What is the difference between a transistor and a resistor?

Here is an explanation about the Resistor to Transistor Transition by Balajee-Seshadri Sir.

      Let us start from a simple circuit.

           In General, Voltage source is not shown in Circuit diagrams. So let us remove the Voltage source from the circuit diagram. Also Voltage and Resistance are given, we need to calculate Current.

          Now let us split the single Resistor in to two Resistors and connect the Resistors(R1 & R2) in Series.

          Since two resistors are connected across a single voltage source, the voltage is divided across the resistors. These voltages are V1 and V2.

          Now let us find the output voltage Vo.

         Now let us  make R2 as Zero ohm. i.e. No resistance, we are going to short circuit the R2. Now the output Vo is 0V, because the output is shorted with Ground. There is no Potential Difference.  

       Now let us make R2 as Infinity Ohm. i.e. Remove R2 and make Open circuit. Since circuit is open no current flow. Hence I = 0A. So V2 = I * R2.
   V2 = 0  *  ∞ = Indeterminate. That means anything. So you can not predict. Then how will you calculate Vo? We may not able to find out V2 but we can find out V1. So Vo = V - V1.  V1 = I * R1 = 0V. So, V0 = V

      Now let us replace R2 with a variable resistor.  If the R2 varies between 0 ohm to ∞ Ohm, then V0 will vary between 0 to V. So V0 can never be less than 0 and More than V.

       When we vary the resistance R2, the current I also varies. Let us assume we can not measure R2 but we can measure I then Vo = V - I * R1. R2 is not used in the calculation.

    Now let us rename the Current I as Iy. 

         Now let us control the variable resistor using Ix. The relation between Ix and Iy is 
                                 Iy = B x Ix

     Now we generate Ix using Vx and Rx. 

                                   Ix = Vx/Rx

         Now let us introduce a battery in between Rx and Ground. So the voltage across resistor Rx is Vx -Ve. So, Ix = (Vx - Ve)/Rx.

       Now let us rename the parts.
                 Vx -> Vb
                 Rx -> Rb
                 Ix  -> Ib

                 Ve -> Vbe
                 Iy  -> Ic
                 R1 -> Rc
                 V1 -> Vc
                 V  -> Vcc

                 B -> β

    Now we change the variable resistor block as Transistor.

              This Transistor acts as a Regulator (Variable Resistor)  or a switch(ON/OFF). 

              The Transistor has 3 operating modes.

               1. Cut-off  (Switch - OFF)
               2- Saturation (Switch - ON)
               3- Active (Regulator).

        1. Cut-off Mode

        Vb < Vbe (Generally 0.7V)
         So Ib = 0A 
              Ic = 0A
              Vc = Ic x Rc = 0V
              Vo = Vcc - Vc = Vcc

      2.  Saturation Mode

        Ic > Ic.max
        Ic.max = Vcc/Rc
        Ic = β Ib
        Ic = Ic.max
        Vc = Ic x Rc = Vcc
        Vo = Vcc - Vc = 0V

       3. Active Mode

        0 < Ic < Ic.max

        Ib = (Vb - Vbe)/Rb
        Ic =  β x Ib
        Vc = Ic x Rc
        Vo = Vcc - Vc
    
         0V < Vo < Vcc

      When a Transistor acts as a Regulator, it is called an Amplifier.
      When a Transistor acts as a Switch, it is called a Gate.

       Transistor in Active Mode - Analog Electronics
       Transistor in Cutoff/Saturation Mode - Digital Electronics