Ac Voltage Applied To A Resistor

When an alternating voltage $v = v_m \sin(\omega t)$ is applied across a resistor $R$, the current through it is $i = i_m \sin(\omega t)$, where $i_m = v_m/R$. The current and voltage are in phase, meaning they reach their maximum, minimum, and zero values simultaneously. Although the average current over a cycle is zero, power is dissipated due to Joule heating ($P_{avg} = i_{rms}^2 R$). The rms (root mean square) values of voltage and current are defined as $V = v_m/\sqrt{2}$ and $I = i_m/\sqrt{2}$ respectively, such that the average power can be expressed as $P_{avg} = IV = I^2R = V^2/R$.

Figure 7.1 shows the circuit, Figure 7.2 illustrates the in-phase relationship between voltage and current, and Example 7.1 calculates resistance, peak voltage, and rms current for a light bulb.

Representation Of Ac Current And Voltage By Rotating Vectors — Phasors

Phasors are rotating vectors used to represent alternating voltages and currents. The magnitude of a phasor represents the amplitude (peak value) of the quantity, and its angle represents its phase. Phasors rotate at the angular frequency $\omega$ of the AC source. The instantaneous value of the AC quantity is the projection of its phasor onto a reference axis (usually vertical). Phasor diagrams simplify the analysis of AC circuits by allowing vector addition of voltages and currents, taking their phase differences into account.

Figure 7.4 illustrates the phasor representation for a resistor, showing the voltage and current phasors in phase.

Ac Voltage Applied To An Inductor

When an AC voltage $v = v_m \sin(\omega t)$ is applied to a pure inductor $L$, the current is $i = i_m \sin(\omega t - \pi/2)$, where the current lags the voltage by $\pi/2$ (90 degrees). The amplitude of the current is $i_m = v_m/X_L$, where $X_L = \omega L$ is the inductive reactance. $X_L$ is the opposition offered by the inductor to the AC current and depends on the frequency and inductance. The average power dissipated in a pure inductor over a full cycle is zero, as the energy stored in the magnetic field during one half-cycle is returned to the circuit in the other half.

Figure 7.5 shows the circuit, Figure 7.6 illustrates the phasor diagram and the phase lag of current, and Example 7.2 calculates inductive reactance and current for an inductor.

Ac Voltage Applied To A Capacitor

When an AC voltage $v = v_m \sin(\omega t)$ is applied to a pure capacitor $C$, the current is $i = i_m \sin(\omega t + \pi/2)$, where the current leads the voltage by $\pi/2$ (90 degrees). The amplitude of the current is $i_m = v_m/X_C$, where $X_C = 1/(\omega C)$ is the capacitive reactance. $X_C$ is the opposition offered by the capacitor to the AC current and depends inversely on frequency and capacitance. Like an inductor, a capacitor also does not dissipate average power over a full cycle, as energy is stored in the electric field during charging and returned during discharging.

Figure 7.7 shows the circuit, Figure 7.8 illustrates the phasor diagram and the phase lead of current, and Example 7.4 calculates capacitive reactance and current. Example 7.3 discusses the behavior of a capacitor with a light bulb in AC and DC circuits.

Ac Voltage Applied To A Series LCR Circuit

Phasor-Diagram Solution

In a series LCR circuit, the voltage and current may not be in phase. Phasor diagrams are used to analyze the circuit. The voltages across the resistor ($V_R$), inductor ($V_L$), and capacitor ($V_C$) have specific phase relationships with the current ($I$): $V_R$ is in phase with $I$, $V_L$ leads $I$ by $\pi/2$, and $V_C$ lags $I$ by $\pi/2$. The total voltage $V$ across the source is the vector sum of these phasor voltages.

The impedance ($Z$) of the series LCR circuit is the total opposition to current flow and is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$, where $R$ is resistance, $X_L = \omega L$ is inductive reactance, and $X_C = 1/(\omega C)$ is capacitive reactance.

The current amplitude is $i_m = v_m/Z$. The phase angle ($\phi$) between the source voltage and the current is given by $\tan\phi = (X_L - X_C)/R$. If $X_L > X_C$, the current lags the voltage; if $X_C > X_L$, the current leads the voltage.

Figure 7.11 and Figure 7.12 illustrate phasor diagrams and the impedance triangle. Example 7.6 calculates current, voltage drops, and resolves the "paradox" of voltage sums using phase analysis.

Resonance

Resonance occurs in a series RLC circuit when the inductive reactance equals the capacitive reactance ($X_L = X_C$). At this resonant frequency ($\omega_0$), the impedance $Z$ is minimum ($Z=R$), and the current amplitude ($i_m = v_m/R$) is maximum.

The resonant angular frequency is $\omega_0 = 1/\sqrt{LC}$, and the resonant frequency is $f_0 = \omega_0/(2\pi)$. Resonance phenomena are crucial in tuning circuits for radios and TVs, where the circuit's natural frequency is matched to the desired signal frequency to maximize the current.

Example 7.8 calculates the impedance, current, phase difference, power, and power factor for a series LCR circuit. Example 7.9 determines the resonant frequency and circuit parameters at resonance.

Power In Ac Circuit: The Power Factor

The instantaneous power ($p$) in an AC circuit is $p = v i$. The average power ($P$) dissipated over a full cycle is given by:

$P = VI \cos\phi = I^2R = \frac{V^2}{R_{eff}}$

where $V$ and $I$ are the rms values of voltage and current, and $\cos\phi$ is the power factor. $\phi$ is the phase difference between the voltage and current. Power is dissipated only in the resistive component of the circuit. For purely inductive or capacitive circuits, $\phi = \pm \pi/2$, $\cos\phi = 0$, and the average power dissipated is zero (termed "wattless current").

A low power factor implies a large current is needed to deliver a given power, leading to higher power loss ($I^2R$) in transmission lines. Capacitors can be used to improve the power factor by reducing the phase difference between voltage and current.

Example 7.7 explains the importance of a low power factor and how capacitors can improve it. Example 7.8 calculates the power dissipated and power factor for a series LCR circuit.

Transformers

A transformer is a device that changes AC voltage from one level to another using the principle of mutual induction. It consists of two coils (primary and secondary) wound on a soft iron core.

For an ideal transformer (negligible resistance and flux leakage), the ratio of the voltages is equal to the ratio of the number of turns:

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$

And the ratio of currents is:

$\frac{I_p}{I_s} = \frac{N_s}{N_p}$

A step-up transformer increases voltage ($N_s > N_p$) and decreases current. A step-down transformer decreases voltage ($N_s < N_p$) and increases current. Transformers are essential for efficient long-distance transmission of electrical power.

Energy losses in real transformers occur due to flux leakage, winding resistance, eddy currents, and hysteresis.

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