Question

The instantaneous voltage through a device of impedance \(20 \Omega\) is \(\varepsilon=80 \sin 100 \pi t\). The effective value of the current is, (a) \(3 \mathrm{~A}\) (b) \(2.828 \mathrm{~A}\) (c) \(1.732 \mathrm{~A}\) (d) \(4 \mathrm{~A}\)

Short answer

The effective value of the current (also known as RMS current) for the given instantaneous voltage function is approximately \(2.828A\) (option b). This can be calculated by first finding the instantaneous current using Ohm's law, and then determining the RMS value using the sinusoidal function formula.

Step-by-step solution

Write down the given information

Instantaneous voltage, \(\varepsilon = 80\sin(100\pi t)\) Impedance, \(Z = 20\Omega\)

Find the instantaneous current using Ohm's law

Ohm's Law states that, \(V = IZ\), where V is voltage, I is current, and Z is the impedance. Instantaneous current is given by, \(i(t) = \frac{\varepsilon (t)}{Z}\) So, \(i(t)= \frac{80\sin(100\pi t)}{20}\) \(i(t)= 4\sin(100\pi t)\)

Calculate the effective (RMS) current

The formula for the effective (RMS) value of a sinusoidal function is given by, \(\text{RMS value} = \frac{(\text{maximum value})}{\sqrt{2}}\) We already have the instantaneous current, so we can determine the maximum current by observing the given function, \(i(t) = 4\sin(100\pi t)\) The maximum value of the current is 4 A (since the maximum value of \(\sin(100\pi t)\) is 1). Applying the formula RMS value, \(I_{rms} = \frac{4}{\sqrt{2}}\) \(I_{rms} = 2\sqrt{2}\) \(I_{rms} ≈ 2.828 A\) From the given options, the effective (RMS) value of the current is: \(I_{rms} = 2.828A\) (option b)

Key concepts

Impedance

Impedance is a critical concept in AC circuits, similar to resistance but applicable to circuits carrying alternating current (AC). Unlike resistance, which only restricts the flow of direct current (DC), impedance also accounts for phase shifts between voltage and current in AC circuits.
Impedance incorporates both resistive and reactive components:

• **Resistive Component (R):** Similar to resistance in DC circuits, it opposes current flow but without causing any phase shift.
• **Reactive Component (X):** This opposes changes in current and voltage, causing phase shifts. It can be capacitive (X_C) or inductive (X_L), depending on the circuit elements involved.

Impedance is represented as a complex number, given as \( Z = R + jX \), where \( j \) is the imaginary unit. The magnitude of impedance, often just referred to as "impedance," is calculated using \( Z = \sqrt{R^2 + X^2} \). For purely resistive circuits, \( Z = R \), while for purely reactive circuits, it's \( Z = X \).
In our problem, the impedance is noted directly as \( 20 \Omega \), indicating it mainly has resistive properties for simplification purposes. This makes calculations straightforward using Ohm's Law.

Ohm's Law

Ohm's Law is a cornerstone of electrical engineering, providing a simple relationship between voltage, current, and resistance. In an AC circuit, the relationship extends to impedance, written as \( V = IZ \), where \( V \) is the voltage, \( I \) is the current, and \( Z \) is the impedance.
The law helps us determine any one of these variables if the other two are known. When dealing with AC circuits:

• **Voltage \( (V) \):** The potential difference applied to the circuit, here given by \( \varepsilon = 80 \sin(100\pi t) \). It's an instantaneous voltage that varies with time.
• **Current \( (I) \):** The flow of electric charge. Calculated by rearranging the formula for a given instantaneous voltage and impedance: \( i(t) = \frac{\varepsilon (t)}{Z} \)
• **Impedance \( (Z) \):** A combination of resistance and reactance, serving here as \( 20 \Omega \).

Using Ohm's Law, we calculated the instantaneous current as \( i(t) = 4 \sin(100\pi t) \). This value shows how current varies over time within the AC circuit, according to the changing instantaneous voltage.

RMS Current

Root Mean Square (RMS) Current is the effective current value used in AC circuit analysis to equate the heating effect of AC to that of DC. It provides a meaningful equivalent of sinusoidal currents, given that their intensity varies over time.
To calculate RMS current, we use the formula for a sinusoidal wave: \( I_{rms} = \frac{I_{max}}{\sqrt{2}} \). Here, \( I_{max} \) is the maximum or peak current, which is derived from the instantaneous current equation.

• **Maximum Current:** Retrieved from the expression \( i(t) = 4 \sin(100\pi t) \), the peak is \( 4 \text{ A} \) since the maximum value of \( \sin(\theta) \) is 1.
• **RMS Calculation:** Substituting into the formula gives \( I_{rms} = \frac{4}{\sqrt{2}} \approx 2.828 \text{ A} \).

Thus, the RMS value allows for easy comparisons and calculations within AC circuits, providing a basis for understanding the steady effect equivalent in terms of power.