INF3410/4411, Fall 2017
Philipp Hä�[email protected]
Excerpt of Sedra/Smith Chapter 7: Integrated CMOS Ampli�erBasics and Initial Frequency Analysis
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
CMOS Current Steering
Computing Rref CMOS Current Steering
(live on whitebord)
Wrong: using the small signal input resistance1
gmfor a desired ID
Correct: Use large signal model!!!
Dynamic Input Currents, Current Ampli�er
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
Intrinsic Gain from Small Signal
Intrinsic Gain vs Bias Current
CS Ampli�er with Current-Source Load
CS Ampli�er Analysis
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
A �rst look at high frequency behaviour
What happens to an ampli�er gain for input signals of di�erentspectra, e.g. when the input is a high frequency/low frequencysignal?
Gain in Decibels
Av =vO
vI
Ap =pO
pL=
vO iO
vI iI= AvAi
Ap,dB = 10 log10 Ap [dB]
Av ,dB = 20 log10 Av [dB]
Arbitrary Signals
Any arbitrary signal can beexpressed as an (in�nite) sum or(in�nite) integral of sine wavesignals of di�erent frequenciesand phases by means of theinverse Fourier transform:
vS(t) =1
2π
∫ ∞−∞
v̂s(ω)e iωtdω
Where the Fourrier transform v̂s is a complex number. How doesthe above equation represent a 'sum of sine waves'?
Cyclic Signals
Cyclic signals can be expressed asdiscreet sums of sine wavesignals, more precisely withharmonics of the cycle frequency.
Example: Approximating Square Wave
Here is how a square wave can beapproximated with the sum of 3sine waves.
v(t) =4V
π
(sinω0t +
1
3sin 3ω0t +
1
5sin 5ω0t + ...
)(1.2)
Spectrum of Square Wave
Thus the spectrum of a square wave (or any cyclic signal) isnon-zero at the harmonics (multiples of the fundamental frequency)only. It can thus be expressed as a sum.
Spectrum of Arbitrary Function
whereas the spectrum of a non-cyclic function can be non-zero forany frequency, e.g. the function in �gure 1.3.
Transfer FunctionA linear circuit network that receives an input signal and producesan output signal is a linear �lter and can be completelycharacterized by it's transfer function T (ω) where ω is a realnumber representing the frequency of the spectrum and T is acomplex number.Per de�nition, T is the transfer function of a �lter if the spectrumof the outout signal Y (ω) is equal to the product of the inputsignal spectrum X (ω) and T (ω)
Y (ω) = T (ω)X (ω) ⇔ Y (ω)
X (ω)= T (ω)
So the magnitude of T (ω) represents a gain and the anglerepresents a phase shift that is applied to the input signal spectrumat each frequency.In the context of circuits one normally writes T (jω) = T (s) whichis the same as above function T (ω) as the term jω consistentlyappears in T when it is derived from analysing circuits and it'ssimpler to just write s in the derivation.
Bode Plot
Transfer functions can be fully described by Bode plots. Forexample:1st Order Low-Pass Filter
Impedances of Linear Circuit Elements
A transfer function of a linear circuit can be conveniently derived bysimply applying Ohm's and Kircho�'s laws to the compleximpedances/admittances of the circuit elements.impedance hspace1cm admittanceresistor: R 1
R
capacitor: 1
sCsC
inductor: sL 1
sL
Ideal linearly dependent sources (e.g. the id = gmvgs sources insmall signal models of FETs) are left as they are.Finally solve for vo
vi
Transfer Function Derivation Variants
The transfer function T (s) of a linear �lter is
I the Laplace transform of its impulse reponse h(t). (Left toFYS3220)
I the Laplace transform of the di�erential equation describingthe I/O realtionship that is then solved for Vout(s)
Vin(s). (Left to
FYS3220)
I The circuit diagram solved quite normally for Vout(s)Vin(s)
by
putting in impedances Z (s) for all linear elements according tothe simple rules on the previous slide. (This lecture!)
Transfer Function
Transfer functions T (s) for linear electronic circuits can be writtenas dividing two polynomials of s.
T (s) =a0 + a1s + ...+ ams
m
1 + b1s + ...+ bnsn
T (s) is often written as products of �rst order terms in bothnominator and denominator in the following root form, which isconveniently showing some properties of the Bode-plots. More ofthat later.
T (s) = a0(1 + s
z1)(1 + s
z2)...(1 + s
zm)
(1 + sω1
)(1 + sω2
)...(1 + sωn
)
Example: CS ampli�er with capacitive load
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
CG ampli�er revisited
Rin of a CG ampli�er
vx = (ix + gmvgs)ro + ixRL ⇒ Rin ≈1
gm+
RL
gmro(7.54)
Important step: vgs = −vx
Rout of a CG ampli�er
vx = (ix − gmvgs)ro + ixRS ⇒ Rout ≈ ro + (gmro)Rs (7.58)
CS with source degeneration no RL
Good as current source, e.g in current mirror, but not so good asampli�er.
RO ≈ (1 + gmRS)ro
gm → g ′m =gm
1 + gmRs
Much higher output resistance. Still nonet-increase in gain!
CS with source degeneration with RL
Av = g ′mRORL
RL + RO
≈ gmroRL
RL + (1 + gmRS)ro
... and more degradation to Av due to load RL!
Cascode Ampli�er
Can be looked upon as a CS and CG in series resulting in a intrinsic
combined gain A = gm1ro1gm2ro2 (i.e. with a large load resistance),or a circuit where the CS serves as high quality voltage controlledcurrent source delivering id ≈ gm1vi and the CG bu�ers thatcurrent to a high output resistance ≈ gm2ro2ro1.
Derivation of intrinsic cascode gain
Cascode Ampli�er with In�nite Load Resistance
Folded Cascode with In�nite Load Resistance
Cascode Ampli�er with Finite Load Resistance (1/2)
The load RL must be of equal magnitude as RO to get the bene�tof the increased gain AV ! So here with a simple pFET we are backto square one.
Cascode Ampli�er with Finite Load Resistance (2/2)
Better: employ a cascoded current source.
Dependency of Gain on RL
Content
Bias Current Steering (book 7.2)
CS con�guration (book 7.3)
CS Low Pass Characteristics (excerpt book 1.2, 1.6)
CG, and cascode con�guration (book 7.4-7.5)
Improved Current Mirrors/Sources (book 7.6)
Cascode Current Mirror
Increased output impedance, but quite a bit of output voltageheadroom necessary ...
Modi�ed Wilson MOS Current Mirror
A CD-CS Ampli�er
Larger bandwidth than simple CS ampli�er(explained later in chapter 9).