Sum of potentials across a resistor in series with parallel RC

[UHchemstu]

Homework Statement

We have built a circuit with an ac source containing a resistor in series with a parallel RC combination, we have measured the voltage across the first resistor and the voltage across the RC combination with a high and a low frequency.
80Hz: V0 = 2,325 V en Vm = 2,780 V V0 +Vm = 5,105 V
50 kHz: V0 = 4,900 V en Vm = 0,156 V V0 +Vm = 5,056 V

(with V0 the voltage across the first resistor and Vm the voltage across the parallel RC combination).
(The power source delivered a voltage with an amplitude of 10V)

We saw that the sum of these voltages remained approx. the same regardless of frequency.
And that at higher frequencies the voltage across the first resistor rises and the voltage across the RC combination becomes less.

Can you explain this behaviour

thanks

Homework Equations

I think i'll need to use the impedance of a parallel RC circuit which is
$$\frac{1}{\sqrt{\frac{1}{R^2}+\frac{1}{Xc^2}}}$$

The Attempt at a Solution

I know how to calculate the impedance of the parallel RC combination and the impedance of the resistor (the impedance of a resistor is it's resistance) but i don't know how to combine these factors into finding an explanation for the behaviour.
I think it might have something to do with the dependence of the capacitor's reactance with frequency which is given by:
$$\frac{1}{j \times \omega \times C}$$

 

[tiny-tim]

Welcome to PF!

Hi UHchemstu ! Welcome to PF!

(have an omega: ω )

Yes, jX_C = 1/jωC, so 1/X_C² = ω²C²

 

[UHchemstu]

I already found the answer;
If what I think is right, then at very high frequencies the reactance of the capacitor goes down drastically,
so almost all of the current flows through the capacitor. The voltage over the parallel resistor and capacitor is then given by V=I*Xc with Xc the reactance of the capacitor, Xc is verry low at high frequencies so the voltage over the parallel RC combination is verry low whilst the voltage over the first resistor becomes higher because the sum of the voltages must be equal to the voltage of the source (the amplitude) and that remains the same regardless of frequency.

 

[tiny-tim]

Yes, that looks right!

(though you might like to add something about what happens to the current, and maybe even the power )

 

[DeeNos]

Your thinking is on the right track. To understand the behavior of the voltages in this circuit, we need to consider the impedance of each component and how it changes with frequency.

First, let's look at the resistor in series. The impedance of a resistor is simply its resistance, which does not change with frequency. So, regardless of the frequency of the AC source, the voltage across the resistor will remain the same.

Next, we need to consider the parallel RC combination. The impedance of this combination is given by the equation you provided, which includes both the resistance and the reactance of the capacitor. The reactance of a capacitor is inversely proportional to frequency, meaning that as the frequency increases, the reactance decreases. This has the effect of decreasing the overall impedance of the parallel RC combination at higher frequencies.

Now, let's consider the behavior of the voltages at different frequencies. At 80Hz, the impedance of the parallel RC combination is relatively high due to the low frequency, so the majority of the voltage is dropped across this component. This results in a lower voltage across the resistor. However, at 50kHz, the impedance of the parallel RC combination is much lower due to the higher frequency, meaning that a larger portion of the voltage is dropped across the resistor. This results in a higher voltage across the resistor and a lower voltage across the parallel RC combination.

Overall, the sum of the voltages across the resistor and the parallel RC combination remains approximately the same regardless of frequency because the impedance of the parallel RC combination decreases as the impedance of the resistor increases, balancing out the voltage distribution between the two components. This behavior is expected in a series circuit, where the total voltage drop is equal to the sum of the individual voltage drops across each component.