hence current density I is given by i=I/2πr² Now the potential gradient is given by Equation (3) Now the potential Vr at a distance r can be found by integerating above equation (3) The potential can be calculated at any point on or below the surface of a homogeneous surface or ground by using above equation. If the sink electrode is at a finite distance from the source as illustrated in the figure below
- 7. The potential at any point is the sum of the potential contributions from each current conducting electrode. Absolute potential are difficult to measure hence normally the potential difference ∆V between any two voltage sensing electrodes is measured and is given by By using the above equation the electric resistivity can be calculated for any electrode configuration . The resistivity should be independent of the both electrode spacing and geometery and same for a uniform material but if the material is non uniform the electric resistivity will vary with the electrode position. Actually the apparent resistivity does not represent the average resistivity of the material and hence negative values can be possible. It does however provide a means of scanning a region for resistivity variations and hence offers the possibility of tomographic imaging. The depth of penetration of the current increases as the separation of the current electrodes is increased and hence it is possible to probe the material to different depths. In general the depth of penetration is limited to about half the electrode separation with features close to the surface having a greater influence on the current path (MC Phillipson et. All) Lock in Amplifier: By using the lock-in amplifier the effects of noise can be subsequently reduced and therefore it improves the signal to noise ratio. The main advantages are that it responds to the frequency of interest, and the reference frequency can be chosen to minimize the effect of 1/f and to avoid strong interfering noise signals.Actually, the lock-in amplifier is a phase sensitive detector which performs the following additional functions: o Phase shifting of input signal with respect to reference signal o Amplification and filtering of input signal. o Narrow bandwidth detection
- 8. Exercise 3: To calculate the apparent resistivity and by plotting the graph between resistivity and position find out if there is any archaeological objects hidden inside the ground. Solution: It is given that Array (current injection probes) separation 2L =6.5 m Voltage probes separation 2l=0.2 m Injected current =0.0002 A The provided data are illustrated in columns 1,2 of the following table. Position of mid point of potential electrodes from LHS of site. DVmeasured (m) (V) 0.5 0.14 1.0 0.14 1.5 0.10 2.0 0.22 2.5 0.60 3.0 0.20 3.5 0.16 4.0 0.10 4.5 0.11 5.0 0.16 5.5 0.30 6.0 1.10 Since the distance ‘x between mid points of current and potential electrodes Is not zero.Hence resistivity can be calculated using the formula: π ( L2 − x 2 ) 2 ∆V ρ= meters*Volts/Ampere 2l ( L2 − x 2 ) I Where L=3.25m, 2l=0.2m, I=0.002A, x=L –Potential Electrode (LHS position)
- 9. To calculate the distance ‘x’ and resistivity we used spreadsheet file and below is the table showing the results x' Position of Distatanc mid point of e from potential the centre electrodes of from LHS of Electrode site. DVmeasured s Resistivity (m) (V) (m) Meter*V/A 0.5 0.14 2.75 545.987137 1.0 0.14 2.25 2128.743182 1.5 0.10 1.75 3242.469482 2.0 0.22 1.25 11542.92394 2.5 0.60 0.75 42358.55263 3.0 0.20 0.25 16299.32189 3.5 0.16 -0.25 13039.45751 4.0 0.10 -0.75 7059.758772 4.5 0.11 -1.25 5771.461968 5.0 0.16 -1.75 5187.951171 5.5 0.30 -2.25 4561.592533 6.0 1.10 -2.75 4289.898934 Table :Calculated Resistivity and distance x values from spreadsheet The graph of the resistivity versus position is shown in figure 4. Resistivity Vs Distance 'x' 45000 0.75, 42358.55263 40000 35000 30000 Resistivity 25000 Series1 20000 15000 10000 5000 0 -4 -3 -2 -1 0 1 2 3 4 Distance 'x' Figure 1: Resistivity Vs Distance ‘x’ Results: From the graph it is clear that between the region -0.25 to +1.25 there is a change in the resistivity of the region and at distance around 0.75 m is a clear indication of some changes in the layer composition.Actually electrical
- 10. resistivity of stones ,rocks and hydrocarbons are much higher than the soil.Therefore between region of -0.25 to +1.25 there may be some valuable archaeological objects hidden beneath the position. Exercise 4:Vertical electrical sounding (VES) Vertical electrical sounding or electrical drilling is a detailed method to find out the information on the vertical succession of different conducting zones and their individual thickness and true resistivity and the method is based on the four point probe method.(Schlumberger array).(http://www.geo- serv.de/geoelec_VES_water.html) In order to achieve vertical electrical sounding the relative spacing between the electrodes should be maintained same and the position of the electrodes should be expanded over a central fixed point . Actually by expanding the electrodes the electric field generated by the injected current electrodes can be increased vertically or down the ground and this method can be used to find the objects deeper inside the ground.The technique can be describe by the following diagram.

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