Whizmath: AC Circuits Basics

1. Introduction to AC Circuits: The Pulse of Modern Electronics

Welcome to Whizmath! In previous lessons, you might have encountered Direct Current (DC) circuits, where current flows in one constant direction. While DC is essential for battery-powered devices and digital electronics, the world outside your small devices, including your home's power outlets and the global power grid, is dominated by Alternating Current (AC).

AC is characterized by current and voltage that periodically reverse direction and vary in magnitude, typically following a sinusoidal pattern. This seemingly simple change from constant to alternating flow opens up a completely new and fascinating realm of circuit behavior, involving components like inductors and capacitors in ways that DC circuits cannot emulate.

Understanding AC circuits is crucial for anyone interested in electrical engineering, electronics, or even just understanding how the power in your home works. AC allows for efficient power transmission over long distances using transformers, forms the basis of radio communication, and is integral to countless electronic devices, from filters to oscillators.

In this lesson, we will lay the foundation for comprehending AC circuits. We'll start by defining AC voltage and current, then explore how familiar components like resistors behave differently, and introduce new concepts like reactance for inductors and capacitors. Finally, we'll examine simple combinations of these components in series (RL, RC, and RLC circuits), paving the way for more advanced circuit analysis. Prepare to unlock the dynamic world of alternating current!

2. AC Voltage & Current: Sinusoidal Fundamentals

Unlike DC, where voltage and current are constant, AC voltage and AC current continuously change their magnitude and periodically reverse their direction. The most common form of AC is a sinusoidal wave.

2.1 Sinusoidal AC Waveforms

A sinusoidal AC voltage or current can be mathematically described as: $$ v(t) = V_{\text{peak}} \sin(\omega t + \phi_v) $$ $$ i(t) = I_{\text{peak}} \sin(\omega t + \phi_i) $$ where:

• $v(t)$ and $i(t)$ are the instantaneous voltage and current at time $t$.
• $V_{\text{peak}}$ and $I_{\text{peak}}$ are the peak (or maximum) voltage and current, respectively.
• $\omega$ (omega) is the angular frequency, measured in radians per second (rad/s).
• $\phi_v$ and $\phi_i$ are the phase angles (or phase shifts) for voltage and current, respectively, indicating their starting point in the cycle relative to a reference.

2.2 Key Parameters of AC Waveforms

• Frequency ($f$): The number of complete cycles per second. Measured in Hertz (Hz). $$ f = \frac{\omega}{2\pi} $$ In North America, standard AC frequency is $60 \text{ Hz}$; in Europe and many other parts of the world, it's $50 \text{ Hz}$.
• Period ($T$): The time taken for one complete cycle. It's the reciprocal of frequency. $$ T = \frac{1}{f} = \frac{2\pi}{\omega} $$ Measured in seconds (s).
• Phase (Angle $\phi$): Represents the position of a point on the waveform cycle. The phase difference between voltage and current is crucial in AC circuits.
• Root Mean Square (RMS) Value: Since AC voltage and current continuously change, we need a way to quantify their "effective" value for power calculations. The RMS value is equivalent to the DC value that would produce the same heating effect in a resistive load. For a sinusoidal waveform: $$ V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}} \approx 0.707 V_{\text{peak}} $$ $$ I_{\text{RMS}} = \frac{I_{\text{peak}}}{\sqrt{2}} \approx 0.707 I_{\text{peak}} $$ The voltage values typically quoted for household electricity (e.g., $120 \text{ V}$ or $230 \text{ V}$) are RMS values.

Understanding these fundamental parameters is the first step to analyzing the behavior of different components in an AC environment.

3. Resistors in AC Circuits: Purely Resistive Behavior

A resistor is the simplest component to analyze in an AC circuit because its behavior is straightforward and identical to its behavior in DC circuits, except that the voltage and current values are continuously changing.

3.1 Ohm's Law for Resistors in AC

When an AC voltage is applied across a resistor, the current through it is still governed by Ohm's Law. If the instantaneous voltage is $v(t) = V_{\text{peak}} \sin(\omega t)$, then the instantaneous current $i(t)$ is: $$ i(t) = \frac{v(t)}{R} = \frac{V_{\text{peak}}}{R} \sin(\omega t) $$ So, $I_{\text{peak}} = V_{\text{peak}}/R$. We can also express this in terms of RMS values: $$ I_{\text{RMS}} = \frac{V_{\text{RMS}}}{R} $$

3.2 Phase Relationship: In Phase

The most important characteristic of a resistor in an AC circuit is the phase relationship between voltage and current. For a purely resistive circuit, the voltage across the resistor and the current through it are always in phase.

• This means that they reach their peak values, zero values, and minimum values at the same instant in time. There is no phase shift between them ($\phi_v - \phi_i = 0$).
• When voltage is maximum positive, current is maximum positive. When voltage is zero, current is zero. When voltage is maximum negative, current is maximum negative.

3.3 Power Dissipation in Resistors

Resistors dissipate electrical energy as heat. In an AC circuit, the instantaneous power $p(t) = v(t) \cdot i(t)$. Since voltage and current are in phase, the power is always positive (or zero), meaning energy is continuously dissipated. The average power dissipated by a resistor in an AC circuit is given by: $$ P_{\text{avg}} = V_{\text{RMS}} I_{\text{RMS}} = I_{\text{RMS}}^2 R = \frac{V_{\text{RMS}}^2}{R} $$ This is identical to the power formula for DC circuits, confirming that RMS values are indeed the "effective" values for power calculations.

While resistors behave simply, inductors and capacitors introduce new concepts: reactance and phase shifts, which are central to AC circuit analysis.

4. Inductors in AC Circuits: Inductive Reactance ($X_L$)

An inductor is a passive electrical component that stores energy in a magnetic field when current flows through it. In DC circuits, an ideal inductor acts like a short circuit once the current stabilizes. However, in AC circuits, its behavior is profoundly different due to the constantly changing current, which induces a back electromotive force (EMF).

4.1 Inductive Reactance ($X_L$)

In AC circuits, an inductor opposes changes in current. This opposition to AC current is called inductive reactance, denoted by $X_L$. It is measured in Ohms ($\Omega$), just like resistance, but it is not a resistance because it doesn't dissipate energy. Instead, it temporarily stores energy in its magnetic field.

The formula for inductive reactance is: $$ X_L = \omega L = 2\pi f L $$ where:

• $X_L$ is the inductive reactance (in Ohms, $\Omega$).
• $\omega$ is the angular frequency of the AC source (in rad/s).
• $f$ is the linear frequency of the AC source (in Hz).
• $L$ is the inductance of the inductor (in Henries, H).

From the formula, we can see that:

• As the frequency $f$ increases, $X_L$ increases. This means inductors offer more opposition to higher frequency AC signals.
• For DC (where $f=0$), $X_L = 0$, meaning an ideal inductor offers no opposition to steady DC current.

4.2 Phase Relationship: ELI the ICE man (Voltage Leads Current)

The most critical characteristic of an inductor in an AC circuit is the phase relationship between voltage and current. For a purely inductive circuit, the voltage across the inductor leads the current through it by $90^\circ$ ($\pi/2$ radians).

A common mnemonic for remembering this is "ELI the ICE man":

• ELI: In an Electrical circuit with an Lnductor, the voltage (E) Leads the current (I).

This means that the voltage across the inductor reaches its peak value a quarter of a cycle before the current reaches its peak value. This phase difference is due to the inductor's opposition to changes in current; it takes time for the current to build up in response to the applied voltage.

While inductors offer opposition to current, they do not dissipate average power over a full cycle in an ideal AC circuit. The energy stored in the magnetic field during one quarter cycle is returned to the circuit in the next quarter cycle. This is related to the concept of reactive power, which we'll discuss later.

5. Capacitors in AC Circuits: Capacitive Reactance ($X_C$)

A capacitor is a passive electrical component that stores energy in an electric field. In DC circuits, an ideal capacitor acts like an open circuit (block) once it's fully charged. However, in AC circuits, its behavior is fundamentally different because the continuous charging and discharging allows current to flow.

5.1 Capacitive Reactance ($X_C$)

In AC circuits, a capacitor opposes changes in voltage. This opposition to AC current is called capacitive reactance, denoted by $X_C$. Like inductive reactance, it is measured in Ohms ($\Omega$) but does not dissipate energy. Instead, it temporarily stores energy in its electric field.

The formula for capacitive reactance is: $$ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} $$ where:

• $X_C$ is the capacitive reactance (in Ohms, $\Omega$).
• $\omega$ is the angular frequency of the AC source (in rad/s).
• $f$ is the linear frequency of the AC source (in Hz).
• $C$ is the capacitance of the capacitor (in Farads, F).

From the formula, we can see that:

• As the frequency $f$ increases, $X_C$ decreases. This means capacitors offer less opposition (effectively act more like a short circuit) to higher frequency AC signals.
• For DC (where $f=0$), $X_C$ approaches infinity, meaning an ideal capacitor acts like an open circuit, blocking steady DC current.

5.2 Phase Relationship: ELI the ICE man (Current Leads Voltage)

For a purely capacitive circuit, the current through the capacitor leads the voltage across it by $90^\circ$ ($\pi/2$ radians).

Using our mnemonic "ELI the ICE man":

• ICE: In an electrical circuit with a capacitor (C), the current (I) Leads the voltage (E). (Note: This mnemonic simplifies the "L" for "Leads" in ICE, even though the component is "C" for capacitor. The key is "I before E" in ICE man).

This means that the current through the capacitor reaches its peak value a quarter of a cycle before the voltage across it reaches its peak value. This phase difference arises because current must flow to charge and discharge the capacitor before the voltage across it can change significantly.

Similar to inductors, ideal capacitors do not dissipate average power over a full cycle in an ideal AC circuit. The energy stored in the electric field during one quarter cycle is returned to the circuit in the next quarter cycle.

6. Series RL Circuits: Resistor and Inductor in Series

Now that we understand the individual behavior of resistors and inductors in AC circuits, let's combine them into a series RL circuit. When a resistor and an inductor are connected in series to an AC voltage source, the current flowing through both components is the same, but the voltages across them are out of phase.

6.1 Impedance ($Z$) of an RL Circuit

In AC circuits, the total opposition to current flow is called impedance, denoted by $Z$. Impedance is a complex quantity that includes both resistance ($R$) and reactance ($X_L$ or $X_C$). Since $V_R$ and $V_L$ are $90^\circ$ out of phase, we cannot simply add their magnitudes. Instead, we use a vector-like sum (phasor addition) to find the total voltage or impedance.

For a series RL circuit, the impedance is calculated as: $$ Z = \sqrt{R^2 + X_L^2} $$ where:

• $Z$ is the total impedance of the circuit (in Ohms, $\Omega$).
• $R$ is the resistance (in Ohms, $\Omega$).
• $X_L$ is the inductive reactance (in Ohms, $\Omega$).

Ohm's Law for AC circuits (using impedance) is then: $$ V_{\text{RMS}} = I_{\text{RMS}} Z $$ or $$ I_{\text{RMS}} = \frac{V_{\text{RMS}}}{Z} $$

6.2 Phase Angle ($\phi$)

In an RL circuit, the total voltage across the circuit ($V_{\text{total}}$) will lead the current ($I$) by a phase angle $\phi$ that is between $0^\circ$ and $90^\circ$. This phase angle is determined by the relative magnitudes of the resistance and inductive reactance. $$ \tan\phi = \frac{X_L}{R} $$ Therefore, the phase angle is: $$ \phi = \arctan\left(\frac{X_L}{R}\right) $$ A positive phase angle indicates that the voltage leads the current, which is characteristic of inductive circuits.

6.3 Behavior at Different Frequencies

• Low Frequencies ($f \to 0$, DC): $X_L \to 0$. The circuit acts predominantly resistive ($Z \approx R$). The current is nearly in phase with the voltage.
• High Frequencies ($f \to \infty$): $X_L \to \infty$. The impedance becomes very large ($Z \approx X_L$), and the circuit acts predominantly inductive, with voltage leading current by nearly $90^\circ$.

RL circuits are fundamental in applications like filtering (e.g., low-pass filters) and motor control.

7. Series RC Circuits: Resistor and Capacitor in Series

Next, let's explore the behavior of a series RC circuit, where a resistor and a capacitor are connected in series to an AC voltage source. Similar to the RL circuit, the current through both components is the same, but the voltages across them are out of phase.

7.1 Impedance ($Z$) of an RC Circuit

For a series RC circuit, the impedance $Z$ combines the resistance $R$ and the capacitive reactance $X_C$. Since $V_R$ and $V_C$ are $90^\circ$ out of phase (with $V_C$ lagging $V_R$), we again use a phasor sum.

The formula for the impedance of a series RC circuit is: $$ Z = \sqrt{R^2 + X_C^2} $$ where:

• $Z$ is the total impedance of the circuit (in Ohms, $\Omega$).
• $R$ is the resistance (in Ohms, $\Omega$).
• $X_C$ is the capacitive reactance (in Ohms, $\Omega$).

Ohm's Law for AC circuits still applies: $$ V_{\text{RMS}} = I_{\text{RMS}} Z $$ or $$ I_{\text{RMS}} = \frac{V_{\text{RMS}}}{Z} $$

7.2 Phase Angle ($\phi$)

In an RC circuit, the total voltage across the circuit ($V_{\text{total}}$) will lag the current ($I$) by a phase angle $\phi$ that is between $0^\circ$ and $-90^\circ$. $$ \tan\phi = \frac{-X_C}{R} $$ Therefore, the phase angle is: $$ \phi = \arctan\left(\frac{-X_C}{R}\right) $$ A negative phase angle indicates that the voltage lags the current, which is characteristic of capacitive circuits. (Equivalently, the current leads the voltage by a positive angle).

7.3 Behavior at Different Frequencies

• Low Frequencies ($f \to 0$, DC): $X_C \to \infty$. The impedance becomes very large ($Z \approx X_C$), and the circuit acts predominantly capacitive, effectively blocking DC current. The voltage lags current by nearly $90^\circ$.
• High Frequencies ($f \to \infty$): $X_C \to 0$. The circuit acts predominantly resistive ($Z \approx R$). The current is nearly in phase with the voltage.

RC circuits are widely used in applications such as timing circuits, high-pass filters, and integrators.

8. Series RLC Circuits: The Full Picture

Combining all three passive components—a resistor (R), an inductor (L), and a capacitor (C)—in series creates a series RLC circuit. This circuit is particularly important because it exhibits the phenomenon of resonance, which we explored in a previous lesson.

8.1 Total Impedance ($Z$) of an RLC Circuit

In a series RLC circuit, the total impedance $Z$ accounts for the resistance and both types of reactance. Since inductive reactance ($X_L$) and capacitive reactance ($X_C$) have opposite phase effects (voltage leads current in an inductor, current leads voltage in a capacitor), they tend to cancel each other out.

The formula for the total impedance of a series RLC circuit is: $$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$ or, substituting the reactance formulas: $$ Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2} $$ where:

• $Z$ is the total impedance (in Ohms, $\Omega$).
• $R$ is the resistance.
• $X_L = \omega L$ is the inductive reactance.
• $X_C = 1/(\omega C)$ is the capacitive reactance.

8.2 Phase Angle ($\phi$)

The phase angle $\phi$ between the total voltage and current in an RLC circuit depends on whether the inductive reactance or capacitive reactance is larger: $$ \tan\phi = \frac{X_L - X_C}{R} $$

• If $X_L > X_C$: The circuit is predominantly inductive, and the voltage leads the current ($\phi > 0$).
• If $X_C > X_L$: The circuit is predominantly capacitive, and the voltage lags the current ($\phi < 0$).
• If $X_L = X_C$: The circuit is purely resistive, and the voltage and current are in phase ($\phi = 0$). This is the condition for resonance.

8.3 Resonance in Series RLC Circuits

A key feature of the RLC series circuit is series resonance. Resonance occurs when the inductive reactance perfectly cancels out the capacitive reactance ($X_L = X_C$). At this resonant frequency ($\omega_0$ or $f_0$), the impedance of the circuit is at its minimum, equal to just the resistance $R$. $$ \omega_0 L = \frac{1}{\omega_0 C} \implies \omega_0 = \frac{1}{\sqrt{LC}} $$ At resonance, the current in the circuit reaches its maximum value for a given voltage, and the circuit behaves like a purely resistive circuit (voltage and current are in phase). This property is fundamental to tuning circuits in radios and other communication systems.

The amount of resistance $R$ in the circuit determines the damping of the resonance. A smaller $R$ means less damping, a sharper and higher resonance peak (higher Q-factor), and better frequency selectivity.

RLC circuits are the workhorses of many electronic applications, from filters and oscillators to tuning circuits and impedance matching networks.

9. Power in AC Circuits: Real, Reactive, Apparent

In AC circuits containing reactive components (inductors and capacitors), the concept of power is more nuanced than in DC circuits. We distinguish between three types of power: real power, reactive power, and apparent power.

9.1 Real Power ($P$)

Also known as active power or average power, this is the actual power dissipated by the circuit as heat or converted into useful work (e.g., light from a bulb, mechanical power from a motor). Real power is only dissipated by the resistive components. $$ P = V_{\text{RMS}} I_{\text{RMS}} \cos\phi $$ where $\cos\phi$ is the power factor. Real power is measured in Watts (W).

9.2 Reactive Power ($Q$)

Reactive power is the power that continuously flows back and forth between the source and the reactive components (inductors and capacitors). It is stored in the magnetic fields of inductors and the electric fields of capacitors during one part of the cycle and returned to the source in another part. Reactive power does not perform any useful work or dissipate energy. $$ Q = V_{\text{RMS}} I_{\text{RMS}} \sin\phi $$ Reactive power is measured in Volt-Amperes Reactive (VAR).

9.3 Apparent Power ($S$)

Apparent power is the product of the total RMS voltage and RMS current in the circuit. It represents the total power delivered by the source, without considering the phase difference between voltage and current. $$ S = V_{\text{RMS}} I_{\text{RMS}} $$ Apparent power is measured in Volt-Amperes (VA).

These three powers are related by the power triangle, similar to impedance: $$ S^2 = P^2 + Q^2 $$

9.4 Power Factor ($\cos\phi$)

The power factor is the cosine of the phase angle between voltage and current ($\cos\phi$). It ranges from 0 to 1 and indicates how effectively the electrical power is being converted into useful work. $$ \text{Power Factor} = \cos\phi = \frac{P}{S} $$

• For purely resistive circuits, $\phi = 0^\circ$, so $\cos\phi = 1$. All apparent power is real power.
• For purely reactive circuits (ideal inductors or capacitors), $\phi = \pm 90^\circ$, so $\cos\phi = 0$. All apparent power is reactive power, and no real power is dissipated.
• For general RLC circuits, $0 < \cos\phi < 1$. A higher power factor means more efficient use of power. Utilities often penalize industries with low power factors because it requires them to supply more apparent power than is actually consumed as real power.

Understanding these power concepts is crucial for designing efficient and cost-effective AC power systems.

10. Applications of AC Circuits: Powering Our World

AC circuits are ubiquitous in modern technology and daily life. Their unique properties, particularly the ability to easily change voltage levels using transformers and the behavior of reactive components, make them indispensable.

10.1 Power Transmission and Distribution

• Transformers: AC voltages can be easily stepped up (increased) for efficient long-distance transmission and then stepped down (decreased) for safe use in homes and businesses. This is the primary reason AC is preferred over DC for power grids. High voltages reduce current ($P=VI$), which in turn minimizes power loss ($P_{\text{loss}} = I^2 R$) over long transmission lines.

10.2 Filtering and Signal Processing

• Filters: RL, RC, and RLC circuits are fundamental building blocks for filters.
  • Low-Pass Filters: Allow low-frequency signals to pass while attenuating high frequencies (e.g., in audio speakers to send low frequencies to the woofer).
  • High-Pass Filters: Allow high-frequency signals to pass while attenuating low frequencies (e.g., sending high frequencies to the tweeter).
  • Band-Pass Filters: Allow a specific range of frequencies to pass (e.g., tuning a radio to a specific station).
  • Band-Stop Filters: Attenuate a specific range of frequencies.
• Equalizers: Used in audio systems to adjust the relative amplitudes of different frequency bands.

10.3 Oscillators and Resonators

• Oscillators: RLC circuits, particularly resonant ones, are used to generate specific frequencies in electronic devices, from radio transmitters and receivers to clock generators in computers.
• Tuning Circuits: The resonance property of RLC circuits allows devices like radios and televisions to select specific broadcast channels.

10.4 Motors and Generators

• AC Motors: The vast majority of electric motors (e.g., in appliances, industrial machinery) run on AC power, leveraging the changing magnetic fields created by alternating current.
• AC Generators (Alternators): Power plants generate AC electricity because it's more efficient for large-scale production and transmission.

10.5 Communication Systems

• Radio, TV, Cell Phones: All rely heavily on AC circuit principles for transmitting and receiving electromagnetic waves at specific frequencies.

From the simple light bulb to complex communication networks, AC circuits form the backbone of modern electrical and electronic systems.

11. Conclusion: The Ubiquitous Nature of AC

You've now taken your first deep dive into the dynamic world of Alternating Current (AC) circuits on Whizmath. You've moved beyond the static nature of DC to embrace the oscillating and phase-shifting characteristics that define AC.

Key takeaways from this lesson include:

• The fundamental properties of sinusoidal AC voltage and current, including peak, RMS, frequency, period, and phase.
• The straightforward behavior of resistors in AC, where voltage and current remain in phase.
• The frequency-dependent opposition of inductors through inductive reactance ($X_L = \omega L$), causing voltage to lead current.
• The frequency-dependent opposition of capacitors through capacitive reactance ($X_C = 1/\omega C$), causing current to lead voltage.
• The concept of impedance ($Z$) as the total opposition to current in AC circuits, combining resistance and reactance.
• The analysis of simple series RL, RC, and RLC circuits, understanding how their impedance and phase angles are calculated.
• A brief introduction to the different types of power in AC: real, reactive, and apparent power, and the power factor.
• The recognition of resonance in RLC circuits as a critical phenomenon for tuning and filtering.

AC circuit theory is the backbone of modern electrical engineering. From the power lines that bring electricity to your home, to the intricate circuits in your smartphone and computer, to advanced communication systems, the principles you've learned here are fundamental. As you progress in your study of electronics, these basic concepts will serve as a solid foundation for understanding more complex systems, including filters, amplifiers, and advanced power electronics.

Electrons dance to the rhythm of AC, powering the innovations of our world.