Transient and Frequency Response of D’ Arsonval Galvanometer

Instructions

Finding galvanometer constants

- First connect the circuit diagram properly through the connecting dots (black dots) according to below instruction.
- Note: Example: connection point 1 - connection point 2 (drag the wire from connection point 1 by pressing left mouse button and release on connection point 2).
- First the galvanometer would be connected in series with the resistance box R (Fig. 1.).
- 9-17, 17-7, 7-15, 8-11,12-21, 19-20, 18-13, 14-22, 23-6, 6-16
- Note: Any wire connection can be deleted by clicking on the connected wire if required.

- Click on 'Check Connection' button to check whether the connection is proper or not.
- Click on the key of 'K₃' to open it.
- Then click on the mcb and switch on the light by clicking on the red switch.
- The reflection from galvanometer (yellow spot) will be seen on the scale.

P, Q, R are the resistance boxes.
- Keeping R to zero value adjust the values of P (0.2 Ω) and Q (1670Ω) by respective rotating knobs to achieve 20 cm deflection on scale.
- Note: To rotate any knob put the mouse cursor on the knob handle (blue line on the knob), a hand symbol will be showing. Press left mouse button, rotate clockwise to increase or anticlockwise to decrease values.
- Note: If the desired value does not appear while rotating the knob in one attempt, try rotating it back and forth to reach the correct value.
- (Note: The numbers, written near the knobs (in Ω or kΩ), shows the multiplying factors of respective knobs.)
- Now open the switch K₂ by removing the connection 13-18 through clicking on it.
- After clicking on the connection 13-18, one alert message will come. Keep ready the stop watch in mobile.
- Click on the confirmation alert message and start the stop watch together.
- Observe the oscillation of the light spot on the scale. Count number of oscillations and stop the watch when the oscillation stops.

- Enter the number of oscillations observed and time of the stop watch in the right hand side box under the scale.
- Click on 'Show Table' button , then on 'Time period'.
- The observation table will be shown at the bottom of the page. Switch off the light.
- Click on the switch 'K₃' to open it and again connect 13-18.
- Switch on the light.
- Disconnect the connection 13-18 for another two times and take observation of time period following step 3.
- After taking three successive observations, calculate the average time period (T) of the galvanometer and note it. It will be required in 'Frequency Response' section.
- Write it in the input box under the table as well as in the corresponding box under the scale. Switch off the light.

- After calculation of average time period, connect 13-18 again and click on the switch 'K₃' to open it.
- Switch on the light. Deflection should be 20 cm now on scale.
- To determine coil resistance R_c (Ω) of the galvanometer, slowly increase the value of R (70 Ω) to get 10 cm deflection on scale.
- This value or R is called R_c. Write it in the corresponding box under the scale. Switch off the light.
- Bring R resistance value to zero.

- Now the galvanometer would be connected in parallel with the resistance box R (Fig. 2.).
- To make the parallel connection, delete connection 14-22, 7-17 by clicking on the wires and connect 14-23, 7-22, 22-17.
- Click on 'Check Connection' button to check whether the connection is proper or not.
- Click on the switch 'K₃' to open it and switch on the light.
- To determine external resistance R_s (Ω) for critical damping of the galvanometer, adjust the values of P (0.2 Ω) and Q (1670 Ω) by respective rotating knobs to achieve 20 cm deflection on scale.
- Now adjust R to obtain oscillatory motion (≈ 100 Ω), then decrease R (≈ 90 Ω) until the motion just ceases to be oscillatory.
- This value of R is R_s.
- Write it in the corresponding box under the scale. Switch off the light.

- Now make the series connection again, as said in step 1. Click on 'Check Connection' button to check whether the connection is proper or not.
- To determine sensitivity S (rad/ampere) of the galvanometer, keep the value of R (R_s + R_c) to a convenient value say 100 Ω.
- Set P to 0.1 Ω , open K₃ and switch on the light.
- Click on 'Show Table' button and then on 'Sensitivity'. The observation table will be shown at the bottom of the page.
- Increase the value of P to 0.2, 0.3, 0.4 and 0.5. Each time after increasing the value click on 'Show Table' button and then on 'Sensitivity' to take the observation.

- At the end of 5 observations click on 'Plot', then on 'Galvanometer current (I_g) Vs. Deflection (θ)'.
- The plot 'Galvanometer current (I_g) Vs. Deflection (θ)' will be shown with I_g in (uA) and θ in radian (hover over the plot section and then click on the camera icon located at the top right corner of the plot to download it).
- From the slope of the plot find the value of sensitivity S (rad/A) as shown in Fig. 3. The formula is as follows :
- Write it in the corresponding box under the scale.

Fig. 3. Calculation of Sensitivity

- Click on 'Calculate' button at the bottom of the left hand side box under the scale.
- The results will be shown in respective input boxes in the right hand side box under the scale.
- Note down the values of galvanometer displacement constant (G), restoring constant (K), moment of inertia (J).
- Click on 'Clear' button.
- Switch off the light. Bring all the resistances to zero value.
- These constants are calculated using the equations below.
$$G = \frac{T(R_c + R_s)}{\pi S} \ nw-m/A/A$$ $$K = \frac{T(R_c + R_s)}{\pi S^2} \ nw-m/rad$$ $$J = \frac{T^3(R_c + R_s)}{4 \pi^3 S^2} \ kg-m^2$$

Transient Response

- First connect the circuit diagram as said in step 1 of 'Finding galvanometer constants' section.
- Click on 'Check Connection' button to check whether the connection is proper or not.
- Click on 'Experiments' button, then 'Transient Response' under it.
- Click on the switch 'K₃' to open it.
- Then switch on the light by clicking on the red switch.
- The reflection from galvanometer (yellow spot) will be seen on the scale.

Adjust the values of P (0.2 Ω) and Q (1670Ω) by respective rotating knobs to achieve 20 cm deflection on scale.

- Now open the switch K₂ by removing the connection 13-18 through clicking on it.
- The under damped response for sudden disconnecting switch K₂ will be shown at the bottom of the page (Fig. 4) (click on 'Download Plot' button to download this plot).
- Calculate θ₁ and θ₂ from the response using the formula below.
$$\theta_1 = (First \ overshoot - \ Steady \ state \ value) \ and \ \theta_2 = (Steady \ state \ value - \ First \ undershoot)$$

Fig. 4. Calculation of θ₁ and θ₂

- Now put those values in the right hand side box under the scale.
- Click on 'Calculate' button in that box to calculate the value of open circuit damping ratio (δ₀)

- Switch off the light, connect 13-18 again and open K₃.
- Switch on the light. Set R to 200 Ω. The underdamped response for this value of R will be shown at the bottom of the page (click on 'Download Plot' button to download this plot).
- The response is having small intervals along Y-axis so that the peak value or θ_peak can be measured properly.

- Enter the values of θ_peak, θ_steady-state (steady-state value of the response) in the left hand side input box under the scale.
- Click on 'Calculate' button in that box to calculate the % overshoot.

- Click on 'Show Table' button, then on '% Maximum overshoot'.
- The observation table will be shown at the bottom of the page.
- Increase the value of R to 300, 400, 500.
- Each time after increasing the value of R follow step 7 and click on 'Show Table' button, then on '% Maximum overshoot' to take the observation.

- After taking 5 observations click on 'Plot' button, then on '% Maximum overshoot Vs. Damping ratio δ'.
- The plot will be shown (hover over the plot section, then click on the camera icon located at the top right corner of the plot to download it).
- Click on 'Clear' button. Switch off the light.

Frequency Response

- Once the transient response of the galvanometer is acquired, it is possible to observe its frequency response.
- First connect the circuit diagram as instructed below.
- 1-17, 17-7, 7-15, 2-11, 12-21, 19-20, 18-13, 14-22, 23-6, 6-16
- Click on 'Check Connection' button to check whether the connection is proper or not.
- Click on 'Experiments' button, then on 'Frequency Response' under it.

- Click on the switch 'K₃' to open it.
- Enter the value of average time period of the galvanometer, calculated in 'Finding galvanometer constants' section in the input box at the bottom of the page.
- Click on 'Calculate' button to calculate the natural frequency of galvanometer (ω_n).

Switch on the light. The light spot will be oscillating since the supply is 2 V ac now. Frequency is now 0.1 Hz.

Click on 'Show Table' button, then on 'Oscillation Amplitude'. The observation table will be shown at the bottom of the page.

- Now increase the frequency of input signal through the frequency knob of function generator to 0.2 Hz.
- Click on 'Sine' button and repeat step 4.
- Now increase frequency by step of 0.1 Hz upto 10 Hz.
- Each time after increasing the value click on 'Sine' button and repeat step 4.

- After taking observations click on 'Plot' button, then on 'Amplitude (cm) Vs. Frequency (Hz)'.
- The frequency response plot will be shown for a particular damping ratio (δ (e.g. 0.47 for R = 500 Ω)).

- Click on 'Clear' button. Switch off the light.
- Change the value of δ (say 0.6, 0.96 ≈ 1) by changing the value of R (say 400, 200 Ω) and follow steps 4-6 again to observe the frequency response for different damping ratios (δ). Compared plot can be observed for five distinct values of δ.