def sourceFreeParallelEquivalentCircuit(R,
C,
L,
find="v",
printEq=True,
**kwargs):
"""Works in radians, not degrees"""
alpha = 1 / (2 * R * C)
w0 = 1 / (L * C)**(1 / 2)
s1 = -alpha + (alpha**2 - w0**2)**(1 / 2)
s2 = -alpha - (alpha**2 - w0**2)**(1 / 2)
eq = list()
if alpha > w0:
print("Overdamped")
eq.append("Eq(v, A1*exp(s1*t) + A2*exp(s2*t))")
if alpha == w0: #C=4*L/R**2
print("Critically Damped")
eq.append("Eq(v, (A1 + A2*t)*exp(-alpha * t))")
if alpha < w0:
print("Underdamped")
# wd = (w0**2 - alpha**2)**(1/2)
# B1 = A1 + A2
# B2 = j*(A1 - A2)
eq.append("Eq(v, exp(-alpha*t) * (A1*cos(wd*t) + A2*sin(wd*t)))")
kwargs.update({
"wd": "(w0**2 - alpha**2)**(1/2)",
"alpha": alpha,
"w0": w0,
"s1": s1,
"s2": s2
})
return solveEqs(eq, find="i", printEq=printEq, **kwargs)
def Value(find, printEq=False, **kwargs):
"""
usage:
measured in Farads (F)
inductor is an short circuit in DC
current cannot change abrumptly in a capacitor
variables:
v=voltage
L=inductance (measured in Henry = Volt/Ampere)
di=change in current
t, t0, t1, dt=time, initial time, future time, change in time
i, i0, di = current, initial current, change in current
w=work (J)
Equations:
v = L * di/dt
i = 1/L * integrate(v,(t,t0,t1)) + i0
p = v * i
p = L * di/dt * i
w = 0.5 * L * i**2
"""
eq = list()
eq.append("Eq(v,L*di/dt)")
eq.append("Eq(v,L*diff(i,t))")
eq.append("Eq(i,1/L*integrate(v,(t,t0,t1))+i0)")
eq.append("Eq(p,v*i)")
eq.append("Eq(p, L*di/dt*i)")
eq.append("Eq(w,0.5*L*i**2)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def naturalResponse(find, printEq=True, **kwargs):
"""
Source-Free
The natural response of a circuit refers to the behavior (in terms of
voltages and currents) of the circuit itself, with no external sources of
excitation.
i(0) = I0
variables:
v_o, v1, v2, v3 = Open loop voltage gain, voltage in
R2, R1 = resistors (view reference image)
tau = The time constant; the time required for the response to
decay to a factor of 1/e or 36.8 percent of its initial value.
"""
eq = list()
eq.append("Eq(tau, L_eq / R_eq")
eq.append("Eq(w0, .5 * L * I0**2")
eq.append("Eq(v_L + v_R, 0")
eq.append("Eq(L * di/dt + R*i, 0")
eq.append("Eq(L * diff(i,t) + R*i, 0")
eq.append("Eq(i, I0*exp(-t/tau))")
eq.append("Eq(v_R, i * R)")
eq.append("Eq(v_R, I0 * R * exp(-2*t/tau))")
eq.append("Eq(p, v_R * i)")
eq.append("Eq(p, I0**2 * R * exp(-2*t/tau))")
eq.append("Eq(w, integrate(p,(t,0,tf)))")
eq.append("Eq(w,.5 * L * I0**2 * (1-exp(-2*t/tau)))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def naturalResponse(find, printEq=True, **kwargs):
"""
Source-Free
The natural response of a circuit refers to the behavior (in terms of
voltages and currents) of the circuit itself, with no external sources of
excitation.
v(0) = V0
variables:
v_o, v1, v2, v3 = Open loop voltage gain, voltage in
R2, R1 = resistors (view reference image)
tau = The time constant; the time required for the response to
decay to a factor of 1/e or 36.8 percent of its initial value.
"""
eq = list()
eq.append("Eq(tau, R_eq * C_eq")
eq.append("Eq(v, V0 * exp(-t / tau))")
eq.append("Eq(i_R, v / R")
eq.append("Eq(i_R, V0 * exp(-t / tau)/R)")
eq.append("Eq(p, v * i_R")
eq.append("Eq(p, V0**2 * exp(-2*t / tau)/R)")
eq.append("Eq(w, integrate(p,(t,0,tf))")
eq.append("Eq(w, integrate(V0**2 * exp(-2*t / tau) / R, (t,0,tf))")
eq.append("Eq(w, .5 * C * V0**2 * (1 - exp(-2*t / tau)))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def projectileMotionEq(find=False,
angleUnitRadians=True,
printEq=False,
**kwargs):
"""variables:
x=position-along-x, y=position-along-y,
alpha0=original trajectory angle,
t=time, g=standard acceleration of gravity
vx=velocity-along-x, vy=velocity-along-y
Note:
Angle solution in radians"""
if not (angleUnitRadians):
dictionary_add_modify(kwargs, alpha0=radians(kwargs["alpha0"]))
if not ("g" in kwargs):
dictionary_add_modify(kwargs, g=scipy.constants.g)
posx = "v0*cos(alpha0)*t"
posy = "v0*sin(alpha0)*t - 0.5*g*t**2"
eq = list()
eq.append(f"Eq(x,{posx})")
eq.append(f"Eq(y,{posy})")
eq.append(f"Eq(vx,diff({posx},t))")
eq.append(f"Eq(vy,diff({posy},t))")
eq.append(f"Eq(ax,diff({posx},t,t))")
eq.append(f"Eq(ay,diff({posy},t,t))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def mach(find="Ma", printEqs=True, **kwargs):
"""
Mach Number = Flow Speed / Sound Speed
"""
eq = list()
eq.append("Eq(Ma, U/a)")
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def summer(find="v_o", printEq=False, **kwargs):
"""variables:
v_o, v1, v2, v3 = Open loop voltage gain, voltage in
Rf, R3, R2, R1 = resistors (view reference image)
"""
eq = list()
eq.append("Eq(v_o,-(Rf/R1*v1 + Rf/R2*v2 + Rf/R3*v3))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def differenceAmplifier(find="v_o", printEq=False, **kwargs):
"""variables:
v_o, v1, v2, v3 = Open loop voltage gain, voltage in
R2, R1 = resistors (view reference image)
"""
eq = list()
eq.append("Eq(v_o,R2/R1*(v2-v1)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def momentumEq(find, printEq=False, **kwargs):
"""variables:
m=mass v=velocity
Usage: Energy Applied on an object
Sample Units: Kg*m/s"""
eq = list()
eq.append("Eq(p,m*v)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def YoungsModulusEq(find="Y", printEq=False, **kwargs):
"""variables:
Y=young's modulus of elasticity
sigma=stress
epsilon=strain
"""
eq = list()
eq.append("Eq(Y, sigma/epsilon)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def Resistance(find, printEq=False, **kwargs):
"""variables:
R=resistance
p=density
l=length
A=cross sectional area"""
eq = list()
eq.append("Eq(R,p*l/A)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def powerEq(find, printEq=False, **kwargs):
"""variables:
P=power W=Work t=time
f=force v=velocity
Base Unit = Watts"""
eq = list()
eq.append("Eq(P, W/t)")
eq.append("Eq(P, F*v)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def orbitVelocity(find="v_orb", printEq=False, **kwargs):
"""variables:
v_orb = gravitational potential energy
G = 6.67e-11 N*m**2/kg**2
m_planet=mass of planet being orbited
r=distance from center of planet being orbited"""
dictionary_add_modify(kwargs, G=6.67e-11)
eq = list()
eq.append("Eq(v,(G*m_planet/r)**(1/2))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def gravitationAttraction(find="F_g", printEq=False, **kwargs):
"""variables:
F_g = force of attraction due to gravity
G = 6.67e-11 N*m**2/kg**2
m1, m2 = mass1, mass2
r = distance between masses"""
dictionary_add_modify(kwargs, G=6.67e-11)
eq = list()
eq.append("Eq(F_g,G*m1*m2/r**2)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def invertingAmplifier(find="v_o", printEq=False, **kwargs):
"""Variables:
v_o, v_in = Open loop voltage gain, voltage in
R2, R1 = resistors (view reference image)
Equation: v_o = -R2/R1 * v_in
"""
eq = list()
eq.append("Eq(v_o,-R2/R1*v_in)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def NodalAnalysis(find="i", printEq=False, **kwargs):
"""variables:
i = current
vh, vl = higher/lower voltage
r = resistor
usage:
Current flows from a higher potential to a lower potential in a resistor"""
eq = list()
eq.append("Eq(i,(v_higher-v_lower)/r)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def forceEq(find, printEq=False, **kwargs):
"""variables:
W=Work F=force p=momentum t=time
x, x0, x1 = distance, start, finish
Usage: Energy Applied on an object"""
eq = list()
eq.append("Eq(F,m*a)")
eq.append("Eq(F,k*x)")
eq.append("Eq(F,p/t)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def workEq(find, printEq=False, **kwargs):
"""variables:
W=Work F=force
x, x0, x1 = distance, start, finish
Usage: Energy Applied on an object"""
eq = list()
eq.append("Eq(W, F*x)")
eq.append("Eq(W,integrate(F,(x,x0,x1)))")
eq.append("Eq(W, P*t)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def gravitationalPotentialEnergy(find="U", printEq=False, **kwargs):
"""variables:
U = gravitational potential energy
G = 6.67e-11 N*m**2/kg**2
m1, m2 = mass1, mass2
r = distance between masses"""
dictionary_add_modify(kwargs, G=6.67e-11)
eq = list()
eq.append("Eq(U,-G*m1*m2/r)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def shearModulus(find="S", printEq=False, **kwargs):
"""variables:
S = shear modulus
sigma_shear = shear stress
dx = along shear force (parallel to surface)
h = height of subject that is subject to shear forces
"""
eq = list()
eq.append("Eq(S, sigma_shear/(dx/h))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def strain(find="epsilon", printEqs=True, **kwargs):
"""
Change in length per unit length
Where:
epsilon = strain
l, l0 = length, original length
"""
eq = list()
eq.append("Eq(epsilon, (l-l0)/l0)")
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def reynolds(find="Re", printEqs=True, **kwargs):
"""
Reynolds Number = Inertia / Viscosity
"""
eq = list()
eq.append("Eq(Re, rho * U * L / mu)")
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
# ref white pg 307
def S_ut(find="S_ut", printEqs=True, **kwargs):
"""
Ultimate tensile strength
Where:
HB = Brinnel Hardness
"""
eq = list()
eq.append("Eq(S_ut, 500*HB +/- 30*HB)") #psi; approx
eq.append("Eq(S_ut,3.45*HB +/- 0.2*HB") #MPa; approx
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def S_ys(find="S_ys", printEqs=True, **kwargs):
"""
Shear Yield Strength
Breaking strength in torsion
Where:
S_ys = shear yield strength
"""
eq = list()
eq.append("Eq(S_ys, 0.577*Sy") #Approximation
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def Capacitance(find="C", printEq=False, **kwargs):
"""
Usage: the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F).
Variables:
C = capacitance (1 Farads = 1 Coulomb/Volt)
epsilon = permitivity of the dielectric material between plates
A = surface area of each plate
d = distance between plates"""
eq = list()
eq.append("Eq(C,epsilon*A/d)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def sigma(find="sigma", printEqs=True, **kwargs):
"""
Stress
Where:
sigma = stress
P = load
A0 = original cross-sectional area
"""
eq = list()
eq.append("Eq(sigma, P / A)")
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def impulseEq(find, printEq=False, **kwargs):
"""variables:
J=impulse
p1,p2=initial,final momentum
F=sum of all forces
t,t0,t1 = time initial,final
Usage: The change in momentum of a particle during a time interval
Sample Units: Kg*m/s"""
eq = list()
eq.append("Eq(J,p2-p1)")
eq.append("Eq(J,integrate(F,(t,t0,t1)))")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def a_rad(printEq=False, **kwargs):
"""
Usage: radial acceleration (direction is towards axis of rotation)
Variables:
v_rad=radial velocity
r = radius
w = magnitude of angular velocity (w_z)"""
find = "a_rad"
eq = list()
eq.append("Eq(a_rad, v_rad**2/r)")
eq.append("Eq(a_rad, w**2*r)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
def U_R(find="U_R", printEqs=True, **kwargs):
"""
Resilience
Where:
U0 = strain energy density per unit volume
sigma = stress
epsilon = strain
"""
eq = list()
eq.append(
"Eq(U_R, integrate(sigma, (epsilon, 0, epsilon_el)))") #Approximation
eq.append("Eq(U_R, 0.5*S_y**2/E") #Approximation
return solveEqs(eq, find=find, printEq=printEqs, **kwargs)
def kinematicsEq(find, printEq=False, **kwargs):
"""variables:
d=distance, d0=initial distance,
v=velocity, v0=initial velocity,
a=acceleration,
t=time"""
eq = list()
eq.append("Eq(d, v*t)")
eq.append("Eq(v, v0 + a*t)") #constant x-acceleration only
eq.append("Eq(d, d0 + v0*t + 0.5*a*t**2)") #constant x-acceleration only
eq.append("Eq(d, d0 + v*t - 0.5*a*t**2)")
eq.append("Eq(v**2, v0**2 + 2*a*d)")
return solveEqs(eq, find, printEq=printEq, **kwargs)
