Impedance in AC Circuits: Videos & Practice Problems

Impedance in AC circuits combines resistance and reactance, similar to how voltage and current interact in series and parallel configurations. In a series RC circuit, the total voltage is calculated using the Pythagorean theorem, leading to the equation for impedance: R^2+(1ωC)^2. Understanding impedance is crucial for analyzing AC circuits effectively.

Impedance in AC Circuits

Impedance is a crucial concept in alternating current (AC) circuits, acting as a measure of how much a circuit resists the flow of current when an AC voltage is applied. It is similar to resistance but also incorporates the effects of reactance, which arises from capacitors and inductors. In AC circuits, elements can be connected in series or parallel, each affecting the current and voltage differently.

In a series circuit, the current remains constant across all components, meaning the current phasors align. Conversely, in a parallel circuit, the voltage remains constant, leading to aligned voltage phasors. For example, in a series circuit with a resistor and a capacitor connected to an AC source, the total voltage across both components (denoted as \( V_{RC} \)) is not simply the sum of the voltages across each component. Instead, the relationship is determined using phasors, which can be added like vectors.

For a series RC circuit, the maximum voltage can be calculated using the Pythagorean theorem, where:

$$ V_{RC} = \sqrt{V_R^2 + V_C^2} $$

Here, \( V_R \) is the voltage across the resistor, and \( V_C \) is the voltage across the capacitor. The maximum current (\( I_{max} \)) can be expressed in terms of the impedance (\( Z \)), which is defined as:

$$ Z = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} $$

where \( R \) is the resistance, \( \omega \) is the angular frequency, and \( C \) is the capacitance. This formula allows us to understand the effective reactance of the entire circuit.

In a series circuit with a resistor and an inductor, the voltage across the resistor is in phase with the current, while the voltage across the inductor leads the current by 90 degrees. The total maximum voltage in this case is also calculated using the Pythagorean theorem:

$$ V_{max} = \sqrt{V_R^2 + V_L^2} $$

Substituting the expressions for the voltages gives:

$$ Z = \sqrt{R^2 + (\omega L)^2} $$

where \( L \) is the inductance. This highlights that the impedance for a series RL circuit differs from that of a series RC circuit, emphasizing the importance of understanding the specific configuration of components in AC circuits.

Overall, impedance serves as a comprehensive measure of how a circuit responds to AC voltage, taking into account both resistance and reactance, and is essential for analyzing and designing AC circuits effectively.

Impedance of a Parallel LR AC Circuit

In a parallel LR circuit, which consists of an inductor (L) and a resistor (R), the impedance can be analyzed through phasor diagrams. In parallel configurations, all components share the same voltage, meaning their voltage phasors are in phase. The current phasor for the resistor is also in phase with the voltage phasor, while the current phasor for the inductor lags the voltage phasor by 90 degrees.

The total current in the circuit is the vector sum of the currents through the resistor and inductor. To find the impedance (Z) of the circuit, we start with the relationship between maximum voltage (V_max) and maximum current (I_max), given by:

$$ I_{max} = \frac{V_{max}}{Z} $$

In a parallel circuit, the maximum voltage across each component is the same, allowing us to express the currents through the resistor and inductor as:

$$ I_R = \frac{V_{max}}{R} $$

$$ I_L = \frac{V_{max}}{X_L} $$

Where the inductive reactance (X_L) is defined as:

$$ X_L = \omega L $$

Using the Pythagorean theorem, the total current can be expressed as:

$$ I_{total} = \sqrt{I_R^2 + I_L^2} $$

Substituting the expressions for the currents, we have:

$$ I_{total} = \sqrt{\left(\frac{V_{max}}{R}\right)^2 + \left(\frac{V_{max}}{X_L}\right)^2} $$

Factoring out V_max gives:

$$ I_{total} = \frac{V_{max}}{Z} = V_{max} \sqrt{\frac{1}{R^2} + \frac{1}{X_L^2}} $$

From this, we can derive the expression for the impedance:

$$ \frac{1}{Z} = \sqrt{\frac{1}{R^2} + \frac{1}{(\omega L)^2}} $$

This equation highlights that in parallel circuits, impedance is often expressed as the reciprocal of the total impedance, emphasizing the relationship between the components. Understanding this concept is crucial for analyzing AC circuits effectively.

What's the impedance of a parallel RC AC circuit?

• A. Z = sqrt(R² - 1/ω²C²)
• B. Z = sqrt(1/R² + ω²C²)
• C. 1/Z = sqrt(1/R² + ω²C²)
• D. 1/Z = sqrt(1/R² + 1/ω²C²)

Answer: (C)

An AC source operates at a maximum voltage of 120 V and an angular frequency of 377 s^-1 . If this source is connected in parallel to a 15 Ω resistor and in parallel to a 0.20 mF capacitor, answer the following questions: 
a) What is the maximum current produced by the source? 
b) What is the maximum current through the resistor? 
c) What is the maximum current through the capacitor?

• A. a) 12 V b) 8A c) 4A
• B. a) 12 V b) 8A c) 9A
• C. a) 7.8 V b) 7.6A c) 0.2A
• D. a) 8.9 V b) 8A c) 4A

Answer: (B)

Here’s what students ask on this topic:

What is impedance in an AC circuit?

Impedance in an AC circuit is a measure of the opposition that a circuit presents to the flow of alternating current (AC). It combines both resistance (R) and reactance (X), which includes inductive reactance (X_L) and capacitive reactance (X_C). Impedance is represented by the symbol Z and is measured in ohms (Ω). The formula for impedance in a series RC circuit is given by:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Understanding impedance is crucial for analyzing how AC circuits behave under different conditions.

How do you calculate the impedance of a series RC circuit?

To calculate the impedance (Z) of a series RC circuit, you need to consider both the resistance (R) and the capacitive reactance (X_C). The formula for impedance in a series RC circuit is:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Here, ω (omega) is the angular frequency of the AC source, and C is the capacitance. This formula is derived using the Pythagorean theorem to combine the resistance and reactance vectors.

What is the difference between impedance and reactance?

Impedance (Z) and reactance (X) are related but distinct concepts in AC circuits. Impedance is the total opposition to AC and includes both resistance (R) and reactance (X). Reactance, on the other hand, is the opposition to AC due to inductors and capacitors only. Reactance can be inductive (X_L) or capacitive (X_C). The formula for impedance in a series RC circuit is:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Impedance is a more comprehensive measure as it accounts for both resistance and reactance.

How does impedance affect the current in an AC circuit?

Impedance (Z) affects the current (I) in an AC circuit by determining the total opposition to the flow of current. According to Ohm's Law for AC circuits, the current is given by:

$\frac{V}{Z}$

where V is the voltage. Higher impedance results in lower current for a given voltage, and vice versa. Impedance combines both resistance (R) and reactance (X), so it provides a complete picture of how the circuit will behave under AC conditions.

What is the impedance of a series RL circuit?

The impedance (Z) of a series RL circuit, which includes a resistor (R) and an inductor (L), is calculated using the following formula:

$\sqrt{R^2+{\omega }^L}$

Here, ω (omega) is the angular frequency of the AC source, and L is the inductance. This formula is derived by combining the resistance and inductive reactance (X_L) vectors using the Pythagorean theorem.