in this video we're going to talk about how to calculate the voltage gain of this npn transistor amplifier so here's the formula that we need to calculate it the voltage gain is equal to the ac collector resistance divided by the sum of the ac emitter resistance and the emitter resistor capital re well to calculate the ac collector resistance it's equal to rc and it's equal to the parallel combination of rc and the load resistor now we don't have a load resistor at the output so in this case the load resistor is infinity so therefore the ac emitter resistance is the same as capital rc the collector resistor for this particular problem so we have rc is 1k or 1000 ohms now we have our e the emitter resistor is 100 ohms what we need to calculate is the ac emitter resistance and before we can do that we need to calculate ie by the way the ac emitted resistance is equal to 25 millivolts divided by ie now before we can calculate the emitter current we need to calculate ve the emitter voltage with respect to ground and before we could do that we need to calculate vb and before that ib so let's calculate the base current first for this particular circuit this is the form that you need to calculate the base current it's vcc minus vbe divided by rb plus beta plus 1 times re now for those of you who might be interested in how i got that formula there's another video that i created on youtube entitled emitter feedback bias circuit which explains how to derive that formula for those of you who are interested check out the links in the description section below of this video because i'll post that video in that area so you can quickly take a look at that if you're wondering where to get this formula now vcc the collector's supply voltage that's 9 volts vbe the voltage between the base and the emitter of the transistor that's typically 0.7 volts the base resistor rb that's a hundred kiloohms beta for this transistor is a hundred so beta plus one that's 101. now re is 100 ohms but because rb is in kiloohms we want the units to match so we're going to convert that to kiloohms to convert ohms into kiloohms divide by a thousand 100 divided by a thousand is point one so re is point one kilo ohms so we have nine minus point seven that gives us a voltage of eight point three volts and then a hundred plus 101 times 0.1 gives us a total ac resistance ac emitted resistance of 110.1 kiloohms so 8.3 divided by 110.1 that's going to be .0754 milliamps when you divide volts by kiloohms you're going to get the current in milliamps now there's many ways in which you can calculate ie given iv as i mentioned before you can calculate vb ve and then ie but this is an easier way of calculating ie from ib instead of going through all those steps ie is the sum of ic and ib and ic is beta times ib so if you factor out ib ie is basically ib times beta plus one so that's going to be five .0754 milliamps times a hundred plus one or 101 so that's ie which i'm gonna write here that's 7.615 milliamps so now that we have the value of ie we can now calculate the ac emitter resistance so it's 25 millivolts divided by 7.615 milliamps so 25 divided by 7.615 this will give us a resistance of 3.28 but we'll round it to 3.3 ohms so now we can calculate the voltage gain the ac emitter resistance is 3.3 and re is a hundred so the voltage gain that is the ratio between the output voltage and the input voltage that's applied here that's going to be a thousand divided by 103.3 and so you could round that to 9.7 so that's going to be the voltage gain of this particular circuit now there's one more thing i do want to mention regarding this particular circuit and that is that our e is significantly larger than the ac emitter resistance when that happens you can approximate the voltage gain and assuming that rl is infinity or much higher than rc the voltage gain is approximately equal to rc over re and that's a quick and simple way to approximate the voltage gain if the load resistance is very very high relative to rc and if re is significantly higher than the ac emitter resistance notice what the approximation will be so if we take our one kiloohm resistor which is a thousand ohms and divided by 100 ohms we'll get a voltage gain of around 10 which is a good approximation of 9.7 so it's a quick and simple way to estimate what the voltage gain will be if you have an emitter resistor that is significantly higher than the ac emitter resistance now in this example problem we have the voltage divider by a circuit beta for the transistor is 200 and we're going to say that rc is 510 ohms for this problem our e is going to be a 220 ohm resistor r2 we're going to set that to 47 kiloohms and r1 is going to be 220 kilo ohms now we do have a load resistor for this problem and we're going to make it 10 kilo ohms cb is the bypass capacitor and for each one we're going to make it 1000 microfarads so that it doesn't introduce any significant capacitive reactants to the circuit go ahead and calculate the voltage gain of this amplifier and given the input voltage of the signal when the source is connected to the circuit calculate the output voltage across the load resistor feel free to pause the video and try this now the first thing that we need to do is we need to calculate the base current and the formula that we need to calculate the base current given a voltage divider by a circuit like what we have here it's vcc times r2 over r1 plus r2 minus vbe divided by r2 times r1 over r1 plus r2 and then plus beta plus 1 times re so let's plug in everything that we have the collector supply voltage is 12 r2 is 47 kilo ohms and the sum of r1 and r2 that's going to be 267 kilo ohms vbe as always will be 0.7 and then r2 times r1 so 47 kilo ohms times 220 kilo ohms divided by the sum of those two which is 267 and then beta plus one beta in this example is 200 so we have 201 times re and we want re to be in kilo ohms so dividing that by a thousand that's going to be .22 kilo ohms so 12 times 47 divided by 267 minus 0.7 that gives us a voltage of 1.412 volts and then 47 kiloohms times 220 kilo ohms divided by 267 kilo ohms that's 38.73 kiloohms and then plus 201 times 0.22 we're going to get 82.95 kilo ohms volts divided by kiloohms will give us the current in milliamps so 1.412 divided by 82.95 this gives us a base current of 0.017 milliamps so that's how we can calculate the base current in this example now that we have the base current we can now calculate the emitter current and as we know the emitted current is equal to beta plus one times ib so beta plus 1 that's 201 ib is 0.017 milliamps so the emitter current is 3.417 milliamps so now that we know the emitter current we can calculate the ac resistance the ac resistance is going to be 25 millivolts divided by the emitter current when you divide milli volts by milliamps you're going to get the resistance in ohms so the ac emitter resistance is 7.32 now the voltage gain is going to be rc over the ac emitter resistance plus the emitter resistor now let's calculate rc rc is going to be the parallel combination of that rc and rl so it's rc times rl divided by the sum of those two because this time we do have a load resistor and we need to take that into account so rc in this example is 510 ohms our l is in kiloohms but we need the units to match so to convert kiloohms to ohms multiply by a thousand so rl is 10 000 ohms and then the sum of rc and rl that's 10 510 ohms so thus rc is 485.3 ohms now notice that we have a bypass capacitor across re and since the bypass capacitor is 1000 microfarads it basically shorts re when an ac current passes through it so re provides resistance to the dc component that goes through it but the ac component will effectively bypass re so thus we can ignore the effect of re it won't have any effect on the voltage gain of the circuit so all we need to do is put the ac emitter resistance of 7.32 here so it's 485.3 divided by 7.32 thus the voltage gain for this circuit is 66.3 so notice the effect of the bypass capacitor it increases the voltage gain of the circuit by decreasing the overall ac emitter resistance now let's calculate the output voltage of this circuit since we know the input voltage the voltage gain is the ratio of the output voltage to the input voltage so to calculate the output voltage multiply the input voltage by the voltage gain so for this circuit we have an input voltage of 15 millivolts and we're going to multiply by 66.3 so the output voltage should be approximately 994.5 millivolts or approximately one volt so that's how you can calculate the output voltage given the voltage gain and the input voltage now one thing i do want to mention regarding the bypass capacitor because i mentioned that adding the bypass capacitor decreases the overall ac resistance which increases the voltage gain of the circuit which is good however there is one drawback sometimes this could lead to distortion because the gain of the entire circuit the voltage gain is not stabilized let me explain why so with the bypass capacitor present the voltage gain is now rc divided by the ac emitted resistance now let's say that rc is a thousand and the ac emitter resistance is four this would be a voltage gain of 250. now as we have variations in the base current due to the ac signal when the base current goes up the emitter current will go up because the base current is proportional to the collector current and the collector current is approximately equal to the emitter current now keep in mind the ac emitter resistance is 25 millivolts divided by ie so it's inversely related to ie which means that as ie goes up the ac meter resistance goes up i mean it goes down now that is inversely related to the voltage gain as the ac meter resistance goes down the voltage gain goes up so let's say if ib doubles in value that means ie is going to double in value approximately and then the ac emitter resistance is going to decrease by a factor of two so if it decreases by a factor of two it's going to drop from four to two which means the voltage gain is 500 notice how much the voltage gains change by from 250 to 500 which means it increased by a hundred percent so that's huge as you can see the gain is not stabilized but if we have a value for capital re notice how the gain is stabilized so let's say initially the ac mid resistance is four and capital re will say it's 50. so this would be a thousand divided by 54. so the gain will be 18.52 now let's say that the ac emitter resistance changes from four to two capital re will be the same and so now the voltage gain is 19.23 so that's an increase of approximately four percent which is not that bad compared to an increase of 100 so add in re without the bypass capacitor it stabilizes the gain now this is a special type of amplifier known as a swamped amplifier which is a gain stabilized amplifier and so we're going to have two resistors at the emitter one is going to be bypass but the other will not be so the circuit looks like this so you have r1 and r2 this is rc this is re and this is capital re and then we have our bypass capacitor and then we have our collector supply voltage and the ground for this particular circuit which is called a swamped amplifier or a gain stabilized amplifier it's going to be equal to rc the voltage gain is rc divided by the ac emitter resistance plus this particular emitter resistance so this one is not taken into account because it's been bypassed by cb so with this circuit the gain is stabilized due to the addition of this resistor so that's how you can calculate the voltage gain of a swamped amplifier keep in mind rc is the parallel combination of capital rc and the load resistor if the load resistor is not there it's infinity rc is equal to capital rc so that's it for this video now you know how to calculate the voltage gain of a transistor amplifier thanks for watching