Measurement of resistors: voltage divider

Introduction

The color coded in resistors are difficult to read. Thus, we will use the ADC in Arduino and voltage dividing circuit to measure the resistances.

Theory: Voltage divider

Voltage divider is a simple circuit with two resistors, one known and other unknown.

Voltage divider is a simple circuit with two resistors, one known and other unknown. According to Kirchhoff voltage law, all potential in the circuit will be used. If we input +5V into a circuit, and the known resistor uses $U_{1}=R_{1}I$, the other has $U_{2}=R_{2}I$ to be used. Also, we know that $U=(R_{1}+R_{2})I$ which gives the current in the circuit to be $I={\frac {U}{R_{1}+R_{2}}}$. Furthermore, this gives that $U_{2}={\frac {R_{2}}{R_{1}+R_{2}}}U$.

The potential (voltage) $U_{2}$ can be read using Arduino's Analog to Digital Converter (ADC) in the analog input ports.

Theory: Arduino circuit

Voltage divider.

The input voltage is $U=+5V$. The input to the A5 pin is called $U_{5}$.

${\begin{aligned}U_{5}&=U-U_{1}\\&=U-{\frac {R_{1}}{R_{1}+R_{2}}}U\\&=U\left(1-{\frac {R_{1}}{R_{1}+R_{2}}}\right)\\&=U\left({\frac {R_{1}+R_{2}}{R_{1}+R_{2}}}-{\frac {R_{1}}{R_{1}+R_{2}}}\right)\\&=U{\frac {R_{2}}{R_{1}+R_{2}}}\\U_{5}(R_{1}+R_{2})&=UR_{2}\\U_{5}R_{2}-UR_{2}&=-U_{5}R_{1}\\R_{2}(U_{5}-U)&=-U_{5}R_{1}\\R_{2}&={\frac {U_{5}}{U-U_{5}}}R_{1}\end{aligned}}$.

or similarly

${\begin{aligned}U_{5}&=U_{2}\\&={\frac {R_{2}}{R_{1}+R_{2}}}U\\A_{U}(R_{1}+R_{2})&=R_{2}U\\A_{U}R_{1}+A_{5}R_{2}&=R_{2}U\\A_{U}R_{2}-UR_{2}&=-U_{5}R_{1}\\R_{2}(A_{5}-U)&=-U_{5}R_{1}\\R_{2}&={\frac {U_{5}}{U-U_{5}}}R_{1}\end{aligned}}$

which gives the same result.

Theory: Arduino program

• 

  $R_{1}=800\Omega$ and $R_{2}=800\Omega$ as a reference data.

• 

  $R_{1}=210k\Omega$ and $R_{2}=1.5M\Omega$

• 

  $R_{1}=1.5M\Omega$ and $R_{2}=2M\Omega$

The simple program to show $R_{2}$. Arduino has 10 bit analog to digital converter, thus there are $2^{10}=1024$ different values for analog pins, such as $A_{5}$.

int raw = 0;
float R1 = 1000;  //Known resistance
float R2 = 0;     //Unknown resistance, we have to measure
void setup(){
  Serial.begin(9600);
}
void loop(){
  raw = analogRead(5);
  if(raw){
    R2 = raw/(1024.0-raw) * R1;
    Serial.print("R2: ");
    Serial.println(R2);
} }

The decimal in 1024.0 makes it to float, and thus the result will be float. Note that the input voltage +5V cancels out, because $U_{5}=U\times A_{5}/1024$ and thus

${\begin{aligned}{\frac {U_{5}}{U-U_{5}}}R_{1}&={\frac {UA_{5}/1024}{U-UA_{5}/1024}}R_{1}\\&={\frac {A_{5}/1024}{1024/1024-A_{5}/1024}}R_{1}\\&={\frac {A_{5}}{1024-A_{5}}}R_{1}\end{aligned}}$