Feb 3: Thevenin Equivalent Circuit

Connection to Power Line vs. Arduino Digital Output

Lego Motor Exercise:
-Observation: Powering the LEGO motor from an Arduino output pin (digital pin 7) had less stalled torque than when powering the motor directly with a power source. This meant that less resistance was needed to stall the motor arm. However, the difference was not significant.


-Voltage between the power and ground busses with an oscilloscope ...
... with motor connected = 5 V 
... without motor connected = 5 V
In addition, on the oscilloscope, the voltage for both cases looked the same: nearly vertical line at 5V. 


-Voltage between the Arduino output pin and ground busses with an oscilloscope ... (We created the code based on the "Blink" code that was already part of the Arduino package): 

... with motor connected = curve shown below with the maximums of these "bumps" at 5 V; not a constant flow of 5 V

... without motor connected = 5V (nearly straight line observed on the oscilloscope)

Conclusion: The output pins (digital) on the Arduino are not very good at powering a LEGO motor (vs. the 5V power line). 


The Thevenin Model
In theory, one could represent any (linear) complex arrangement of batteries and resistors inside a "black box" with an equivalent circuit:

Thevenin equivalent of an input and output

*Note: One can think of this as a voltage divider! This is powerful because it simplifies complex circuits and is general, meaning that one can use the Thevenin equivalent circuit for the output and just change the input (the load).

Exercises

The Thevenin Equivalent Circuit of a Power Supply
-Summary: Applied 5V to one bus and ground to the other and used a multimeter to monitor the change in voltage between the power and ground busses as we connected and disconnected a 47 ohm resistor between the busses. 
-Data: 
Voltage (when 47-ohm resistor connected) = 5.0 V = V-load
Voltage (when 47-ohm resistor disconnected) = 5.1 V (open circuit) = V-th
-Calculations:
V-load = (Rload/(Rth+Rload))* V-th
Rth = 1.4 ohms
-Result: 

Thevenin equivalent circuit for the 5 V supply on the Arduino board

The Thevenin Equivalent Circuit of an Arduino Digital Output Pin
-Summary: Used the code from before to use the Arduino digital output pin 7 at HIGH. 
-Data: 
Voltage (when 47-ohm resistor connected) = 3.1 V = V-load
Voltage (when 47-ohm resistor disconnected) = 5.1 V = V-th
-Calculations:
V-load = (Rload/(Rth+Rload))* V-th
Rth = 31 ohms
-Result: 

Thevenin equivalent circuit for the Arduino output pin

The Thevenin Equivalent Circuit of a LEGO Motor
-Summary: Determined the Thevenin equivalent circuit of a non-spinning LEGO motor. 
-Data: 
Voltage (when 47-ohm resistor connected) = 3.1 V = V-load
Voltage (when 47-ohm resistor disconnected) = 5.1 V = V-th
-Result: 

V load = 2.2 V
V th = 5.1 V
R th = 31 ohms
R load = 23 ohms

*To check our work, we measured the resistance between the two pins of the motor directly, which was 29 ohms, which was relatively close to 23 ohms.


Important Formulas (for determining V-th and R-th for an artbitrary circuit) 
*See notes for more detailed explanations.

V-th = V-oc
(oc = open circuit)

R-th = (V-oc / I-sc)
(sc = short circuit)


Thevenin Equivalent Circuit for a Voltage Divider

R-th = (V-oc / I-sc) = (R1*R2)/(R1+R2)
*Note: R-th = R1 || R2


Voltage Divider (Exercise)

-Powered the circuit using the Arduino board 5 V pin. Measured the open circuit output voltage (which was 2.5 V = V-oc = V-th) When we attached a 10 k load, V-load = 1.7 V.

Two-thirds of the voltage drop is across the 10 k resistor, which corresponds to (2/3)(2.5 V) = 1.7 V, which agrees with our experimental result.

-Short circuit current = I-sc = 0.5 mA (which makes sense because 5 V/10k ohms = 0.5 mA).

(Measured current by using an ammeter to find I-sc.)

Calculations:

R-th = 2.5V / 0.0005 ohms = 5k Ohms
(Check: V-th = R-th (I-sc) = 2.5 V)

From I-sc and V-oc, the Thevenin equivalent circuit is: 

(equivalent circuits)

We then built the Thevenin equivalent circuit using a variable regulated DC supply as the voltage source. We used a 4.7k Ohm resistor (nearly 5k, which was R-th) and set the input voltage as 2.5 V (V-th) using the DC supply and voltmeter.

When we attached a 10k load, just as we did with the original voltage divider, we got identical results: V-load = 1.7 V and I-sc = 0.5 mA.