1. 8. Frequency Response of Amplifiers
ECE 342
8. Frequency Response of Amplifiers
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois

2. Low-Pass Circuit
In frequency domain:

3. Low-Pass Circuit
At very high frequencies, the single time constant (STC) transfer function goes as
At high frequencies, slope of curve is –20 dB

4. High-Pass Circuit

5. Octave & Decade If f2 = 2f1, then f2 is one octave above f1
If f2 = 10f1, then f2 is one decade above f1
2 GHz is one octave above 1 GHz
10 GHz is one decade above 1 GHz

6. Frequency Response
3-dB points are points where the magnitude is divided by 21/2 (power is halved) |1+j|= 21/2
AdB= 20log1.414 = 3 dB
Amplifier has intrinsic gain Ao
Low-pass characteristics is:
High-pass characteristics is:
Overall gain A(f) is

7. Octave & Decade
Overall gain A(f) is

8. Model for general Amplifying Element
Cc1 and Cc2 are coupling capacitors (large)  mF
Cin and Cout are parasitic capacitors (small)  pF

9. Midband Frequencies Coupling capacitors are short circuits
Parasitic capacitors are open circuits

10. Low Frequency Model Coupling capacitors are present
Parasitic capacitors are open circuits

11. Low Frequency Model

12. Low Frequency Model

13. Example Rout = 3 kW, Rg=200 W, Rin=12 kW, RL=10 kW
Cc1=5 mF and Cc2=1 mF

14. High Frequency Model Assume coupling capacitors are short
Account for parasitic capacitors
Potential Thevenin equivalent for input as seen by Cin

15. High Frequency Model

16. High Frequency
Overall gain is: or

17. Important Remarks
An arbitrary network’s transfer function can be described in terms of its s-domain representation
s is a complex number s = s + jw
The impedance (or admittance) or networks can be described in the s domain

18. Transfer Function Representation
The coefficients a and b are real and the order m of the numerator is smaller than or equal to the order n of the denominator
A stable system is one that does not generate signal on its own.
For a stable network, the roots of the denominator should have negative real parts

19. Transfer Function Representation
In general, the transfer function of an amplifier can be expressed as
Z1, Z2,…Zm are the zeros of the transfer function
P1, P2,…Pm are the poles of the transfer function
s is a complex number s = s + jw

20. 3dB Frequency Determination
Designer is interested in midband operation
However needs to know upper 3-dB frequency
In many cases some conditions are met:
Zeros are infinity or at very high frequencies
One of the poles (wP1) is at much lower frequency than other poles (dominant pole)
If the conditions are met then FH(s) can be approximated by:

21. 3dB Frequency Determination
If the lowest frequency pole is at least 4 times away from the nearest pole or zero, it is a dominant pole
If there is no dominant pole, the 3-dB frequency wH can be approximated by:

22. High-Frequency Behavior - Example
sedr42021_0613.jpg

23. Open-Circuit Time Constants
The coefficients a and b are related to the frequencies of the zeros and poles respectively.
b1 can be obtained by summing the individual time constants of the circuit using the open-circuit time constant method

24. Open-Circuit Time Constant Method
The time constant of each capacitor in the circuit is evaluated. It is the product of the capacitance and the resistance seen across its terminals with:
All other internal capacitors open circuited
All independent voltage sources short circuited
All independent current sources opened
The value of b1 is computed by summing the individual time constants

25. Open-Circuit Time Constant Method
An approximation can be made by using the value of b1 to determine the 3dB upper frequency point wH
If the zeros are not dominant and if one of the poles P1 is dominant, then
Assuming that the 3-dB frequency will be approximately equal to wP1