3.2 Complex Impedances (8 marks) a) What is the impedance of the circuit in figu

Question

3.2 Complex Impedances (8 marks) a) What is the impedance of the circuit in figure 3b? Write your answer in cartesian form. Note: You may leave your answer in terms of R, w and C or plug in numbers, but make sure to write in Cartesian form, Le a tjb with a, and b purely real. b) What is the impedance for DC signals? Hint: what is w for a constant voltage c) If connected to a 10 V, com 2m x 50 kHz sinusoidal wave v(t) 10V cos(wt) the current will also be a sine wave: i(t)-fosin(wt + ). Find io, and . Hints: (i) You can represent your impedance from part (b) in polar form and your voltage as toejut to make your life WAY easter! (n) You moy represent your phase angle n terms of , ie = ZT and round r to 2 significant figur es b) a) 1.0 R 10 k2 0.8 L=1 mF 0.6 0.4 0.2 0.0 3.0 0.0 0.5 1.0 Time [/T)

Explanation / Answer

part a:

capacitive impedance of C=Xc=1/(s*C)

where s=j*w

it is in parallel with R.

total impedance=R*(1/(sC))/(R+(1/sC))

=R/(sRC+1)

inductive impedance=s*L

it is in series with s*L

so net impedance=(R/(sRC+1))+s*L=(s^2*R*L*C+s*L+R)/(s*R*C+1)

using s=j*w

net impedance=(R-w^2*R*L*C+j*w*L)/(j*w*R*C+1)

multiplying with 1-j*w*R*C

net impedance=(R-w^2*R*L*C+j*w*L)*(1-j*w*R*C)/(1+w^2*R^2*C^2)

part b:

for constant signal , w=0

net impedance=R

part c:

given :

w=2*pi*50*10^3 rad/s

net impedance=1.0132*10^(-5) + i*3.1384*10^2 ohms

as real part is very less as compared to imaginary part, net impedance=i*313.84 ohms

so current =voltage/impedance

=10/(i 313.84)

=-i*0.031863

in time domain, current will be 0.031863*cos(w*t-90 degrees)

=0.031863*sin(w*t) A

so i0=0.031863 A

phi=0 degrees

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