dear students this is the continuation of previous lecture video effect of negative feedback or the properties of negative feedback in this video we are going to cover the last three properties bandwidth extension increase the input resistance and decreased output resistance okay let's start with bandwidth extension property as we know that bandwidth is the difference between two cutoff frequencies those are lower cutoff frequency fel and upper cutoff frequency of you so it is the range of frequencies between fl and fu due to the introduction of the negative feedback in an amplifier the bandwidth is extended or increased okay so for that we can consider the diagram this is the frequency response of the amplifier with feedback this one is the frequency response of the amplifier without feedback from this diagram we come to know that the bandwidth of the amplifier with feedback is increased from without feedback range so here fel and fu both represent the lower cutoff frequency and upper cutoff frequency for that amplifier without feedback here fpl dash and fu does represent lower cutoff frequency and upper cutoff frequency for the amplifier with feedback so in case of lower cutoff frequency the value is decreased due to that negative feedback in case of upper cutoff frequency the value is increased due to this negative feedback so we are going to prove this property the first step is to find the lower cutoff frequency of feedback amplifier f l dash okay the gain at low frequency can be expressed as a l is equal to am by 1 minus j f l by f so here am represents the mid band gain a l represents the gain at low frequency without feedback here fvl is the lower cutoff frequency without feedback do you all understand this is the relation between a l and am so in this low cut-off frequency gain the ratio is f l by f here this f is the operating frequency for that amplifier okay next we are going to find out the gain of the amplifier with feedback at lower cutoff frequency it is represented as a l f that is equal to a l by 1 plus a l beta as we know the value of a l we can substitute that value in this formula then we can get a l f is equal to am by 1 minus j fl by f that is the value of a l divided by 1 plus a l is replaced with the value a m by 1 minus j f l by f into beta okay so next we have to take this value as a common one here for that we can multiply this one with this value so alf is equal to a m by 1 minus j f e l by f divided by 1 s multiplied by this value so 1 minus j f l by f plus am beta divided by this value then we can divide this denominators okay then we can get the value as am divided by 1 we can write in this order 1 plus am beta minus j f l by f do you all understand this okay in the next step we can take this 1 plus am beta as a common one from this denominator if we are going to take this value as common then this value can be divided by this one so we can write a m divided by 1 plus a m beta as a common one so this term becomes 1 minus j this f l by f is divided by this value so f l by f into 1 plus a m beta do you all understand okay so this expression can be written as a l f is equal to a mf divided by 1 minus j f l dash by f where am f is nothing but mid band gain with feedback its value is am divided by 1 plus am beta so this value can be written as am f with feedback okay then the denominator is having that value 1 minus j f l dash by f where this f e l dash is nothing but f l divided by 1 plus a m beta that is the lower cut off frequency with feedback do you all understand now so this term fel divided by 1 plus am beta can be replaced with the term f l does okay equation f l dash is equal to f l by 1 plus a m beta clearly shows that the f l dash value is always less than f l here f l is the low cutoff frequency without feedback f l dash is the low cutoff frequency with feedback so with feedback means at low frequency range it is decreased okay that means the negative feedback decreases the lower cutoff frequency by the factor 1 plus am beta so here you can see this this is the frequency response means this one is for without feedback so its range is cfl here the frequency low cutoff frequency for this feedback amplifier is decreased by fl dashes decreased by the factor 1 plus a beta so next we are going to find out the upper cutoff frequency of a feedback amplifier f u does okay sometimes it can also be represented as f h dash can be represented either as high cut off frequency or upper cutoff frequency both are same okay for that we can consider the gain at upper cutoff frequency without feedback okay without feedback means mid band gain divided by 1 plus j f by f u so you have to remember for this high cutoff frequency its value in the denominator is plus and f by fu okay then the gain at upper cutoff frequency with feedback is obtained as a hf is equal to a h by a 1 plus a h into beta so this can also be represented as u u as well as h both are same high cut off frequency or upper cutoff frequency okay next we are going to replace the value of this a u as am by 1 plus j f by f u similarly in the denominator 1 plus a u is replaced with the value am by 1 plus j f by f u into beta then we can take the lcm value here 1 plus j f by f u plus a m beta on the whole divided by this value the denominators are divided each other then we can get a u f is equal to what am by 1 plus a m beta plus j f by f u for further simplification we can take this 1 plus a m beta as a common one in the denominator then the term becomes am by 1 plus a m beta into this term becomes 1 plus j f by f u which is divided by 1 plus a m beta so this expression can be written as a uef is equal to that is the gain of feedback amplifier at upper cutoff frequency is equal to am f divided by 1 plus j f by f u dash so here we can group this value so this is nothing but mid band gain with feedback that is represented as a m f here this fu multiplied with this 1 plus am beta is represented as f u dash okay so this is the gain of the feedback amplifier at upper cutoff frequency range is equal to fu into 1 plus am beta from this equation we can understand that that fu dust that is the upper cutoff frequency with feedback is always greater than fu so it is increased okay the upper cutoff frequency of the feedback amplifier is increased by the factor 1 plus am beta thus the overall bandwidth is increased do you all understand the next property is increased input resistance due to negative feedback in general an amplifier should have high input resistance and low output resistance so that it will not load the preceding stage so in order to avoid loading effect it should always have high input resistance okay so for that we can consider the basic structure of feedback amplifier here that actual input given to this amplifier is vi and the current flowing through this amplifier is represented as i i okay so here we can consider ri is the input impedance of the amplifier without feedback okay if there is no feedback simple amplifier circuit in this case we are giving v i is the input and v out here the current flowing through this amplifier is i i okay so in that case that input impedance is v i by i i so next we are going to introduce negative feedback to this circuit at the time what will happen due to this negative feedback the concept vf is introduced at this feedback signal the negative feedback in an amplifier increases the input resistance we are going to prove this concept so here input impedance of the amplifier with feedback is denoted as rief that is equal to v s by i i so here we can include this source voltage also do you understand whenever the feedback is introduced in amplifier we can include the source voltage okay so here we know that vi is equal to what now va is equal to vs minus bf from this we can get the value of this bs that's what given here okay as we know that va the actual input to the amplifier is equal to source voltage minus feedback voltage since it is a negative feedback so from this we can get vs is equal to vi plus vf substitute this value in this formula r i f is equal to v i plus b here v f can be replaced with the value beta into v naught correct then this v naught can be replaced with the value a into v i so in this one v i is common we can take it outside then r i f is equal to v i into 1 plus a beta divided by i i as we know that v i by ii is nothing but ra we can replace this value as ri finally we can get the input impedance with feedback is equal to input impedance without feedback multiplied with the factor 1 plus a beta thus the input impedance or resistance is increased by the factor 1 plus a beta in a negative feedback amplifier do you all understand that so next one is decreased output resistance here we can consider a small signal equivalent circuit of this amplifier so inside this amplifier we are having this input resistance ri output resistance rbo so here the current is i i the amplified output is here avi okay current through the servo is io ro is the output impedance or output resistance without feedback rof represents the output impedance with feedback so this rbo is the without feedback value inside this amplifier however we are going to introduce the negative feedback then this output impedance value is decreased so we are going to prove that concept here so first we have to find out the voltage drop across this arbo voltage drop across this ro means what io that is the voltage drop across this ro that is equal to here it is v not that is the overall output voltage minus a b i okay so that is the voltage drop across this r o do you all understand this one so i o r is equal to b naught minus a b i next we have to assume this v s is equal to 0 if you are not giving input or the source signal what will happen if you are not giving source supply signal only giving this feedback voltage then this vi becomes minus vf because it is negative feedback so this vi becomes minus vf then this term becomes v naught plus a b yeah do you all understand this the voltage drop across r naught is the ioro that is equal to v o minus a b i here we can assume v s is equal to 0 then v i becomes minus v then v i can be replaced with that value minus v of means minus into minus plus a b f v f is nothing but beta into v naught then we can take this v naught as the common one so v o into 1 plus a beta then we can move this 1 plus a beta to this side and here i know to this side we can get b naught by i naught is equal to r naught by 1 plus a beta v naught by i naught is equal to r o f that is the output impedance with feedback nothing but ro by 1 plus a beta the output resistance is decreased by the factor 1 plus a beta