Resistors in AC Circuits: Videos & Practice Problems

In AC circuits, the current is sinusoidal, and Ohm's law states that the voltage across a resistor is the product of current and resistance. The maximum voltage can be expressed as $I^{\max }=RI\left(t\right)$. For a 10-ohm resistor with an RMS voltage of 120 volts, the RMS current is 12 amps, and the maximum current is approximately 17 amps. Multiple resistors can be combined into an equivalent resistance for analysis.

Resistors in AC Circuits

In alternating current (AC) circuits, the current produced by the source is sinusoidal, meaning it varies over time in a wave-like pattern. According to Ohm's Law, the voltage across a resistor as a function of time can be expressed as the product of the current through the resistor and its resistance. This relationship can be mathematically represented as:

$$ V(t) = I(t) \cdot R $$

For a sinusoidal current, the voltage across the resistor can be specifically described as:

$$ V(t) = I_{\text{max}} \cdot R \cdot \cos(\omega t) $$

Where \( I_{\text{max}} \) is the maximum current, \( R \) is the resistance, and \( \omega \) is the angular frequency. For example, if a 10-ohm resistor is connected to an outlet with a root mean square (RMS) voltage of 120 volts, we can calculate the RMS current using the formula:

$$ I_{\text{rms}} = \frac{V_{\text{rms}}}{R} $$

Substituting the values gives:

$$ I_{\text{rms}} = \frac{120 \text{ volts}}{10 \text{ ohms}} = 12 \text{ amps} $$

To find the maximum current, we use the relationship between maximum current and RMS current:

$$ I_{\text{max}} = \sqrt{2} \cdot I_{\text{rms}} $$

Thus, the maximum current is:

$$ I_{\text{max}} = \sqrt{2} \cdot 12 \text{ amps} \approx 17 \text{ amps} $$

When dealing with multiple resistors in an AC circuit, the approach remains consistent with direct current (DC) circuits. The resistors can be combined into a single equivalent resistor, allowing the same equations to be applied for analysis. This foundational understanding of resistors in AC circuits simplifies the problem-solving process significantly.

The voltage across a resistor is found to be given by v _R (t) = (10 V) cos[(120 s⁻¹)t]: 
a) At what frequency does the AC course operate? 
b) If the resistance is 12 Ω, what is the maximum current in this circuit? 
c) What is the RMS voltage of the AC source?

• A. a) 19.1 Hz;
  b) 0.83 A;
  c) 7.1 V
• B. a) 120 V;
  b) 1.2 A;
  c) 2.2 V
• C. a) 754 Hz;
  b) 120 A;
  c) 14 V

Answer: (A)

Resistors in Parallel in an AC Circuit

In an AC circuit with multiple resistors in parallel, each resistor shares the same voltage from the source. For a 10 ohm resistor in such a configuration, the maximum voltage across it is equal to the maximum voltage of the AC source, which is 5 volts. To find the maximum current through the 10 ohm resistor, we apply Ohm's Law, which states that current (I) is equal to voltage (V) divided by resistance (R). Thus, the maximum current (I₁₀) can be calculated as:

$$I_{10} = \frac{V_{10}}{R} = \frac{5 \text{ volts}}{10 \text{ ohms}} = 0.5 \text{ amps}$$

However, the question asks for the current as a function of time, not just the maximum current. The current in an AC circuit varies with time and can be expressed as:

$$I(t) = I_{max} \cdot \cos(\omega t)$$

Here, I_max is the maximum current (0.5 amps), and ω is the angular frequency. In this case, the angular frequency is given as 200 radians per second. Therefore, the current through the 10 ohm resistor at any point in time can be expressed as:

$$I(t) = 0.5 \text{ amps} \cdot \cos(200 \text{ s}^{-1} \cdot t)$$

This equation describes how the current through the 10 ohm resistor oscillates over time, reflecting the nature of alternating current.

Here’s what students ask on this topic:

What is the relationship between voltage and current in a resistor in an AC circuit?

In an AC circuit, the relationship between voltage and current in a resistor is governed by Ohm's law, which states that the voltage across the resistor (V) is equal to the current (I) flowing through it times the resistance (R). Mathematically, this is expressed as:

$V=IR$

Since the current in an AC circuit is sinusoidal, the voltage across the resistor will also be sinusoidal and in phase with the current. The instantaneous voltage can be expressed as:

$V\left(t\right)=I\left(t\right)R$

How do you calculate the RMS current in a resistor connected to an AC source?

To calculate the RMS (Root Mean Square) current in a resistor connected to an AC source, you use the RMS voltage and the resistance of the resistor. The formula is:

$I{\mathrm{RMS}}_{}=\frac{V_{{\mathrm{RMS}}_{}}}{R}$

For example, if a 10-ohm resistor is connected to an AC source with an RMS voltage of 120 volts, the RMS current would be:

$I{\mathrm{RMS}}_{}=\frac{120}{10}=12 A$

What is the maximum current in a resistor connected to an AC source with a given RMS voltage?

The maximum current (I_max) in a resistor connected to an AC source can be calculated from the RMS current (I_RMS) using the following relationship:

$I{\max }_{}=\sqrt{2}I{\mathrm{RMS}}_{}$

For example, if the RMS current is 12 amps, the maximum current would be:

$I{\max }_{}=\sqrt{2}12 A \approx 17 A$

How do you combine multiple resistors in an AC circuit?

Combining multiple resistors in an AC circuit is done in the same way as in a DC circuit. For resistors in series, the equivalent resistance (R_eq) is the sum of the individual resistances:

$R{\mathrm{eq}}_{}=R$₁+R₂+…+R_n

For resistors in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances:

$\frac{1}{R}=\frac{1}{R}$₁+1R₂+…+1R_n

Once combined into a single equivalent resistor, you can apply the same AC circuit analysis techniques as for a single resistor.

What is the significance of the RMS value in AC circuits?

The RMS (Root Mean Square) value in AC circuits is significant because it provides a measure of the effective or equivalent DC value of an AC voltage or current. The RMS value allows for the comparison of AC and DC circuits in terms of power dissipation. For a sinusoidal AC waveform, the RMS value is calculated as:

$V{\mathrm{RMS}}_{}=\frac{V}{{\max }_{}}$

and

$I{\mathrm{RMS}}_{}=\frac{I}{{\max }_{}}$

The RMS value is crucial for calculating power in AC circuits, as power is proportional to the square of the RMS voltage or current.