Analysis of Amplitude Modulated Wave

$v_m = V_m \sin(\omega_m t)$ (8.1) and $v_C = V_C \sin(\omega_c t)$ (8.2), with $\omega_m = 2\pi f_m$, $\omega_c = 2\pi f_c$ and $f_c \gg f_m$. Here $V_m$ and $V_C$ are the peak amplitudes and $f_m, f_c$ the frequencies of the message and carrier (with the carrier phase ignored); these two expressions …

$v_{AM} = V_C(1 + m_a \sin\omega_m t)\sin(\omega_c t)$ (8.5), obtained from $v_{AM} = A\sin\omega_c t$ with the modulated amplitude $A = V_C(1 + m_a \sin\omega_m t)$. Here $m_a = V_m/V_C$ is the modulation index; expanding this product with a product-to-sum identity gives the AM w …

$m_a = V_m/V_C$: the ratio of modulating-signal amplitude to carrier amplitude. It must lie between 0 and 1 (maximum 1.0); a value greater than 1 causes over-modulation and envelope distortion. Because it fixes both the quality and the strength of the transmitted signal, the modulation …

$v_{AM} = V_C\sin\omega_c t + \tfrac{m_a V_C}{2}\cos(\omega_c-\omega_m)t - \tfrac{m_a V_C}{2}\cos(\omega_c+\omega_m)t$ (8.6): the carrier at $f_c$, the lower sideband at $f_c - f_m$ and the upper sideband at $f_c + f_m$, the …

The sum frequency $f_c + f_m$ (upper sideband) and difference frequency $f_c - f_m$ (lower sideband) produced by modulation. Each has amplitude $m_a V_C/2$; the carrier carries no information, so all t …

$BW = f_{USB} - f_{LSB} = (f_c+f_m)-(f_c-f_m) = 2 f_m$ (8.7): twice the highest modulating frequency. Here $f_{USB} = f_c + f_m$ is the upper side frequency and $f_{LSB} = f_c - f_m$ the lower one, so an AM channel needs a bandwidth of twice the highest modulating frequency; a medium-wave A …

$M = \dfrac{V_m}{V_C}\times 100$ (8.9), the modulation index expressed as a percentage. Here $V_m$ is the modulating-signal amplitude and $V_C$ the carrier amplitude, so the percentage is simply the modulation index scaled by 100. For undistorted AM it stays at or below 100%; a higher v …

$m_a = \dfrac{V_{max}-V_{min}}{V_{max}+V_{min}}$ (8.14), with $V_C = (V_{max}+V_{min})/2$. This is the standard method of measuring $m_a$ from an oscilloscope trace. Here $V_{max}$ and $V_{min}$ are the peak and trough amplitudes of the AM envelope, read directly from an oscilloscope trace, so no separate k …

$m_a = \dfrac{I_{max}-I_{min}}{I_{max}+I_{min}}$ (8.15), used when the AM wave is measured as an antenna current. Here $I_{max}$ and $I_{min}$ are the maximum and minimum envelope currents; this mirrors the voltage form $m_a = (V_{max}-V_{min})/(V_{max}+V_{min})$ and is used when the AM wave is …

Figure 8.2.2 (Frequency spectrum of AM wave): amplitude-versus-frequency graph with the carrier line at $f_c$ (height $V_C$) flanked by the lower sideband at $f_c - f_m$ and upper sideband at $f_c + f_m$, each of height $m_a V_C/2$; …

Figure 8.2.3 (Amplitude modulated wave): a modulated carrier whose upper envelope traces $(V_C + V_m\sin\omega_m t)$ and lower envelope its mirror image; the right side brackets the peak envelope $V_{max}$ and trough $V_{min}$, and the left marks the unmodulated carrier level $V_C$. Used to …

Figure 8.2.4 (Amplitude modulation for $m_a = 0$): with no modulating signal the AM wave equals the unmodulated carrier of constant amplitude $A$. What to notice: the peak amplitude stays constant and the envelope is a flat, unchanging line — with no message applied there is nothing to trace, so the output is in …

Figure 8.2.5 (Amplitude modulation for $m_a = 0.5$): carrier of amplitude $A$ plus a 0.5A signal gives an AM wave peaking at 1.5A whose envelope keeps a distinct waist and does not touch zer …

Figure 8.2.6 (Amplitude modulation for $m_a = 1$): carrier plus an equal-amplitude signal gives an AM wave peaking at 2A whose envelope pinches down to touch the zero axi …

Figure 8.2.7 (Amplitude modulation for $m_a = 1.5$): a 1.5A signal over-modulates the carrier; the envelope peaks at 2.5A and is flattened near zero with the negative peaks clipped, producing …