Working out freq for circuit to be in phase

[craka]

Homework Statement

AC circuit Vs is 100V 50Hz and is used as reference.
there are two impedences in parallel, made up of a 5ohm resistor and an inductor reactance = +j2 ohm that is parallel with 3 ohm resistor and a inductor reactance = -j3 ohm (though I though negative imaginary part was for capactance?)

voltage across the 3 ohm resistor is 45 Volts

The branch with the resistor and inductor is called I1 , branch with resistor and capacitor is I2
Is is the total current from supply before the parallel branch.

Please see attachment of circuit

Homework Equations

I2 = v/I angle = arctan x/r

Is = Vs/Zt = Vs * 1/ (Z1 parallel Z2) = Z1Z2/(Z1+Z2)

The Attempt at a Solution

I2 = 45/3 =15A angle = artan -3/3 = -45 degrees

magnitude of Vs = 15 * sqrt(3^2+3^2) = 63.64 volts

Z1 = 5+2j = 5.385 angle 21.8 deg Z2 = 3-3j = 4.234 angle -45 deg

Z1//Z2 = {(5.385 angle 21.8) * ( 4.234 angle -45)}/ {(5+2j)+(3-3j)}

Z1//Z2 = 22.849 angle -23.2 deg / 8.06 angle -7.125

Z1//Z2 = 2.835 angle -16.075 deg

Is = 63.64/(2.835 angle -16.075) = 22.45 angle -16.08 degFrom here is where I'm lost on how to calculate the frequency needed to get this circuit to have current and voltage in phase (zero degrees)

 

[CEL]

For voltage and current to be in phase, the impedance must be a pure resistance, so the imaginary part of your impedance must be zero.
Since you know the impedances of capacitor and inductor at 50Hz, you can calculate their values.
Write Z1, Z2 and Z1//Z2 as functions of the known values of resistances, inductances and capacitances and the unknown value of frequency. Then calculate the frequency for which the imaginary part of the impedance is zero.

 

[craka]

I had worked out the inductance values of components for the known frequency of 50Hz howerver I'm unsure what you mean by "write Z1, Z2 and Z1//Z2 as functions of the known values of resistance, and inductance capacitance and the unknonwn value of frequency".

 

[CEL]

I had worked out the inductance values of components for the known frequency of 50Hz howerver I'm unsure what you mean by "write Z1, Z2 and Z1//Z2 as functions of the known values of resistance, and inductance capacitance and the unknonwn value of frequency".

If you call f the frequency, $$Z_1=R_1 + 2\pi f L_1$$.
You have a similar expression for $$Z_2$$ and for the parallel of the two. The only unknown is f, that you calculate by making the imaginary part of the impedance equal to zero.

 

[craka]

So I have Z1 = 5 + 2pifL and Z2 = 3 + 2pifl but what is the expression for Z1//Z2 with using the above value of Z1 and Z2 ??

 

[CEL]

So I have Z1 = 5 + 2pifL and Z2 = 3 + 2pifl but what is the expression for Z1//Z2 with using the above value of Z1 and Z2 ??

No,
$$Z_1 = 5 +j2\pi f L$$
$$Z_2 = 3+\frac{1}{j2\pi f C}$$

and
$$Z_1 // Z_2 = \frac{Z_1 Z_2}{Z_1 + Z_2}$$

 

[craka]

from that i get this

$$\begin{array}{l} z_1 //z_2 = \frac{{(5 + j2\pi fL) \times (3 + j\frac{1}{{2\pi fC}})}}{{(5 + j2\pi fL) + (3 + j\frac{1}{{2\pi fC}})}} \\ = \frac{{15 + j\frac{5}{{2\pi fC}} + j6\pi fL + \frac{L}{C}}}{{8 + j2\pi fL + j\frac{1}{{2\pi fC}}}} \\ \end{array}$$
Again I'm stuck. Could you please give me some more assistance thankyou.

 

[CEL]

from that i get this

$$\begin{array}{l} z_1 //z_2 = \frac{{(5 + j2\pi fL) \times (3 + j\frac{1}{{2\pi fC}})}}{{(5 + j2\pi fL) + (3 + j\frac{1}{{2\pi fC}})}} \\ = \frac{{15 + j\frac{5}{{2\pi fC}} + j6\pi fL + \frac{L}{C}}}{{8 + j2\pi fL + j\frac{1}{{2\pi fC}}}} \\ \end{array}$$
Again I'm stuck. Could you please give me some more assistance thankyou.

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator. In this way only the numerator will be complex.
Equate the imaginary part of the numerator to zero and calculate the value of f.