 session on Hartley oscillator. Learning outcomes are at the end of session students will be able to analyze Hartley oscillator. Contains are like this. As we have seen the basic principle behind LC oscillator and Hartley oscillator is nothing but a special type of LC oscillator which consists of amplifier, then 3 impedances Z1, Z2, Z3 which forms the tank circuit and this tank circuit allows you to determine the frequency of oscillations as well as it will determine the amplitude of the oscillations. So, this Z1, Z2, Z3 will be inductor and capacitors to form the tank circuit. For Colpitt's oscillator that we have observed in the previous lecture, this Z1, Z2 where your capacitor and Z3 were inductor. Now in this case Z1 and Z2 are inductor and Z3 is capacitor. Now if you simplify the amplifier in its H model it will look like this where Hre is the input impedance of amplifier, Hfeib will be the equivalent current source for the transistor. When in Z2 replaced by the inductor and Z3 is replaced by capacitor to form the Hartley oscillator. So, this is the circuit diagram for Hartley oscillator. Best voltage divider bias technique is there to get the biasing of amplifier. RF choke is used to block AC and pass DC signal. Capacitor is used to couple the output voltage and this is coupled to the tank circuit. Output voltage is given to this tank circuit and feedback voltage is through this L2 which will be again provided at the input of amplifier. AC output oscillations are observed at the secondary winding of this inductor. L2 provides the positive feedback which gives the oscillations because this amplifier is providing you 180 degree phase shift of input signal. Same 180 degree phase shift is provided through this tank circuit. So, total phase shift is 360 degree which is same as in phase with input signal. So, we can say L2 provides the positive feedback. RF coil permits the DC and it blocks AC signal and here output is taken across inductor. As you want to vary the frequency of oscillations L1 and L2 are mutually coupled and therefore, it forms the attrotransformer. So, these can be simultaneously. As we have discussed tank circuit is formed with L1, L2 and C. One more thing when you derive the portion of the circuit of Huttley oscillator you will get this impedances Z1, Z2, Z3. So, you have the ZL like this as you know the equation for gain is minus hf e by h i into ZL. Voltage is Z2 h i by Z2 plus h i plus Z3 into i1 and this is Vf that is Z2 h i by Z2 plus h i into i1 and beta is Vf by Vm. So, all these equations we have came across the concept of single stage amplifier. So, Z3 as you know it is a capacitor. So, it is minus j by omega C, Z1 is j omega L1 plus j omega M, Z2 is j omega L2 plus j omega M. So, when you replace the values in the equation of V out and Vf of Z1, Z2 and Z3 by these values due to the capacitor inductor used over here. So, this is Z3, Z1 and Z2. So, here you will come across the equation like this h i e into Z1 plus Z2 plus Z3 plus 1 plus hf e into Z1, Z2 plus Z2, Z3 equal to 0. So, this is the equation of gain over here and now we are putting these values over here. Then you will come across the equation which consists of some real part and some imaginary part. So, here red color indicates the imaginary part that is L1 plus L2 plus 2M minus 1 by omega square C minus omega square into L2 plus M into bracket L1 plus M into 1 plus hf e minus 1 by omega square C bracket complete equal to 0. So, here we will come across the equation consisting of this imaginary part and the real part. So, here if we equate this imaginary part to 0 you will obtain the frequency of oscillations. So, that frequency of oscillation can be calculated like this L1 plus L2 plus 2M equal to 1 by omega square C. So, we can say omega square equal to 1 by L equivalent into C. So, here L1 plus L2 plus 2M is nothing but L equivalent which is the combination of L1 L2. Finally, value of f equal to 1 by 2 pi under root L equivalent into C. So, this is equation for frequency of oscillations for Hartley oscillator. Now, if you equate this part to 0 you will obtain the value of gain for Hartley oscillator. Now, any part that is L1 plus M into 1 plus hf e equal to 1 upon omega square C you know the value of omega square that is 1 upon L1 plus L2 plus 2M into C. This is the value of omega square that we have derived from the previous equation plus hf e equal to L1 plus L2 plus 2M divided by L1 plus M and here L1 plus M plus L2 plus M is written separately and here you have cancellation of this term. So, we will get the value as 1 plus L2 plus M by L1 plus M. So, here 1 get cancelled hf C should be greater than or equal to L2 plus M by L1 plus M and therefore, we can say condition for sustained oscillation is that every beta should be greater than or equal to 1 as this is the value of beta. So, we can say gain should be greater than or equal to L1 plus M by L2 plus M and this is due to the value of beta which is L2 plus M by L1 plus M. So, hence we have come across the two equations that is gain and the frequency of oscillations. Now recall in Hartley oscillator feedback is obtained from where and the answer is it is obtained from the inductor L2 and to ensure that circuit oscillates at what particular condition of gain and your answer is it is greater than or equal to L1 plus M by L2 plus M applications of Hartley oscillator. It is used to provide a sine wave of desired frequency by changing the value of tank circuit elements we can change that and we can also get the high frequency of oscillations and wide frequency of oscillations with the Hartley oscillator. As you know large frequency that is the range of frequency in megahertz with the help of Hartley oscillator. So, it is mostly used as a local oscillator in radio receivers as well as it is used for the RF oscillators as it works with the megahertz frequency range. References are like this. Thank you.