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Derive expression and solve v and vout in polar

(See attached file for full problem description)

Refer to diagram (attached)

1.Derive an algebraic expression for the transfer function of the circuit assuming that no external load is connected between terminals A&B. Simplify the expression as far as possible but do not attempt to rationalise it. Refer to the two reactances as Xc & Xl.

2. Calculate voltages V & Vout in polar form and draw a phasor diagram showing these voltages together with the source voltage phasor.

(See attached file for full problem description)


Solution Preview

I'm going to use a text notation to draw equivalent circuits:
(-) means series connection
(||) means parallel connection
(ang) means the angle in phasor notation

The problem circuit is (in phasor notation)

V - XC - {500 || (2000 + XL) }

Vout is the voltage across the impedance XL.
Other given values:
V = 10 ang 0. (Eq. a)
XC = 1/jwC = 1/j*50e3*20e-9 = -j1000. (Eq. b)
XL = jwL = j*50e3*25e-3 = j1250. (Eq. c)

Write equivalent circuits using parallel and series circuit equations. Series elements add algebraically: eg. Xeq = X1 + X2. Parallel elements add like resistors in parallel: eg. Xeq = X1*X2/(X1+X2)

The original circuit can be rewritten:

V - XC - XP

where XP = 500*(2000+XL)/(XL+500+2000) (Eq. 1)

Now we can write the current flowing from the source
I = V/Z = V / (XC + XP). (Eq. 2)

Use a current divider to find the current in each of the two branches of the original ...

Solution Summary

The following is a detailed solution with equations. The solution is in the designated box and in an attachment. There is a diagram in the attachment as well.