3_phase_pll_dq_detector.txt

Consider the output of a ramp phase θ

abc = 3 phase voltages
dq = real and imaginary state vector of 3-phase voltages

A = amplitude

Va = Asin(wt)
Va = Asin(wt-2π/3)
Va = Asin(wt+2π/3)

Vd = +2/3 [ cos(θ)Va + cos(θ-2π/3)Vb + cos(θ+2π/3)Vc ]
Vq = -2/3 [ sin(θ)Va + sin(θ-2π/3)Vb + sin(θ+2π/3)Vc ]

# Solving for Vq


# In control loop
We use Vq as an error signal

E(s) = Vq(s)
U(s)/E(s) = (Kp*s + Ki)/s

θ(t) = [Wc + u(t)] t
θ(s) = [Wc + U(s)]/s

# Responses
Open   loop: G_OL(s) = A * (Kp*s + Ki)/s * 1/s = A(Kp*s + Ki)/s^2
Closed loop: G_CL(s) = G_OL(s) / [1 + G_OL(s)]
	     G_CL(s) = A(Kp*s + Ki)/(s^2 + AKp*s + AKi)
	     
# Designing the closed loop response
If Ki/Kp >> 0, then we can ignore the pole

D(s) = s^2 + 2ζWn*s + Wn^2

Wn = sqrt(AKi)
ζ = AKp/(2Wn) = AKp/(2sqrt(AKi) 
ζ = sqrt(A/Ki) * Kp/2

# Design parameters
ζ = 1/sqrt(2) = 0.707		# Critically damped response
Wn = 100Hz 			# As a reasonable naturally frequency for 3-phase 50Hz
A = 240 * sqrt(2) = 339.4 V 	# For our grid voltage