1.2 AC Fundamentals, Sine Waves, Peak, RMS, and Phase Relationships

1.2 AC Fundamentals, Sine Waves, Peak, RMS, and Phase Relationships

Alternating current (AC) is the fundamental medium of radio communications. While direct current provides the steady biasing voltages required to power active devices, all radio frequency (RF) signals, modulated carrier waves, audio signals, and electrical mains power exist as time-varying alternating waveforms. Analysing and designing amateur radio equipment requires mastery of sinusoidal parameters, non-sinusoidal harmonic spectra, and reactive AC power distribution.

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1. Nature and Generation of Alternating Current

Electromagnetic Generation of Sine Waves

Alternating current reverses its direction of flow at regular recurring intervals. The canonical mathematical waveform is the sinusoid, which arises naturally from uniform circular motion. When a planar conductive coil of wire rotates with constant angular velocity ($\omega$) within a uniform magnetic field ($B$), the magnetic flux ($\Phi$) linking the coil varies as:

$$\Phi(t) = B \cdot A \cdot \cos(\omega t)$$

According to Faraday's Law of Electromagnetic Induction, the induced electromotive force ($e$) across the coil terminals is proportional to the negative time rate of change of magnetic flux:

$$e(t) = -N \frac{d\Phi}{dt} = N \cdot B \cdot A \cdot \omega \cdot \sin(\omega t)$$

Defining the peak induced voltage as $V_{\text{pk}} = N B A \omega$, the general equation for a sinusoidal alternating voltage with arbitrary initial phase angle $\phi$ is:

$$v(t) = V_{\text{pk}} \sin(\omega t + \phi)$$

Period, Frequency, and Angular Velocity

• Cycle: One complete alternation consisting of one positive half-cycle and one negative half-cycle, returning to the identical starting state.
• Period ($T$): The time duration in seconds required to complete one full cycle of the waveform.
• Frequency ($f$): The number of complete cycles executed per second, expressed in Hertz ($\text{Hz}$):

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

• Angular Frequency ($\omega$): The rotational speed of the generating vector, measured in radians per second ($\text{rad/s}$). Because one full circle equals $2\pi$ radians ($360^\circ$):

$$\omega = 2\pi f = \frac{2\pi}{T}$$

Worked Example 1.4: Frequency, Period, and Angular Frequency across Amateur Bands

Amateur radio operates across frequency bands spanning from low frequency (LF) to microwave frequencies:

1. Australian AC Mains Power: $f = 50\text{ Hz}$.

$$T = \frac{1}{50\text{ Hz}} = 0.020\text{ s} = 20.0\text{ ms}, \quad \omega = 2\pi \times 50 \approx 314.16\text{ rad/s}$$

1. 40-Metre Amateur Band: $f = 7.100\text{ MHz}$ ($7.100 \times 10^6\text{ Hz}$).

$$T = \frac{1}{7.100 \times 10^6\text{ Hz}} \approx 1.4085 \times 10^{-7}\text{ s} = 140.85\text{ ns}$$ $$\omega = 2\pi \times 7.100 \times 10^6 \approx 4.4611 \times 10^7\text{ rad/s}$$

1. 2-Metre Amateur VHF Band: $f = 146.0\text{ MHz}$ ($1.460 \times 10^8\text{ Hz}$).

$$T = \frac{1}{1.460 \times 10^8\text{ Hz}} \approx 6.849 \times 10^{-9}\text{ s} = 6.849\text{ ns}$$ $$\omega = 2\pi \times 1.460 \times 10^8 \approx 9.1734 \times 10^8\text{ rad/s}$$

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2. Voltage and Current Amplitudes

Because an alternating waveform continuously changes amplitude, a single numerical value cannot describe its magnitude without specifying the metric.

Peak Voltage ($V_{\text{pk}}$) and Peak-to-Peak Voltage ($V_{\text{p-p}}$)

• Peak Amplitude ($V_{\text{pk}}$ or $V_p$): The maximum instantaneous displacement from the zero-volt reference line to either the positive crest ($+V_{\text{pk}}$) or negative trough ($-V_{\text{pk}}$).
• Peak-to-Peak Amplitude ($V_{\text{p-p}}$): The total vertical excursion between the positive maximum and negative minimum:

$$V_{\text{p-p}} = 2 \cdot V_{\text{pk}}$$

Oscilloscopes display waveforms directly in peak-to-peak volts, making this metric fundamental to laboratory measurement.

Root Mean Square (RMS) Value ($V_{\text{rms}}$) — Effective Value

The Root Mean Square (RMS) value is the single most important metric in AC power engineering. It defines the heating equivalence of an alternating current compared to direct current.

Physical Principle: An AC voltage with an amplitude of $V_{\text{rms}}$ applied across a pure resistive load produces exactly the same average thermal power dissipation as a constant DC voltage of the identical numerical value.

Mathematically, the RMS value is found by squaring the instantaneous function, integrating over one complete period, dividing by the period, and taking the square root:

$$V_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T v(t)^2 \, dt}$$

For a pure sinusoidal wave $v(t) = V_{\text{pk}} \sin(\omega t)$, evaluating this integral yields:

$$V_{\text{rms}} = \frac{V_{\text{pk}}}{\sqrt{2}} \approx 0.7071 \cdot V_{\text{pk}}$$ $$V_{\text{pk}} = \sqrt{2} \cdot V_{\text{rms}} \approx 1.4142 \cdot V_{\text{rms}}$$ $$V_{\text{p-p}} = 2\sqrt{2} \cdot V_{\text{rms}} \approx 2.8284 \cdot V_{\text{rms}}$$

AC voltage and current ratings normally use RMS values. RF power specifications identify mean power or peak envelope power (PEP); neither is an RMS power rating.

Average Value ($V_{\text{avg}}$)

Over one complete symmetric cycle ($0$ to $2\pi$), the true mathematical average of a pure sine wave is identically zero because the positive and negative half-cycles cancel completely. In practical electronics, the term "average value" refers to the rectified average (the average over a single half-cycle or of a full-wave rectified wave):

$$V_{\text{avg}} = \frac{1}{\pi} \int_0^\pi V_{\text{pk}} \sin(\theta) \, d\theta = \frac{2}{\pi} \cdot V_{\text{pk}} \approx 0.6366 \cdot V_{\text{pk}}$$

Relating average and RMS values for a sine wave:

$$V_{\text{avg}} = \frac{2\sqrt{2}}{\pi} \cdot V_{\text{rms}} \approx 0.9003 \cdot V_{\text{rms}}$$

Form Factor and Crest Factor

These dimensionless ratios characterize the geometric shape of an alternating waveform:

• Form Factor: The ratio of RMS value to the rectified average value:

$$\text{Form Factor} = \frac{V_{\text{rms}}}{V_{\text{avg}}} = \frac{V_{\text{pk}} / \sqrt{2}}{2 V_{\text{pk}} / \pi} = \frac{\pi}{2\sqrt{2}} \approx 1.1107$$

Analog meters that measure average voltage but are calibrated to read RMS incorporate this $1.11$ scaling factor. Consequently, average-responding meters display inaccurate readings when measuring non-sinusoidal waveforms.

• Crest Factor (Peak Factor): The ratio of peak amplitude to RMS value:

$$\text{Crest Factor} = \frac{V_{\text{pk}}}{V_{\text{rms}}} = \frac{V_{\text{pk}}}{V_{\text{pk}} / \sqrt{2}} = \sqrt{2} \approx 1.4142$$

Worked Example 1.5: Australian Mains and Transmitter RF Output

1. Australian AC Mains: Nominally rated at $230\text{ V}_{\text{rms}}$ at $50\text{ Hz}$.

   • Peak voltage: $V_{\text{pk}} = 230\text{ V} \times \sqrt{2} \approx 325.27\text{ V}$.
   • Peak-to-peak voltage: $V_{\text{p-p}} = 2 \times 325.27\text{ V} \approx 650.54\text{ V}$.
   • Rectified average voltage: $V_{\text{avg}} = 230\text{ V} \times 0.9003 \approx 207.07\text{ V}$. Safety note: Select reservoir capacitors for the maximum operating DC-plus-ripple peak, allowing for mains tolerance and the specified transient duty. The nominal $325\text{ V}$ peak calculation alone does not establish a safe voltage rating.

2. $100\text{ W}$ Transmitter Output into a $50\ \Omega$ Load:

   • RMS voltage: $V_{\text{rms}} = \sqrt{P \cdot R} = \sqrt{100\text{ W} \times 50\ \Omega} = \sqrt{5000} \approx 70.71\text{ V}$.
   • Peak voltage: $V_{\text{pk}} = 70.71\text{ V} \times \sqrt{2} = 100.00\text{ V}$.
   • Peak-to-peak voltage: $V_{\text{p-p}} = 2 \times 100.00\text{ V} = 200.00\text{ V}$.
   • Peak RF current: $I_{\text{pk}} = V_{\text{pk}} / R = 100.00\text{ V} / 50\ \Omega = 2.00\text{ A}$.

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3. Non-Sinusoidal Waveforms and Fourier Harmonic Composition

In radio communications, signals are rarely pure unmodulated sine waves. Transmitters generate pulses, digital modulations, square waves, and sawtooth ramps. Fourier analysis expresses a periodic waveform as its DC mean, which may be zero, plus harmonically related sinusoidal components at the fundamental frequency ($f$) and integer harmonics ($2f, 3f, 4f, \dots$).

Common Non-Sinusoidal Waveforms

1. Symmetrical Square Wave (50% Duty Cycle)

A symmetrical square wave switches instantaneously between $+V_{\text{pk}}$ and $-V_{\text{pk}}$ with equal half-cycle durations. Its Fourier series expansion is:

$$v(t) = \frac{4 V_{\text{pk}}}{\pi} \left( \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \frac{1}{7}\sin(7\omega t) + \dots \right)$$

Key characteristics:

• Contains only odd harmonics ($3f, 5f, 7f, \dots$); all even harmonics are identically zero.
• Harmonic amplitudes roll off inversely with harmonic order ($A_n = 1/n$). The third harmonic has $33.3\%$ of fundamental amplitude; the fifth harmonic has $20\%$.
• Effective RMS value equals peak value: $V_{\text{rms}} = V_{\text{pk}}$.
• Crest factor $= 1.0$; Form factor $= 1.0$.

2. Symmetrical Triangle Wave

A triangle wave ramps linearly between positive and negative peaks with mirror symmetry:

$$v(t) = \frac{8 V_{\text{pk}}}{\pi^2} \left( \sin(\omega t) - \frac{1}{9}\sin(3\omega t) + \frac{1}{25}\sin(5\omega t) - \frac{1}{49}\sin(7\omega t) + \dots \right)$$

Key characteristics:

• Contains only odd harmonics ($3f, 5f, 7f, \dots$).
• Harmonic amplitudes decrease as the square of harmonic order ($A_n = 1/n^2$). The third harmonic has only $1/9 \approx 11.1\%$ of fundamental amplitude.
• $V_{\text{rms}} = V_{\text{pk}} / \sqrt{3} \approx 0.5774 \cdot V_{\text{pk}}$.
• Crest factor $= \sqrt{3} \approx 1.732$; Form factor $= 2 / \sqrt{3} \approx 1.155$.

3. Sawtooth Wave (Asymmetric Ramp)

A sawtooth waveform rises linearly and falls back instantaneously:

$$v(t) = \frac{2 V_{\text{pk}}}{\pi} \left( \sin(\omega t) - \frac{1}{2}\sin(2\omega t) + \frac{1}{3}\sin(3\omega t) - \frac{1}{4}\sin(4\omega t) + \dots \right)$$

Key characteristics:

• Contains both even and odd harmonics ($2f, 3f, 4f, 5f, \dots$).
• Harmonic amplitudes fall off as $1/n$.
• $V_{\text{rms}} = V_{\text{pk}} / \sqrt{3} \approx 0.5774 \cdot V_{\text{pk}}$.
• Frequently used in oscilloscope horizontal timebase sweeps and spectrum analyser sweep generators.

4. Pulse Train (Narrow Duty Cycle)

A train of rectangular pulses with duration $\tau$ and period $T$ (duty cycle $D = \tau / T \ll 1$) produces a dense comb of both even and odd harmonics extending out to a spectral null at $f_{\text{null}} = 1/\tau$. Fast switching edges in switch-mode power supplies (SMPS) and digital microcontrollers act as comb generators, creating broadband RF interference (RFI) across the HF and VHF amateur bands.

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4. Phase Angle, Phase Difference, and Vector Representation

Phase Angle and Phase Shift

When two alternating waveforms of the same frequency are compared, their zero-crossings and peaks may not coincide in time. This displacement is measured as a phase angle ($\theta$ or $\phi$), expressed in degrees or radians:

• In-Phase ($\theta = 0^\circ$): The two waveforms pass through zero in the same direction and reach positive and negative peaks at the identical instant.
• Quadrature ($\theta = 90^\circ$ or $\pi/2\text{ rad}$): One waveform reaches its maximum positive peak at the precise instant the other crosses zero. Quadrature signals (designated In-phase $I$ and Quadrature $Q$) form the analytical basis of modern Software Defined Radios (SDR), image-reject mixers, and digital vector modulations (QPSK, QAM).
• Anti-Phase / Phase Inversion ($\theta = 180^\circ$ or $\pi\text{ rad}$): The waveforms have precisely opposite polarities at every instant. When combined in equal amplitude, they cancel completely.

Leading vs. Lagging Relationships

Consider voltage and current waveforms:

$$v(t) = V_{\text{pk}} \sin(\omega t), \quad i(t) = I_{\text{pk}} \sin(\omega t - \theta)$$

Because the current waveform reaches its positive peak $\theta / \omega$ seconds later than the voltage waveform, current lags voltage by $\theta$ degrees (or equivalently, voltage leads current by $\theta$ degrees). In capacitive circuits, current leads voltage; in inductive circuits, current lags voltage.

Phasors

A phasor is a planar vector rotating counter-clockwise at angular velocity $\omega$. Its length corresponds to the RMS (or peak) amplitude of the AC waveform, and its angle relative to the positive horizontal axis represents instantaneous phase. Phasors allow sinusoidal differential equations to be simplified into algebraic complex-number arithmetic.

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5. Power in AC Circuits: True, Apparent, Reactive, and Power Factor

Assume sinusoidal steady-state voltage and current for this power triangle. In a pure resistance, they are in phase and instantaneous power is nonnegative. With distorted current, power factor remains $P/S$ but may differ from $\cos\theta$.

In circuits containing reactive elements (inductors and capacitors), voltage and current are displaced by phase angle $\theta$. Instantaneous power $p(t) = v(t) \cdot i(t)$ becomes negative during portions of each cycle, indicating that energy stored in reactive magnetic or electric fields is flowing back toward the source.

The Three Types of AC Power

1. True Power ($P$ — Active / Real Power): The rate at which energy is permanently dissipated in resistive elements, doing actual work (such as heating a resistor or radiating an electromagnetic wave from an antenna). Measured in Watts ($\text{W}$):

$$P = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \cos\theta$$

1. Apparent Power ($S$): The simple product of terminal RMS voltage and RMS current, representing the total capacity required of transformers, generators, and distribution wiring. Measured in Volt-Amperes ($\text{VA}$):

$$S = V_{\text{rms}} \cdot I_{\text{rms}}$$

1. Reactive Power ($Q$): The peak rate at which energy sloshes back and forth between the source and reactive magnetic/electric fields without performing net work. Measured in Volt-Amperes Reactive ($\text{VAR}$):

$$Q = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \sin\theta$$

The Power Triangle

Because reactive power is orthogonal ($90^\circ$ out of phase) to true power, these quantities form a right-angled triangle where Apparent Power is the hypotenuse:

$$S = \sqrt{P^2 + Q^2}$$

Power Factor ($\text{PF}$)

The Power Factor represents the fraction of apparent power converted into useful real work:

$$\text{PF} = \frac{P}{S} = \cos\theta$$

• Purely resistive circuit: $\theta = 0^\circ \implies \text{PF} = \cos(0^\circ) = 1.0$ (Unity power factor).
• Purely inductive circuit: $\theta = +90^\circ \implies \text{PF} = \cos(90^\circ) = 0.0$ lagging.
• Purely capacitive circuit: $\theta = -90^\circ \implies \text{PF} = \cos(-90^\circ) = 0.0$ leading.

Worked Example 1.6: Linear Power Supply Transformer Load

A heavy high-voltage transformer in an amateur linear power supply connects to the $230\text{ V}_{\text{rms}}$, $50\text{ Hz}$ mains. For this sinusoidal equivalent-load example, it draws a sinusoidal current of $8.0\text{ A}_{\text{rms}}$ lagging the voltage by $\theta = 36.87^\circ$.

1. Calculate Apparent Power ($S$):

$$S = V_{\text{rms}} \cdot I_{\text{rms}} = 230\text{ V} \times 8.0\text{ A} = 1840\text{ VA}$$

1. Calculate the Power Factor ($\text{PF}$):

$$\text{PF} = \cos(36.87^\circ) = 0.800$$

1. Calculate True Power ($P$):

$$P = S \cdot \cos\theta = 1840\text{ VA} \times 0.800 = 1472\text{ W}$$

1. Calculate Reactive Power ($Q$):

$$Q = S \cdot \sin\theta = 1840\text{ VA} \times \sin(36.87^\circ) = 1840 \times 0.600 = 1104\text{ VAR}$$

1. Verify using the Pythagorean power relationship:

$$\sqrt{P^2 + Q^2} = \sqrt{1472^2 + 1104^2} = \sqrt{2166784 + 1218816} = \sqrt{3385600} = 1840\text{ VA}$$

This demonstrates that although $1472\text{ W}$ of actual power is delivered, the mains supply wiring must support $1840\text{ VA}$ of current capacity due to reactive field storage.