EE 105 Midterm 1 Formula
no = po = ni^2 , where no = e- concentration at thermal
equilibrim, po = hole concentration at thermal equilibrim, ni=10^10 cm^-3
(27)intrinsic concentration at 300k (26)
n-type
no^2 – Nd*no – ni^2 =0
no = (Nd +- sqrt(Nd^2 + 4* ni^2))/2
no = Nd/2 + Nd/2*sqrt(1+4ni/(Nd^2)) ~= Nd (29)
po = ni^2/no ~= ni^2/Nd (29)
p-type
po ~= Na (31)
no = ni^2/po ~= ni^2/Na (31)
n = ((Nd-Na)/2 + (Nd-Na)/2*sqrt(1+4ni/((Nd-Na)^2))
~= Nd (Brian Review)
ni = 3.9*10^16 T(3/2)e^(-0.605eV/kT) cm^-3, Equilibrium
Intrinsic Concentration,where T is in K, k is Boltzman’s constant =
8.62^10^-5 eV/K
p = q(po – no + Nd – Na)=0, where p is charge
neutrality, no = e- concentration at thermal equilibrim, po = hole
concentration at thermal equilibrium, Nd = donor concentration, Na=acceptor
concentration (HW3)
vdn = -un*E, where vdn is e- drift velocity, un is
mobility of e-, E is electrical field (37)
vdp = up*E, where vdp is hole drift velocity, up is
mobility of hole, E is electrical field (V/m)
if E >> Esat, -vdn=vdp=10^7 cm/s (37)
Net charge (delta Q) = -q*n*A*vdn*(delta t), where q =
e- density, n = e- concentration (HW3), t=thickness
Jn,drift = -q*n*vn , where J(A/cm^2), current density,
where vn is –ve, vn=e- drift velocity(40)
Jp, drift = q*p*vp (HW3)
Jdrift = Jn,drift + Jp, drift
R(sheet) = (q*un*Nd*t)^-1, where q is charge of
electron, un is mobility of e-, Nd is donor density, t is thickness, unit is
ohm/square (61)
N(sheet) = L/W
Rtotal = N(sheet) * R(sheet) (HW3)
pn ~= 1/(q*Nd*un) = (q*Nd*un +
(q*ni^2*up)/Nd)^-1, pn is the n-type resistivity, q = charge of e-, Nd=donor
concentration, un=mobility of e-. [m ohm-cm] (59)
pp ~= 1/(q*Na*up) = (q*Na*up +
(q*ni^2*un)/Na)^-1, pp is the p-type resistivity, q = charge of e-, Na=donor
concentration, up=mobility of hole. [m ohm-cm](59)
Qn,avg = 1/2*(Qn(y=0) +Qn(y=L)) = 1/2*[-Cox(Vgs – Vtn) – Cox(Vgs – Vds – Vtn)], Average Channel Charge, where L position from source (same as the
position for drain), C is coulomb, 1e- = 1.6x10^-19 C, [pC/um^2] (202)
# of e- = (Qn,avg)(W)(L)/(-q), where Qn is
charge density, q is charge of e-, W and L is width and length of the
MOSFET,(HW3)
MOSFET
Id = (W/L) un * Cox [Vgs – Vtn- (Vds/2)]Vds, where
Vgs-Vds>Vtn (Triode), un is mobility, Cox is capacitance of the oxide (203)
Rds = Vds/Id = (L/W) [1/ (un * Cox (Vgs – Vtn-
(Vds/2))] = (L/W) Rsheet, (HW3)
Rsheet = 1/ (un * Cox (Vgs – Vtn- (Vds/2)), (HW3)
R,sheet = p/t, where p=resistivity, t=
thickness, [ohm/square]
R = (p/t)*(L/W)= p*L/A, where
p=resistivity(u ohm cm), t=thickness, A = area (cm^2) (HW3)
Vo = Vi * (1-e^-(t/RC)), Voltage of Capacitor as time
(t) >0(HW3)
Vth = Vth + gamma [ sqrt(-2 phi,p – Vbs) – sqrt(-2
phi,p) ], where gamma = (V^(1/2)), (Boser Review)
Lamda = lamda,o *(Lo/L) (Boser Review)
Units: Yotta=1E24, Zetta=1E21, Exa=1E18, Peta, 1E15,
Tera=1E12, Giga=1E9, Mega-1E6, Kilo=1E3, milli=1E03, u(micro)=1E-6, nano=1E-9,
pico=1E-12, femto=1E-15
Group III Doper: B,
Group IV Doper: C, Si, Ge
Group V Doper: N, P, As
EE 105 Midterm 2 Formula
Id = un*Cox*(W/2L)(Vgs – Vth)^2 (1+lamda*Vds)(HW6)
gm =
2(Id)/Vdsat = 2(Id)/(Vgs - Vth)=
d(Id)/d(Vgs) |q point= un*Cox(W/L)(Vgsq-Vth) = 2* sqrt[un*Cox(W/2L) *Idq] =
sqrt[2*un*Cox(W/L) *Idq]= sqrt(2un*Cox)*sqrt(W/L)*sqrt(Idq), where Vgs is Large
signal, Vgsq is Vgs at the q point , Idq = Id at the q point, dy/dx = partial
y/partial x (HW6)
ro = 1/(lamda*Id) = d(Id)/d(Vds) |q point
(HW6)
Id = (W/L) (un*Cox) (Vgs –Vtn)^2 (HW6)
In Triode
Id = un*Cox(W/L)(Vgs - Vtn - Vds/2) Vds
gds = Ids*(lamda,n)
ro = 1/ gds = 1/(lamda,n*Vds)
ids = d(Ids)/d(Vgs)*vgs + d(Ids)/d(Vds)*vds
+ d(Ids)/d(Vbs)*vbs
ids = gm*vgs + gds*vds + gmb*vbs
diode-connected transistors = transistor
that bulk and source tied together = resistance of 1/(gm + gmb) // ro
Large Signal Shortcut
Rdrain = Ro + Rsource +(gm + gmb)*Ro*Rsource
Rsource = (Ro + Rdrain) / (gm + gmb)Ro
GM Rin Rout
CS gm Inf ro//roc
CG -1 1/(gm+gmb) [ro+gm*ro*Rs]//roc
CD gm/(gm+gmb) Inf 1/(gm+gmb)
Iout = (W/L)_2 / (W/L)_1 * Iref
Two Port Model
-----Rin------Rout
high Vin Iout
low
Iin
Vout
Find Gm or Av
1. Calculate Rout
2. Calculate Gm
3. Av = -Gm*Rout
Find Ai or Rm
1. Calculate Rout
2. Calculate Ai
3. Rm = -Ai*Rout
Vb and Vg has to be small signal ground