9. frequency responsearies.ucsd.edu/najmabadi/class/ece102/11-f/notes/ece102...typical frequency...
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9. Frequency Response
Reading: Sedra & Smith: Sec. 1.6, Sec. 3.6 and Sec. 9 (MOS portions),
(S&S 5th Ed: Sec. 1.6, Sec. 3.7 (capacitive effects), Sec. 4.8, Sec. 4.9, ,Sec. 6. (Frequency response sections, i.e., 6.4, 6.6, …),
Sec. 7.6
ECE 102, Fall 2011, F. Najmabadi
Typical Frequency response of an Amplifier
Up to now we have ignored the capacitors. To include the capacitors, we need to solve the circuit in the frequency domain (or use Phasors). o Lower cut-off frequency: fL o Upper cut-off frequency: fH o Band-width: B = fH − fL
Classification of amplifiers based on the frequency response
AC amplifier (capacitively-coupled) DC amplifier (directly-coupled) fL = 0
Tuned or Band-pass amplifier (High Q)
Cc1 open: vi = 0 → vo = 0 Contributes to fL
Example:
How to find which capacitors contribute to the lower cut-off frequency
Consider each capacitor individually. Let f = 0 (capacitor is open circuit): o If vo (or AM) does not change, capacitor does NOT contribute to fL o If vo (or AM) → 0 or reduced substantially, capacitor contributes to fL
CL open: No change in vo Does NOT contribute to fL
How to find which capacitors contribute to the higher cut-off frequency
Cc1 short: No change in vo Does NOT contribute to fH
CL short: vo = 0 Contributes to fH
Consider each capacitor individually. Let f → ∞ (capacitor is short circuit): o If vo (or AM) does not change, capacitor does NOT contribute to fH o If vo (or AM) → 0 or reduced substantially, capacitor contributes to fH
Example:
Impact of various capacitors depend on the frequency of interest
f → ∞ All Caps are short. Used to find high-frequency C.
f → 0 All Caps are open. Used this to find low-frequency C.
Mid-band: High-f caps are open Low-f caps are short.
Computing fH : High-f caps are included. Low-f caps are short
Computing fL: High-f caps are open. Low-f caps included.
Impendence of capacitors (1/ωC)
Constructing appropriate circuits Example:
Cc1 contributes to fL CL contributes to fH
Mid-band: High-f caps are open Low-f caps are short.
Computing fH : High-f caps are included. Low-f caps are short
Computing fL: High-f caps are open. Low-f caps included.
Low-Frequency Response
Typical Low-frequency response of an amplifier
Each capacitors gives a pole. All poles contribute to fL (exact value of fL from computation or simulation)
A good approximation for design & hand calculations: fL ≈ fp1 + fp2 + fp3 + …
If one pole is at least a factor of 4 higher than others (e.g., fp2 in the above figure), fL is approximately equal to that pole (e.g., fL ≈ fp2 in above within 20%)
321
x x x ppp
Msig
o
ss
ss
ssA
VV
ωωω +++=
Example: an amplifier with three poles
(Set s = jω to find Bode Plots)
Low-frequency response of a CS amplifier
Cc1 open: vi = 0 → vo = 0
Cc2 open: vo = 0
Cs open: Gain is reduced substantially (from CS amp. To CS amp. With RS)
)||||(
x x x 321
LDomsigG
GM
pppM
sig
o
RRrgRR
RA
ss
ss
ssA
VV
+−=
+++=
ωωω
See S&S pp689-692 for detailed calculations (S&S assumes ro → ∞ and RS → ∞ )
,)]/||/1(||[
1
)||(1 ,
)(1
2
23
11
moLDmSsp
LoDcp
sigGcp
grRRgRC
RrRCRRC
+≈
+=
+=
ω
ωω
All capacitors contribute to fL (vo is reduced when f → 0 or caps open circuit)
Finding poles by inspection
1. Set vsig = 0 2. Consider each capacitor separately, e.g., Cn (assume others
are short circuit!)
3. Find the total resistance seen between the terminals of the capacitor, e.g., Rn (treat ground as a regular “node”).
4. The pole associated with that capacitor is
5. Lower-cut-off frequency can be found from fL ≈ fp1 + fp2 + fp3 + …
nnpn CR
fπ2
1=
* Although we are calculating frequency response in frequency domain, we will use time-domain notation instead of phasor form (i.e., vsig instead of Vsig ) to avoid confusion with the bias values.
Example: Low-frequency response of a CS amplifier
Examination of circuit shows that ALL capacitors contribute to the low-frequency response.
In the following slides with compute poles introduced by each capacitor. (Compare with the detailed calculations.)
Then fL ≈ fp1 + fp2 + fp3
Example: Low-frequency response of a CS amplifier
1. Consider Cc1 :
∞
)( 21
11
sigGcp RRC
f+
=π
2. Find resistance between Capacitor terminals
Terminals of Cc1
Example: Low-frequency response of a CS amplifier
1. Consider CS :
Terminals of CS
)]/||/1(||[ 21
2moLDmSS
p grRRgRCf
+=
π
2. Find resistance between Capacitor terminals
moLD
m
grRRg
/)||(/1 +
moLD
m
grRRg
/)||(/1 +
moLD
m
grRRg
/)||(/1 +
Example: Low-frequency response of a CS amplifier
1. Consider Cc2 : Terminals of Cc2
)||( 21
23
oDLcp rRRC
f+
=π
2. Find resistance between Capacitor terminals
High-Frequency Response
In addition to the impact of external capacitors, amplifier gain falls off due to the internal capacitive effects of transistors
Capacitive Effects in pn Junction
Majority Carriers Charge stored is a function of applied
voltage. We can define a “small-signal”
capacitance, Cj
In reverse-bias region, analysis show (see S&S pp154-156):
V0 : Junction built-in voltage Cj0 : Capacitance at zero reversed-bias
voltage. m : grading coefficient (1/2 to 1/3). For forward-bias region: Cj ≈ 2Cj0
QR VVR
Jj dV
dQC=
=
mR
jj VV
CC
)/1( 0
0
+=
Capacitive Effects in pn Junction
Minority Carriers Excess minority carriers are stored in p and n sides of the
junction. The charge depends on the minority carrier “life-time” (i.e.,
how long it would take for them to diffuse through the junction and recombine.
Gives Diffusion Capacitance, Cd
Cd is proportional to current (Cd = 0 for reverse-bias)
T
DTd V
IC ⋅=τ
Small Signal Model for a diode
Forward Bias Reverse Bias
02 jj CC ⋅≈
T
DTd V
IC ⋅=τ
0=dC
mR
jj VV
CC
)/1( 0
0
+=
rD
Cj + Cd
Junction capacitances are small and are given in femto-Farad (fF)
1 fF = 10−15 F
Capacitive Effects in MOS 1. Capacitance between Gate and channel (Parallel-plate capacitor) appears as 2 capacitors: between gate/source & between gate/drain
3. Junction capacitance between Source and Body (Reverse-bias junction)
4. Junction capacitance between Drain and Body (Reverse-bias junction)
2. Capacitance between Gate & Source and Gate & Drain due to the overlap of gate electrode (Parallel-plate capacitor)
MOS High-frequency small signal model
MOS high-frequency small signal model For source connected to body
(used by S&S) Accurate Model
(we use this model here)
Generally, transistor internal capacitances are shown outside the transistor so that we can use results from the mid-band calculations.
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