Ehh, Smith charts are good for transmission lines and RF stuff, not so much for general circuits. You're better off working the algebra.

Yep, I have done resistance networks. Which is why my approach was to use the impedances to go for the voltage divider. I'm a little confused though, is this the (a) correct approach?

So I have Ztotal = 50 - j5 (+ C)

then Vo(t) = Vs(t) * C/(50-j5)+C

But I'm not sure where to go from here (if this is correct)

What is C? -- I think you forgot to multiply by j*w (or s). First step (easy to forget!): convert component values to impedances (Zc = 1 / (j*w*C), etc.). Then substitute those into the node/loop equations and continue.

Generally, frequency is w (lowercase omega, but good luck getting that to work on this forum..), so that time-domain voltage is written as A * sin(w*t). So here, w = 10^6 rad/s.

Let's label the resistor R, the "C" capacitor C1 and the "0.2uF" capacitor C2.

Total impedance Zt = Zr + Zc1 + Zc2

= R + 1 / (j*w*C1) + 1 / (j*w*C2)

= R - (j / w) * (1/C1 + 1/C2)

(Note the use of 1/j == -j, an identity of the multiplicative rule.)

Current = I = Vs / Z

= Vs / [R - (j / w) * (1/C1 + 1/C2)]

(and so on, however far you wish to simplify it)

Voltage on C2 = Vo = Zc2 * I

= Vs * Zc2 / (Zr + Zc1 + Zc2)

Voltage on C1 + C2 = Vc = (Zc1 + Zc2) * I

= Vs * (Zc1 + Zc2) / (Zr + Zc1 + Zc2)

Vo / Vc = 1/2 = capacitor divider gain (as requested) =

Zc2 * I / ((Zc1 + Zc2) * I)

= Zc2 / (Zc1 + Zc2)

The 1/(j*w) stuff is common to all terms, so we can continue and write...

= (1/C2) / (1/C1 + 1/C2)

= 1 / [C2 * (C2 + C1)/(C1*C2)]

= C1*C2 / [C2 * (C2 + C1)]

= C1 / (C1 + C2)

Which is, by value, the opposite of the voltage divider equation, because capacitance is inversely proportional to impedance. An important thing to keep in mind when you're doing capacitors in series (uses resistors in parallel formula) and parallel (they add) and such!

Since we want 1/2 = C1 / (C1 + C2), then

2*C1 = (C1 + C2)

C1 = C2

Which is what we intuitively expect.

I skipped writing out all the currents and voltages exactly, because it's a lot of bother for no value, but again, you're welcome to expand any of the equations along the way by substituting in the relevant bits.

By the way, it's mildly interesting to note what they

*didn't* say: when asking for a ratio of node voltages, note that that ratio itself is, in general, a complex number. When analyzing a filter, for example, usually one is interested in Vo/Vi == H(w), the transfer function (gain) of the system. The resulting form is a rational polynomial in j*w. By...manifest destiny, rather than sheer luck, we are given H(w) = 1/2 -- a function that is constant with w. But, that is, of course, the only function we can have here, because the frequency response of two capacitors in series is exactly proportional, and anything else nearby is a red herring (or busywork).

Another thing, perhaps less "mildly interesting" and more "pedantic": note that the question is phrased entirely in the time domain. Lowercase script v and i, functions of time. In that case, you shouldn't write the impedances at all (which are a frequency domain, steady state form), but the differential equations of the components (I = C * dV/dt, etc.). And, in that case, the transfer function "vo(t) / vi(t)" == h(t) could be tremendously more complex than the mere ratio 1/2, or of a polynomial in frequency; in general, one cannot divide functions, one must perform a convolution (written "vo(t) = vi(t) star h(t)") instead! It seems all the more bizarrely providential that, out of the space of possible functions used with convolution, the function "1/2" should be chosen!

Tim