1.8 AC Circuits, Frequency, Wavelength, and RMS Values

1.8 AC Circuits, Frequency, Wavelength, and RMS Values

Unlike direct current, Alternating Current (AC) continually varies in magnitude and periodically reverses direction. In radio communications, high-frequency AC currents generated by transmitters travel along transmission lines and radiate from antennas as electromagnetic waves. A thorough understanding of AC sine waves, frequency, wavelength, RMS voltage, and RF skin effect is mandatory for ACMA Standard Theory candidates.

---

1. AC Sinusoidal Waveforms, Frequency, and Period

The fundamental waveform generated by an ideal AC alternator or RF signal generator is a sine wave (sinusoid). A sine wave represents the smooth, continuous mathematical function of an angle over time:

$v(t) = V_{\text{pk}} \sin(2 \pi f t + \phi)$

Definitions:

• Cycle: One complete repetition of a periodic waveform (from $0^\circ$ through $+90^\circ$ peak, $180^\circ$ zero crossing, $-90^\circ$ negative peak, back to $360^\circ$).
• Period ($T$): The time taken in seconds to complete one single cycle.
• Frequency ($f$): The number of complete cycles occurring per second. The SI unit of frequency is the Hertz (Hz), named after Heinrich Hertz ($1\text{ Hz} = 1\text{ cycle/second}$).

Frequency and Period are inversely proportional:

$f = \frac{1}{T} \quad \vert \quad T = \frac{1}{f}$

Metric Frequency Units in Radio:

• Hertz (Hz): $10^0\text{ Hz}$
• Kilohertz (kHz): $10^3\text{ Hz}$ (e.g., AM broadcast band $531 - 1602\text{ kHz}$, HF voice channel bandwidth $2.8\text{ kHz}$)
• Megahertz (MHz): $10^6\text{ Hz}$ (e.g., Australian 80m amateur band $3.5\text{ MHz}$, 2m band $144\text{ MHz}$)
• Gigahertz (GHz): $10^9\text{ Hz}$ (e.g., 23cm amateur band $1.24\text{ GHz}$, microwave links)

---

2. Radio Wavelength and Electromagnetic Velocity

Radio waves are electromagnetic waves traveling through space at the speed of light ($c$). In free space (vacuum or air), velocity is approximately:

$c \approx 299,792,458\text{ m/s} \approx 3.00 \times 10^8\text{ m/s}$

Wavelength ($\lambda$)

Wavelength ($\lambda$, Lambda) is the physical distance between two consecutive identical points on a wave (e.g., peak to peak or trough to trough) measured along the direction of propagation.

$\lambda = \frac{c}{f}$

Where:

• $\lambda$ is wavelength in metres (m)
• $c$ is speed of light ($3.00 \times 10^8\text{ m/s}$)
• $f$ is frequency in Hertz (Hz)

Practical Engineering Formula (Frequencies in MHz):

$\lambda_{\text{metres}} = \frac{300}{f_{\text{MHz}}}$

Antenna Sizing Applications

Antenna physical dimensions are directly proportional to wavelength:

• Half-Wave Dipole Antenna: Total theoretical electrical length $\lambda / 2$. $L_{\text{half-wave (m)}} \approx \frac{142.5}{f_{\text{MHz}}} \quad \text{(accounting for conductor end-effect velocity factor } \approx 0.95\text{)}$
• Quarter-Wave Vertical Antenna: Length $\lambda / 4$. $L_{\text{quarter-wave (m)}} \approx \frac{71.25}{f_{\text{MHz}}}$

Worked Example 1.5: 7 MHz 40-Meter Band Antenna Calculation

Calculate the free-space wavelength for an amateur radio transmission at $7.100\text{ MHz}$, and determine the physical length of a resonant quarter-wave ($\lambda/4$) mobile whip antenna.

Solution: $\lambda = \frac{300}{7.100\text{ MHz}} = 42.25\text{ metres}$ $L_{\text{quarter-wave}} = \frac{71.25}{7.100\text{ MHz}} \approx 10.04\text{ metres}$

---

3. AC Voltage Parameters: Peak, Peak-to-Peak, and RMS Values

Because sine wave voltage varies continuously over time, engineers define several specific voltage levels:

+Vpk ------------+ (Positive Peak)
                 |   \
                 |    \      Sine Wave
                 |     \_____
0V --------------+--------------+-------------
                       /     |
                      /      | 
-Vpk -----------------+      + (Negative Peak)
|<----- Vpp = 2 * Vpk ------>|

1. Peak Voltage ($V_{\text{pk}}$): The maximum instantaneous voltage measured from the zero reference line to the maximum crest.
2. Peak-to-Peak Voltage ($V_{\text{pp}}$): The total voltage displacement measured from the negative trough to the positive crest: $V_{\text{pp}} = 2 \times V_{\text{pk}}$
3. Root-Mean-Square Voltage ($V_{\text{rms}}$): The effective AC voltage value that produces the exact same heating power in a purely resistive load as an equivalent DC voltage.

For pure sine waves:

$V_{\text{rms}} = \frac{V_{\text{pk}}}{\sqrt{2}} \approx 0.707 \times V_{\text{pk}}$ $V_{\text{pk}} = \sqrt{2} \times V_{\text{rms}} \approx 1.414 \times V_{\text{rms}}$ $V_{\text{pp}} = 2 \times 1.414 \times V_{\text{rms}} \approx 2.828 \times V_{\text{rms}}$

1. Average Voltage ($V_{\text{avg}}$): The mathematical average over one rectified half-cycle: $V_{\text{avg}} = \frac{2}{\pi} \times V_{\text{pk}} \approx 0.637 \times V_{\text{pk}}$

Australian Mains Voltage Note

In Australia, standard mains wall sockets supply nominal $230\text{ V AC RMS}$ at $50\text{ Hz}$.

• Peak voltage $V_{\text{pk}} = 230 \times 1.414 = 325.2\text{ V}$.
• Peak-to-peak voltage $V_{\text{pp}} = 2 \times 325.2 = 650.4\text{ V}$.

---

4. Skin Effect at Radio Frequencies

At direct current ($0\text{ Hz}$) and low AC power frequencies ($50\text{ Hz}$), electric current is distributed uniformly across the entire cross-sectional area of a solid wire conductor.

However, as frequency increases into the radio frequency spectrum (RF), alternating magnetic flux created within the interior of the conductor generates counter-electromotive forces (eddy currents). These internal eddy currents oppose current flow in the centre of the conductor, forcing the moving electrons outward toward the outer perimeter.

This physical phenomenon is called Skin Effect.

Consequences of Skin Effect in RF Systems:

1. Increased Effective Resistance: Because current is restricted to a thin outer layer (skin depth $\delta$), the effective cross-sectional area is greatly reduced, causing the RF resistance ($R_{\text{ac}}$) to be significantly higher than the DC resistance ($R_{\text{dc}}$).
2. Conductor Geometry: To minimise skin effect losses at RF, antenna conductors and inductor coils often use large-diameter tubing, silver-plated copper wire, or multi-strand Litz wire (at lower frequencies).