Voltage divider biasing common emitter amplifier is one of the core contents in analog circuit curriculum, and almost all of traditional textbooks apply approximate calculation method to estimate all characteristic parameters. In calculating quiescent point, transistor base current is generally ignored to get the approximate base potential and emitter current, then other operating parameters, and AC small signal parameters can be acquired. The main purpose of this paper is to compare traditional and Thevenin equivalent methods and to get the difference of the two methods. A Formula is given to calculate the error of the traditional method. Example calculating reveals that the traditional method can generate an error about 10%, and even severe for small signal amplifier with higher quiescent point.
An amplifier is one of the most important contents of electronic circuit systems. The main reason is that almost all the analog signals from the sensors are very weak and could not drive loads directly. The main function of the amplifiers is amplifying the weak signals so that the signal can become strong enough for practical applications. How to improve the characteristics of amplifiers is always one ongoing problem. Lots of scholars have carried out a lot of work from different directions.
We can classify the research into two different fields. The first region is the study of new material devices. In this region, H. C. Lin and G. M. Rebeiz [
As shown in
If we can determine that the transistor is biased in active forward region, we can continue the design or analysis of dynamic situation using hybrid ∏ equivalent circuit.
As we mentioned above, we can use different methods to calculate the quiescent point currents and voltages. Then, three questions are standing in front of us, they are:
1) If we ignore the base current as we described in method 1, what is the significant level of the effect to the quiescent point currents and voltages?
2) How can we get a formula so that we evaluate the error range of the first method?
3) If there is a relation between the error of the quiescent point currents and voltages and the quiescent point currents and voltages?
In this paper, we will give the answers for the three questions above.
For the integrity of this paper, let us give a short brief review for the first approach to evaluate the quiescent point currents and voltages.
For the circuit shown in
The voltage between collector and emitter VCE can be calculated as shown in Equation (3). Base current IB can be evaluated as in Equation (4).
V B ≈ R 2 R 1 + R 2 ⋅ V CC (1)
I C ≈ I E = V B − V BE R E (2)
V CE = V CC − I C R C − I E R E ≈ V CC − I C ( R C + R E ) (3)
I B = I E 1 + β = V B − V B E R E ( 1 + β ) (4)
If we do not ignore the base current IB, we can use Thevenin equivalent methods to find quiescent point currents and voltages. The basic equivalent circuit is shown in
V T H = R 2 R 1 + R 2 V C C (5)
R T H = R 1 ∥ R 2 (6)
V T H = I B Q R T H + V B E + I E Q R E = I B Q R T H + V B E + ( 1 + β ) I B Q R E (7)
I B Q = V T H − V B E R T H + ( 1 + β ) R E (8)
By comparing the quiescent point current IB and IBQ according to Equations ((4) and (8)), we can get the absolute error IBQ_Err of IB due to the approximate calculating as shown in Equation (9).
I B Q _ E r r = I B Q − I B = V T H − V B E R T H + ( 1 + β ) R E − V B − V B E ( 1 + β ) R E (9)
Notice that V T H − V B E equals to V B − V B E , hence I B Q _ E r r can be written as Equation (10)
I B Q _ E r r = ( V T H − V B E ) ( 1 R T H + ( 1 + β ) R E − 1 ( 1 + β ) R E ) (10)
So we can get the relative error of quiescent point current IB from the first method as shown in Equation (11)
E r r o r = I B Q _ E r r I B Q = | 1 R T H + ( 1 + β ) R E − 1 ( 1 + β ) R E 1 R T H + ( 1 + β ) R E | = R T H ( 1 + β ) R E (11)
From Equation (11) we can see that bigger RTH means bigger error which is caused by first method. According to the parameters in
IB (μA) | IB error | IC (mA) | IC error | VCE (V) | VCE error | ||
---|---|---|---|---|---|---|---|
Circuit from [ | Method 1 | 27.0 | 5.4 μA | 2.70 | 0.54 mA | 3.52 | 1.29 V |
Method 2 | 21.6 | 25% | 2.16 | 25% | 4.81 | 26.8% | |
Circuit from [ | Method 1 | 27.5 | 3.4 μA | 1.65 | 0.2 mA | 7.75 | 0.9 V |
Method 2 | 24.1 | 14.1% | 1.45 | 14.1% | 8.65 | 10.4% |
R2 (KΩ) | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|
IBQ (μA) | 2.66 | 5.84 | 8.82 | 11.60 | 14.22 | 16.67 |
IB (μA) | 2.96 | 6.63 | 10.18 | 13.61 | 16.95 | 20.18 |
Error (%) | 11.36 | 13.41 | 15.40 | 17.33 | 19.19 | 21.00 |
R2 (KΩ) | 11 | 12 | 13 | 14 | 15 | 16 |
IBQ (μA) | 18.99 | 21.17 | 23.24 | 25.19 | 27.05 | 28.81 |
IB (μA) | 23.31 | 26.35 | 29.31 | 32.18 | 34.97 | 37.68 |
Error (%) | 22.76 | 24.46 | 26.12 | 27.72 | 29.28 | 30.80 |
From the results as shown in the two examples above and Equation (11), if we want the relative error of IB lower, we need lower resistance ratio between RTH and (1 + β)RE. In other words, the biasing resistances should be smaller. As we all know, smaller biasing resistance means higher biasing current which cause higher DC power consumption. Furthermore, it can decrease the input resistance and hence the source voltage amplification factor will be decreased. This kind of effects causes lower performance of amplifiers.
In order to further reveal the relation between the value of relative error of IBQ and quiescent point IBQ, we keep the resistance of R1 and sweep the resistance of R2 from 5 KΩ to 16 KΩ, get the value of IB from first method and IBQ from the second method, and the relative error as shown in
From
It should be noted that the reason which causes the error is the ignorance of the current through the biasing resistors in the first method. When the resistance called RTH increases, the current IBQ will decays. On the other hand, the increasing of the degradation resistor called RE can make the error of IBQ stable because of feedback effect.
In this paper, we compared the two common-used methods to the solution of voltage-divider common-emitter amplifier, giving the formula of the difference of the quiescent point base current. From the formula and some simulation results we can know that the higher quiescent point means more significant error of quiescent base current. The contribution of this paper can improve the understood feature of this theory and avoid significant errors in determining the quiescent of bipolar junction transistor amplifiers and furthermore, help the designers to character the features of amplifiers for the AC analysis. We will discuss some more complex circuits for the two methods studied in this paper in the future to avoid errors in the design process.
This work is supported by the Project of Education Department of Henan Province ( 12A 510020), 2015 National College Students’ Innovative Entrepreneurial Training Program from Department of China (12129), the Science and Technology Project of Henan Province (172102210455), and Project of Graduate Student Research Innovation Fund from Xinyang Normal University (2016KYJJ35).
Chen, X.W., Xue, J.J., Xie, S.B., Huang, W.X., Wang, P., Gong, K. and Zhong, L.J. (2017) Error Analysis of Approximate Calculation of Voltage Divider Biased Common-Emitter Amplifier. Circuits and Systems, 8, 247-252. https://doi.org/10.4236/cs.2017.810017