RCR (series-parallel) Circuit Analysis

[itsthewoo]

Homework Statement

If the input AC voltage amplitude is Vo and the angular frequency w, find the impedance of the circuit below and the amplitude of the current that the generator provides. What is the behavior of the circuit at very low and at very high frequencies?

This is a circuit with AC source connected to a resistor R that is in series with a resistor R' and capacitor C that are in parallel. No values are given.

Homework Equations

Z(w) = f(w) + j*g(w)

|Z(w)| = sqrt(f^2(w) + g^2(w))

Vo = Io(w) * Z(w)

f(w) = ?, g(w) = ?

The Attempt at a Solution

Z(w) = R + (1/R' + jwC)^-1

Z(w) = R + ( (1 + R'jwC) / R' )^-1

Z(w) = R + R' / (1 + R'*jwC)

I'm stuck after this point; I can't think of any algebra that let's me isolate a function that ends up as f(w) + j*g(w). Help?

 

[CEL]

Multiply the numerator and the denominator of your fraction by the complex conjugate of the denominator. In this way you have a real denominator. Then you can separate the real from the imaginary part of the equation.

 

[Mene]

I would suggest approaching this problem by first simplifying the expression for Z(w) and then using the given equations to find the impedance and current amplitude.

To simplify Z(w), you can use the fact that the denominator of the second term can be rewritten as (1 + jwRC) by multiplying by the complex conjugate. This will result in a simpler expression for Z(w) that can be written as a single complex number.

Once you have the simplified expression for Z(w), you can use the given equations to find the impedance and current amplitude. For example, you can use the equation |Z(w)| = sqrt(f^2(w) + g^2(w)) to find the magnitude of Z(w) and then use the equation Vo = Io(w) * Z(w) to solve for the current amplitude Io(w).

At very low frequencies (w → 0), the capacitor will act as an open circuit and the circuit will behave as a simple series circuit with only the resistors. This means that the impedance will be equal to the sum of the resistances (R + R') and the current amplitude will be Vo / (R + R').

At very high frequencies (w → ∞), the capacitor will act as a short circuit and the circuit will behave as a simple parallel circuit with only the capacitor and resistor R'. This means that the impedance will be equal to R' and the current amplitude will be Vo / R'.

Overall, the behavior of the circuit will depend on the value of the frequency w. At low frequencies, the impedance will be dominated by the resistors and the current amplitude will be small. At high frequencies, the impedance will be dominated by the capacitor and the current amplitude will be larger. This behavior can be visualized by plotting the magnitude of Z(w) as a function of frequency.