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136 2. Tutorial
Let’s apply solution-based approximation to the transfer function of the common-emitter amplifier
calculated in the preceding section. First, we extract the list of design-point values from the
DAEObject using GetDesignPoint.
In[16]:= dpceamp = GetDesignPoint[ceampsta]
Out[16]=
C1 1. 10 7 , C2 1. 10 6 , R1 100000.,
RC 2200., R2 47000., Rpi$Q1 2200., Ro$Q1 36400.,
Cbc$Q1 4.39 10 12 , Cbe$Q1 7.01 10 11 , gm$Q1 0.0808,
Cbx$Q1 0., RE 1000., V1 1., RL 47000., Simulator PSpice
Then we let ApproximateTransferFunction remove all numerical insignificant terms from the
coefficients of the frequency variable s, thereby allowing for a maximum design-point error of
in each coefficient. Note that a coefficient error does not imply error on the magnitude of
the transfer function in the design point. Ideally, the coefficient error should be a common factor
and thus cancel from numerator and denominator, yielding zero overall error in the design point.
In practice, however, there is no direct correlation between the overall error and the maximum
coefficient error. The former is usually lower than the latter but may, in some cases, even be larger.
In[17]:= voutsimp = ApproximateTransferFunction[ceampvout,
s, dpceamp, 0.1] // Together
Out[17]=
C1 C2 gm$Q1 R1 R2 RC RL Rpi$Q1 s 2 V1 C1 C2 Cbc$Q1 gm$Q1 R1 R2 RC RE
RL Rpi$Q1 s 3 V1 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RE RL Rpi$Q1 s 4 V1
R1 R2 gm$Q1 R1 RE Rpi$Q1 gm$Q1 R2 RE Rpi$Q1 C2 R1 R2 RL s
C1 gm$Q1 R1 R2 RE Rpi$Q1 s C2 gm$Q1 R1 RE RL Rpi$Q1 s
C2 gm$Q1 R2 RE RL Rpi$Q1 s C1 C2 gm$Q1 R1 R2 RE RL Rpi$Q1 s 2
C1 C2 Cbe$Q1 R1 R2 RE RL Rpi$Q1 s 3 C1 C2 Cbc$Q1 gm$Q1 R1 R2 RC
RE RL Rpi$Q1 s 3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RE RL Rpi$Q1 s 4
Here, the approximation process reduces the original transfer function to only significant terms.
We can check the quality of the approximation by evaluating it with the design-point values and
comparing it graphically with the original function.
In[18]:= voutexactn[s_] = Together[ceampvout /. dpceamp]
Out[18]=
470. s 2 6.68943 10 8 3.00693 s 2.54816 10 9 s 2
1.15576 10 15 5.9939 10 13 s
1.52545 10 11 s 2 1543.08 s 3 1.19764 10 6 s 4
In[19]:= voutsimpn[s_] = voutsimp /. dpceamp
Out[19]=
8.63878 10 6 s 2 0.0379242 s 3 3.29021 10 11 s 4
3.08307 10 10 1.53259 10 9 s
3.92672 10 6 s 2 0.041331 s 3 3.29021 10 11 s 4
For the following BodePlot we choose a linear scaling of the magnitude values to get a better
impression of the true magnitude error. It turns out that the simplified expression (dashed line)
approximates the original transfer function (solid line) quite well, although it comprises only about
one tenth of the terms of the latter.