 Hello everyone, welcome to lecture on differential amplifier with two opamp. Before starting with the actual session, at the end of this session, students will be able to calculate different parameters of it, like voltage gain, input resistance and output resistance bandwidth. In previous session, we already seen the differential amplifier with single opamp. In other case, we are going to study for two opamp for differential amplifier. Now before starting with actual session, let us pause the video and think about what is the gain expression of the inverting amplifier or differential amplifier with one opamp. This is nothing but the ad equals to VU upon Vxy equals to minus RF upon R1, where ad is nothing but gain of the differential amplifier, which is and V0 is nothing but the output voltage and Vxy is nothing but the differential input voltage in the case of a differential amplifier. The same equation can be used for the inverting amplifier. In that case, the ad instead of ad, you have to use a, V0 is going to be remain same and instead of Vxy, it becomes VE because there is only single input source in the inverting amplifier case. In differential, it has two inputs where that is why the differential of that two input is nothing but the total input voltage of the differential amplifier and equals to nothing but minus RF upon R1. Minus sign indicates that the whatever the output we are getting that is a having inverse of your input. So that gain is calculated or you can say adjusted by the RF by R1. RF is nothing but the resistance in the feedback path. R1 is the resistance in series with the input source. Now let us start with the differential amplifier with opamp. If you see it has two opamps, first is a A1, second one is A2. So this figure shows differential amplifier with opamps. If you see two inputs are there Vx and Vy, Vx is connected to A2 to the non-inverting terminal, Vy is connected again to A1 to the non-inverting terminal. So RF is nothing but the feedback resistance. So if you see it has two stages, first stage is A1 which is nothing but your non-inverting amplifier and for that non-inverting amplifier, this is a Vz as you can say as output and this is a feedback resistance of that non-inverting amplifier stage that is R3. Different stage is nothing but the differential amplifier. If you see this is act as a input to the inverting terminal and one is Vx which is directly connected to the non-inverting terminal. So it is a two inputs, so that is why it is called differential amplifier stage. This one is having only single input that is a Vy to the non-inverting terminal. So that is why it is called as a non-inverting amplifier stage. So it has two stages, stage one, stage two. So if you see in these two stages it has an equal gain. You already know the equation for the non-inverting amplifier gain equation which is different than the gain equation of the differential amplifier. So let us calculate for the voltage gain of a differential amplifier with two opamps. So as we seen there are two stages which is having different gain equations. So that we can find total gain of the differential amplifier circuit with a two opamp by calculating individual gain of the two stages and addition of that. So the output of first stage we already seen which is nothing but the non-inverting amplifier which is nothing but Vz. So equation of non-inverting amplifier we have Vz equals to 1 plus R3 upon R2 divided by Vy. In this case Vy is a input to the non-inverting amplifier stage, Vz is nothing but the output, R3 is nothing but the feedback resistance of that particular stage, R2 is a one more resistance. So equation of general equation for the non-inverting amplifier is nothing but 1 plus R3 upon R2 into V in this case V in is nothing but the Vy. So this is a one equation of the output voltage for first stage. Now by applying superposition theorem to the second stage we can find out the output voltage of second stage. So second stage we already know it is a nothing but the differential amplifier. So Vo equals to minus RF upon R1 into Vz plus 1 plus RF upon R1 into Vx. So in that case this one is a 1 plus RF upon R1, RF is a feedback resistor, R1 is nothing but the input resistance and Vx is a input source which is connected to non-inverting terminal of the opamp of second stage differential amplifier and minus RF upon R1 into Vz is nothing but the voltage equation at the inverting terminal that is why the sign is over here is negative. So minus RF upon R1 into Vz, Vz is nothing but the input voltage to the second stage plus 1 plus RF upon R1 into Vx. Now if you see this equation having Vz we already calculated equation for the Vz. So by substituting this equation in this one we will get Vo equals to minus RF upon R1 in the bracket 1 plus R3 upon R2 into Vy plus 1 plus RF upon R1 into Vx. Now this is again due to the Vx, Vy and this is again due to the Vx. So in this equation we have conditions that if R1 equals to R3 and RF equals to R2 that becomes equation Vo equals to 1 plus RF upon R1 common into Vx minus Vy. So that Vx minus Vy is nothing but the difference of two inputs connected to differential amplifier. So that treated as a Vxy we already know that. So therefore the final equation becomes Vo equals to 1 plus RF upon R1 into Vxy that Vxy comes over here. So Vo upon Vxy equals to nothing but 1 plus RF upon R1. So this is nothing but the gain of the differential amplifier with two open. Now let us say second parameter that is input resistance. So input resistance RIF of the differential amplifier is nothing but the resistance determined in looking into the either one of the two input terminals with other one grounded means if you find the input resistance total input resistance of the differential amplifier you have to ground the other terminal and find the resistance of the from first terminal. Next stage ground the first terminal and find the input resistance from the other terminal. So by doing that when Vx equals to 0 means voltage Vx source input source which is connected to 0. So first stage which is nothing but the non-inverting amplifier we have the equation for the input resistance of the non-inverting amplifier from the previous sessions that equation is nothing but the RIF Y here Y indicates that the Y is connected as a inputs are in this one because we already grounded the Vx source which is having 0 voltage. So RIF input resistance due to voltage source Vy equals to nothing but RI into 1 plus AB. Here RI is nothing but open loop input resistance of the opamp 1 plus A is nothing but the gain of the amplifier B is nothing but the gain of the feedback circuit. So for that we have the equation B equals to R2 upon R2 plus R3. So this is a standard equation for non-inverting amplifier similarly when Vy equals to 0 in second stage of the differential amplifier of opamp that also becomes a non-inverting amplifier. For same input resistance equation we have RIF X, X is because of we already grounded Y input source Vy. So due to input source Vx we finding the input resistance RIF X equals to RI into 1 plus AB. Similarly for A is nothing but the gain of the amplifier B is nothing but the gain of the feedback circuit which is given by the equation R1 upon R1 plus RF. Since we already seen that R1 equals to R3 and RF equals to R2 in that case in this condition whatever that we calculated input resistance due to RIF Y and RIF X these are not equal to each other. So because of that the loading effect of the input source Vx and Vy may occurs. So in other words that the output signal may be smaller in the amplitude than expected means if you apply input signals so whatever you are expecting output that is not amplified it is actually smaller than the input one whatever your expectations. Next parameter is the output resistance with the feedback the output resistance of two configuration must be identical. In other words you can say that the output resistance of the differential amplifier should be same as that of a non-inverting or inverting amplifier. So for that we have equation ROF is nothing but equals to RO upon 1 plus A by AD where RO is nothing but the output resistance of the opamp. AD is nothing but the closed loop voltage gain of differential amplifier and A is nothing but the open loop voltage gain of the opamp. Next parameter is bandwidth with feedback as in case of the inverting and non-inverting amplifiers the bandwidth of differential amplifier is also depends on the closed loop gain of the amplifiers. So for that we have equation FF equals to unity gain bandwidth divided by closed loop gain that is AD here FF is for frequency with feedback. So unity gain bandwidth which is also called as a UGB so for that we have equation UGB equals to A into F0 where A F0 is nothing but the open loop break frequency of the opamp which is already we seen in the previous sessions. So this final equation is for the bandwidth with feedback of the opamp. These are the references thank you.