Solving for Angular Frequency w: Help!

[chopficaro]

help me out guys i have a test on Wednesday, and I am stuck on a problem, it seems I am supposed to solve for angular frequency w but I am getting a 4th power equation for it which is unsolvable

a 10k ohm resistor and a 100uF capacitor are in parallel, determine the angular frequency w where the absolute value of the input impedance is 2k ohms

z=1/(1/10000+1/(jwc))
z=1/(1/10000+1/(jw(.0001))
z=(1/10000-1/(jw(.0001))/((1/10000+1/(jw(.0001))(1/10000-1/(jw(.0001)))

ok so I've conjugated the denominator of z so that the real part and the imaginary part are separated, we have a quadratic equation for w in the denominator

the absolute value of z is the sum of the squares of the real part and the imaginary part

|z|=2000=sqrt((1/10000)/((1/10000+1/(jw(.0001))(1/10000-1/(jw(.0001)))^2+(-1/(jw(.0001))/((1/10000+1/(jw(.0001))(1/10000-1/(jw(.0001)))^2)

now we have a 4th power equation for w in the denominator which is unsolvable, there must be something I am doing wrong

 

[tiny-tim]

hi chopficaro!

(have a mu: µ and a couple of omegas: ω Ω )

a 10k ohm resistor and a 100uF capacitor are in parallel, determine the angular frequency w where the absolute value of the input impedance is 2k ohms

z=1/(1/10000+1/(jwc))
z=1/(1/10000+1/(jw(.0001))
z=(1/10000-1/(jw(.0001))/((1/10000+1/(jw(.0001))(1/10000-1/(jw(.0001))) …

why are you making it so complicated?

you know that |Z| = 2000, so |1/Z| = 0.0005,

and 1/Z = 1/10⁴ + 1/1/jω10^-4 = 10^-4(1 + jω) …

carry on from there ​

 

[chopficaro]

i think what u are doing gives the real part of z, not the absolute value, which is the magnitude sqrt(real^2 +imaginary^2)
i don't think ur method takes into account the coefficient of j

 

[tiny-tim]

hi chopficaro!

(try using the X² icon just above the Reply box )

i think what u are doing gives the real part of z, not the absolute value …

no, it should give you the whole of Z …

then you get the magnitude from that ​

 

[chopficaro]

if we vary the imaginary part, your solution for |z| doesn't change, its as though were setting z to a scalar resistance, its the real part not the magnitude, the magnitude is the absolute vale, that's what we are looking for

 

[chopficaro]

ok i got it

we NEED to cojugate the denominator if he gives us the real or immaginary part of z

if he gives us the magnitude |z| or the angle, we want to put the two in parallel by the rule of Zeq=Z1*Z2/(Z1+Z2)

so for a capacitor and a resistor in parallel we get

Z1=R

Z2=1/jwc

Zeq=R1(1/jwc)/(R1+(1/jwc))

Zeq=R1/(jwc(R1+1/jwc))

Zeq=R1/(R1jwc+1)

then INSTEAD OF CONJUGATING we put the numerator and denominator in polar form

remember to take j out of the imaginary part when converting to polar

Zeq=(R1<90)/denominator

denominator=(R1jwc+1) to polar = sqrt((R1wc)^2+1^2)<arctan(R1wc/1)

and if we cary out the division we get out impedance in polar form, and the magnitude is |z|

|Zeq|<angle=(R1/sqrt((R1wc)^2+1^2))<90-arctan(R1wc/1)

the problem is what is w given |Zeq| so we plug in a value for |Zeq| and we may have to use the quadratic formula to find w, but maybe not

|Zeq|=(R1/sqrt((R1wc)^2+1^2))

R1/|Zeq|=sqrt((R1wc)^2+1^2)

(R1/|Zeq|)^2=(R1wc)^2+1^2

(R1/|Zeq|)^2-1=(R1wc)^2

remember here that the sqrt can be + or -

+-sqrt((R1/|Zeq|)^2-1)=(R1wc)

+-sqrt((R1/|Zeq|)^2-1)/(R1C)=w !