Notebook

In [1]:

from sympy import *
init_printing()

http://een.iust.ac.ir/profs/Abrishamifar/Analog%20Integrated%20Circuit%20Design/Data%20For%20Design/Useful%20equations/Equations.pdf

In [2]:

Id, W,L, VDS, VGS, Vnth, VnA, nTechConst=symbols("I_D, W, L, V_DS, V_GS, V_thn, V_An, k_n'")
Id, W,L, VDS, VGS, Vnth, VnA, nTechConst

Out[2]:

$$\left ( I_{D}, \quad W, \quad L, \quad V_{DS}, \quad V_{GS}, \quad V_{thn}, \quad V_{An}, \quad k_{n'}\right )$$

In [3]:

NLinFull=nTechConst*(W/L)*((VGS-Vnth)*VDS-VDS**2/2)*(1+VDS/VnA)
NLinFull

Out[3]:

$$\frac{W k_{n'}}{L} \left(1 + \frac{V_{DS}}{V_{An}}\right) \left(- \frac{V_{DS}^{2}}{2} + V_{DS} \left(V_{GS} - V_{thn}\right)\right)$$

In [4]:

NLinSimp=nTechConst*(W/L)*((VGS-Vnth)*VDS-VDS**2/2)
NLinSimp

Out[4]:

$$\frac{W k_{n'}}{L} \left(- \frac{V_{DS}^{2}}{2} + V_{DS} \left(V_{GS} - V_{thn}\right)\right)$$

In [5]:

NSatFull=nTechConst/2*W/L*(VGS-Vnth)**2 *(1+VDS/VnA)
NSatFull

Out[5]:

$$\frac{W k_{n'}}{2 L} \left(1 + \frac{V_{DS}}{V_{An}}\right) \left(V_{GS} - V_{thn}\right)^{2}$$

In [6]:

NSatSimp=nTechConst/2*W/L*(VGS-Vnth)**2
NSatSimp

Out[6]:

$$\frac{W k_{n'}}{2 L} \left(V_{GS} - V_{thn}\right)^{2}$$

In [7]:

NIdFull=Piecewise(
    (0, VGS<Vnth),
    (NLinFull, And(VGS>=Vnth, VDS<VGS-Vnth)),
    (NSatFull, And(VGS>=Vnth, VDS>=VGS-Vnth))
)
NIdFull

Out[7]:

$$\begin{cases} 0 & \text{for}\: V_{GS} < V_{thn} \\\frac{W k_{n'}}{L} \left(1 + \frac{V_{DS}}{V_{An}}\right) \left(- \frac{V_{DS}^{2}}{2} + V_{DS} \left(V_{GS} - V_{thn}\right)\right) & \text{for}\: V_{GS} \geq V_{thn} \wedge V_{DS} < V_{GS} - V_{thn} \\\frac{W k_{n'}}{2 L} \left(1 + \frac{V_{DS}}{V_{An}}\right) \left(V_{GS} - V_{thn}\right)^{2} & \text{for}\: V_{DS} \geq V_{GS} - V_{thn} \wedge V_{GS} \geq V_{thn} \end{cases}$$

In [8]:

NIdSimp=Piecewise(
    (0, VGS<Vnth),
    (NLinSimp, And(VGS>=Vnth, VDS<VGS-Vnth)),
    (NSatSimp, And(VGS>=Vnth, VDS>=VGS-Vnth))
)
NIdSimp

Out[8]:

$$\begin{cases} 0 & \text{for}\: V_{GS} < V_{thn} \\\frac{W k_{n'}}{L} \left(- \frac{V_{DS}^{2}}{2} + V_{DS} \left(V_{GS} - V_{thn}\right)\right) & \text{for}\: V_{GS} \geq V_{thn} \wedge V_{DS} < V_{GS} - V_{thn} \\\frac{W k_{n'}}{2 L} \left(V_{GS} - V_{thn}\right)^{2} & \text{for}\: V_{DS} \geq V_{GS} - V_{thn} \wedge V_{GS} \geq V_{thn} \end{cases}$$

In [9]:

NCutoffCond=VGS<Vnth; NCutoffCond

Out[9]:

$$V_{GS} < V_{thn}$$

In [10]:

NLinearCond=VDS<VGS-Vnth; NLinearCond

Out[10]:

$$V_{DS} < V_{GS} - V_{thn}$$

In [11]:

NSatCond=VDS>=VGS-Vnth; NSatCond

Out[11]:

$$V_{DS} \geq V_{GS} - V_{thn}$$

In [12]:

NLinFull.subs({})

Out[12]:

$$\frac{W k_{n'}}{L} \left(1 + \frac{V_{DS}}{V_{An}}\right) \left(- \frac{V_{DS}^{2}}{2} + V_{DS} \left(V_{GS} - V_{thn}\right)\right)$$

In [13]:

class NMOS():
    def __init__():
        pass
    
    @staticmethod
    def LinFull(subs={}):
        return NLinFull.subs(subs)
    
    @staticmethod
    def LinSimp(subs={}):
        return NLinSimp.subs(subs)
    
    @staticmethod
    def SatFull(subs={}):
        return NSatFull.subs(subs)
    
    @staticmethod
    def IdFull(subs={}):
        return NIdFull.subs(subs)
    
    @staticmethod
    def IdSimp(subs={}):
        return NIdSimp.subs(subs)
    
    @staticmethod
    def CutoffCond(subs={}):
        return NCutoffCond.subs(subs)
    
    @staticmethod
    def LinearCond(subs={}):
        return NLinearCond.subs(subs)
    
    @staticmethod
    def SatCond(subs={}):
        return NSatCond.subs(subs)

NMOS.LinFull(subs={VnA:5})print(NMOS.LinFull()) print(NMOS.LinSimp()) print(NMOS.SatFull()) print(NMOS.IdFull()) print(NMOS.IdSimp()) print(NMOS.CutoffCond()) print(NMOS.LinearCond()) print(NMOS.SatCond())

In [14]:

VSD, VSG, Vpth, VpA, pTechConst=symbols("V_SD, V_SG, V_thp, V_Ap, k_p'")
VSD, VSG, Vpth, VpA, pTechConst

Out[14]:

$$\left ( V_{SD}, \quad V_{SG}, \quad V_{thp}, \quad V_{Ap}, \quad k_{p'}\right )$$

In [15]:

PLinFull=pTechConst*(W/L)*((VSG-abs(Vpth))*VSD-VSD**2/2)*(1+VSD/abs(VpA))
PLinFull

Out[15]:

$$\frac{W k_{p'}}{L} \left(- \frac{V_{SD}^{2}}{2} + V_{SD} \left(V_{SG} - \left|{V_{thp}}\right|\right)\right) \left(\frac{V_{SD}}{\left|{V_{Ap}}\right|} + 1\right)$$

In [16]:

PLinSimp=pTechConst*(W/L)*((VSG-abs(Vpth))*VSD-VSD**2/2)
PLinSimp

Out[16]:

$$\frac{W k_{p'}}{L} \left(- \frac{V_{SD}^{2}}{2} + V_{SD} \left(V_{SG} - \left|{V_{thp}}\right|\right)\right)$$

In [17]:

PSatFull=pTechConst/2*W/L*(VSG-abs(Vpth))**2 *(1+VSD/abs(VpA))
PSatFull

Out[17]:

$$\frac{W k_{p'}}{2 L} \left(V_{SG} - \left|{V_{thp}}\right|\right)^{2} \left(\frac{V_{SD}}{\left|{V_{Ap}}\right|} + 1\right)$$

In [18]:

PSatSimp=pTechConst/2*W/L*(VSG-abs(Vpth))**2
PSatSimp

Out[18]:

$$\frac{W k_{p'}}{2 L} \left(V_{SG} - \left|{V_{thp}}\right|\right)^{2}$$

In [19]:

PIdFull=Piecewise(
    (0, VSG<abs(Vpth)),
    (PLinFull, And(VSG>=abs(Vpth), VSD<VSG-abs(Vpth))),
    (PSatFull, And(VSG>=abs(Vpth), VSD>=VSG-abs(Vpth)))
)
PIdFull

Out[19]:

$$\begin{cases} 0 & \text{for}\: V_{SG} < \left|{V_{thp}}\right| \\\frac{W k_{p'}}{L} \left(- \frac{V_{SD}^{2}}{2} + V_{SD} \left(V_{SG} - \left|{V_{thp}}\right|\right)\right) \left(\frac{V_{SD}}{\left|{V_{Ap}}\right|} + 1\right) & \text{for}\: V_{SG} \geq \left|{V_{thp}}\right| \wedge V_{SD} < V_{SG} - \left|{V_{thp}}\right| \\\frac{W k_{p'}}{2 L} \left(V_{SG} - \left|{V_{thp}}\right|\right)^{2} \left(\frac{V_{SD}}{\left|{V_{Ap}}\right|} + 1\right) & \text{for}\: V_{SD} \geq V_{SG} - \left|{V_{thp}}\right| \wedge V_{SG} \geq \left|{V_{thp}}\right| \end{cases}$$

In [20]:

PIdSimp=Piecewise(
    (0, VSG<abs(Vpth)),
    (PLinSimp, And(VSG>=abs(Vpth), VSD<VSG-abs(Vpth))),
    (PSatSimp, And(VSG>=abs(Vpth), VSD>=VSG-abs(Vpth)))
)
PIdSimp

Out[20]:

$$\begin{cases} 0 & \text{for}\: V_{SG} < \left|{V_{thp}}\right| \\\frac{W k_{p'}}{L} \left(- \frac{V_{SD}^{2}}{2} + V_{SD} \left(V_{SG} - \left|{V_{thp}}\right|\right)\right) & \text{for}\: V_{SG} \geq \left|{V_{thp}}\right| \wedge V_{SD} < V_{SG} - \left|{V_{thp}}\right| \\\frac{W k_{p'}}{2 L} \left(V_{SG} - \left|{V_{thp}}\right|\right)^{2} & \text{for}\: V_{SD} \geq V_{SG} - \left|{V_{thp}}\right| \wedge V_{SG} \geq \left|{V_{thp}}\right| \end{cases}$$

In [21]:

PCutoffCond=VSG<abs(Vpth); PCutoffCond

Out[21]:

$$V_{SG} < \left|{V_{thp}}\right|$$

In [22]:

PLinearCond=VSD<VSG-abs(Vpth); PLinearCond

Out[22]:

$$V_{SD} < V_{SG} - \left|{V_{thp}}\right|$$

In [23]:

PSatCond=VSD>=VSG-abs(Vpth); PSatCond

Out[23]:

$$V_{SD} \geq V_{SG} - \left|{V_{thp}}\right|$$

In [24]:

class PMOS():
    def __init__():
        pass
    
    @staticmethod
    def LinFull(subs={}):
        return PLinFull.subs(subs)
    
    @staticmethod
    def LinSimp(subs={}):
        return PLinSimp.subs(subs)
    
    @staticmethod
    def SatFull(subs={}):
        return PSatFull.subs(subs)
    
    @staticmethod
    def IdFull(subs={}):
        return PIdFull.subs(subs)
    
    @staticmethod
    def IdSimp(subs={}):
        return PIdSimp.subs(subs)
    
    @staticmethod
    def CutoffCond(subs={}):
        return PCutoffCond.subs(subs)
    
    @staticmethod
    def LinearCond(subs={}):
        return PLinearCond.subs(subs)
    
    @staticmethod
    def SatCond(subs={}):
        return PSatCond.subs(subs)

print(PMOS.LinFull(subs={VpA:5})) print(PMOS.LinSimp()) print(PMOS.SatFull()) print(PMOS.IdFull()) print(PMOS.IdSimp()) print(PMOS.CutoffCond()) print(PMOS.LinearCond()) print(PMOS.SatCond())

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