CHPTR 31 Notes: AC Circuits

Generators and Energy Conversion

• Generators convert mechanical energy into electrical energy.

• An alternating current (AC) generator is depicted.

AC and DC Currents

• Devices such as light bulbs and toasters operate effectively on both AC and direct current (DC).

• Motors can be more easily adapted to operate with AC currents than with DC.

Circuit Symbols and Time Varying Voltages

• The circuit symbol for an AC source is presented.

• The time-varying voltage is defined mathematically as:
    - $V(t) = V imes ext{cos}( heta)$ where $heta = ext{ω}t$.

• In the U.S., the frequency of AC is given as $f = 60 ext{Hz}$ which corresponds to angular frequency $ext{ω} = 377 ext{rad/s}$.

• Here, $V$ represents the "voltage amplitude" or peak voltage.

Root Mean Square (RMS) Values

• The root-mean-square (rms) value of sinusoidal voltage is given by:
    - $V_{rms} = rac{V}{ ext{√2}}$.

• The rms value of sinusoidal current is defined as:
    - $I_{rms} = rac{I}{ ext{√2}}$.

• Therefore, for a specification of 120 volts AC, the rms voltage $V_{rms} = 120 ext{volts}$, leading to a peak voltage amplitude $V = 120 ext{V} imes ext{√2} ext{ or } 170 ext{V}$.

Instantaneous Current and Voltage in Resistors

• The instantaneous current $I(t)$ and voltage $V(t)$ across a resistor are in phase, leading to the relationships:
    - $I(t) = I_{max} ext{cos}( ext{ω}t)$
    - $V(t) = V_{max} ext{cos}( ext{ω}t)$
    - Indicates that $V = I imes R$ at any instant.

Inductive Reactance

• The instantaneous voltage across an inductor is illustrated as:
    - $V_L = L rac{di}{dt}$ and is given by:
      - $V_L(t) = L ext{ω} ext{sin}( ext{ω}t)$.

• This depicts that the voltage leads the current by 90 degrees, meaning:
    - $I_L(t) = I_{max} ext{sin}( ext{ω}t)$.
    - Therefore, the relationship is captured between voltage and current as:
      - $rac{V_L}{I_L} = X_L = ext{ω}L$.

Capacitors and Their Reactance

• The instantaneous current, charge, and voltage across a capacitor are given as:
    - $V(t) = rac{1}{C} imes q(t)$, where current $I(t)$ leads voltage by 90 degrees:
      - $I(t) = I_{max} ext{sin}( ext{ω}t)$,
      - $V(t) = V_{max} ext{cos}( ext{ω}t)$.

• Indicating the phase relationship:
    - $ext{φ} = -90°$.

• The expression for capacitive reactance is:
    - $X_C = rac{1}{ ext{ω}C}$.

Currents in Capacitors and Inductors

• Current lags voltage across an inductor, represented as or recursively defined:
    - $V = IX_L$.

• Conversely, current leads voltage across a capacitor:
    - $V = IX_C$.

• Implying:
    - For a circuit, the current through each element remains the same at any given instant.

Voltage Relationships in RLC Circuits

• For any circuit moment, the total voltage across R, L, and C combines to equal the source voltage:
    - $V_{R} + V_{L} - V_{C} = V_{s}$.

• Voltage amplitudes related can be summarized as:
    - $V_R^2 + (V_L - V_C)^2 = V^2$.

Impedance in Series RLC Circuits

• The impedance of a series RLC circuit is defined as:
    - $Z = ext{√}(R^2 + (X_L - X_C)^2)$,
      - Where the phase angle $ext{φ}$ is calculated as:
      - $ext{tan}( ext{φ}) = rac{X_L - X_C}{R}$.

Resonance in RLC Circuits

• For radio signals tuned to a frequency of $101.5 ext{kHz}$, maximum current $I$ is achieved when impedance $Z$ is minimal:
    - $V = IZ$.

• Resonance occurs when the inductive reactance equals the capacitive reactance:
    - $X_L = X_C$,
    - This leads to the condition:
      - $ext{ω}_0 L = rac{1}{ ext{ω}_0 C}$.

• The resonance frequency can be computed as:
    - $ext{ω}_0 = rac{1}{ ext{√(LC)}}$
      - Divide this result by $2 ext{π}$ to convert to frequency in Hertz.

Transformers and Power Transmission

• Losses in power transmission are minimized by transmitting at the highest possible voltages.

• Voltage step-up and step-down transformers are utilized for efficient power transmission.

• The voltage across the secondary coil of a transformer is defined as:
    - $rac{V_1}{V_2} = rac{N_1}{N_2}$ where $N_1$ and $N_2$ are the number of turns in the primary and secondary coils respectively.