I'm reading a piece about Ohm's law and its relation to complex numbers. There is a piece of text about an inverse relationship, which I don't understand because it's about a capacitor. Can someone help me? I would personally conclude that the relationship is directly proportional rather than inversely proportional.
The text:
Ohm's law
Ohm's law relates the current i through a resistor R and the voltage u across the resistor R:u = i.R
Ohm's law for sinusoidal currents:
The above also applies to sinusoidal currents:
Suppose current () = . (2. π. f. + ) with t the time: • I is the amplitude • f is the frequency • φ is the phase shift
Again, we can represent the current as a revolving vector.
The imaginary value (projection on the imaginary axis) gives the sinusoidal current. We use the “bold” I to indicate that it is a complex number.
The voltage then becomes: u(t) = . (2. π. f. + ). = . . (2. π. f. + )
• We see that the frequency and phase are preserved.
• The amplitude changes and becomes the amplitude of the current times R.
The voltage is again a spinning vector with the same frequency and phase, but the magnitude has changed. After all, we know that multiplying by a real number only changes the magnitude.
Voltage when the resistor passes through a capacitor
What will the voltage be if we pass the resistor through a capacitor?
Capacity:
There is a relationship between the charge q on the capacitor and the voltage u across the capacitor. This relation is the capacitance C of the capacitor: = /
The voltage across the capacitor cannot suddenly change. For the voltage to change, the charge on the capacitor must also change. This charge is provided by the current flowing to it, thereby building up the charge. So the voltage will only change later than the current. The phase of the voltage will have a negative phase with respect to the current. It can be shown that the phase difference is 90°. The voltage thus lags behind the current by 90°.
From = q / u we get = q / C
It applies:
• The larger the capacitor, the more charge q is required for the voltage u to change.
• The charge q is proportional to the current i ( ~ ). The greater the current, the more charge flows to the capacitor.
So the voltage u will be inversely proportional to the current i. This inverse proportionality is easy to express: ~ / C
Calculation with complex numbers:
We have yet to obtain the phase difference of 90°. This is not possible with real numbers, but it can be done with complex numbers. For this we introduce the notion of “pure imaginary number”. Purely imaginary numbers are imaginary numbers whose real part is 0.
We know that multiplying a complex number by a pure positive imaginary number shifts the phase by +90°. If we divide by a purely positive imaginary number, the phase will rotate by -90°. We must have this.
We then obtain the expression: = / (.)
This is equivalent to u = i.R , but R is replaced by 1/ (j.C) This is called the impedance Z of the capacitor. Z is a complex number.
So for sinusoidal voltages holds: U = I ∙ Z
Why is the bold text true?
The relationship between $u$ and $i$ expressed in $u \sim i/C$, where $i$ is a variable rather than the imaginary unit, says that $u$ and $i$ are directly proportional, not inversely proportional.
You are right. The statement in your source text is wrong.