Demonstration of equation for Impedance and phase angle on a RLC series circuit
Let’s consider a RLC series circuit,
being i(t)=Imcos(wt+ϕ) the intensity flowing
through resistor (R), inductor (L) and capacitor
(C). Drop of potential (voltage) on each of
such devices will be:
uR(t) = RIm cos (wt)
uL(t) = LwIm cos (wt +π/2)
uC(t) = (1/Cw)Im cos (wt -π/2)
And drop of potential on terminals of RLC dipole (total drop of potential):
u(t) = uR (t)+ uL (t)+ uC (t)= RIm cos (wt)+ LwIm cos (wt +π/2)+ (1/Cw)Im cos (wt -π/2)
(1)
As uR, uL , and uC are sinusoidal functions, their addition will be a sinusoidal function too, being
amplitude (Um) and initial phase (ϕ) unknown: u(t) =Um cos (wt+ϕ)
(2)
By expanding (1) with equation for cosine(A+B) (cos(A+B)=cosAcosB-sinAsinB) and taking in
account that sin(π/2)=1 and cos(π/2)=0:
u(t) = RIm cos (wt)+ LwIm (cos (wt)cos(π/2)- sin(wt)sin(π/2))+(1/Cw)Im(cos (wt)cos(π/2)+sin(wt)sin(π/2)=
RIm cos (wt)- LwImsin(wt)+(1/Cw)Imsin(wt) (3)
And expanding (2):
u(t) =Um (cos(wt)cos(ϕ)-sin(wt)sin(ϕ)) (4)
Identifying coefficients of sin(wt) and cos(wt) between (3) and (4):
sin(wt):
-LwIm +(1/Cw)Im=-Umsin(ϕ) (5)
cos(wt):
RIm= Umcos(ϕ) (6)
Lw −
By dividing both equations (5)/(6):
R
1
Cw = tgϕ
Squaring (5) and (6) and adding it, taking in account that cos2(ϕ)+ sin2(ϕ)=1:
(R2+(Lw-(1/Cw))2Im2=Um2(cos2(ϕ)+ sin2(ϕ))⇒
U m = I m R 2 + (Lω −
1 2
)
Cω