# Amplitude Modulation

 This tutorial is addressed to HF2LI lock-in amplifier users that have purchased both HF2-MF multi-frequency and HF2-MOD AM/FM modulation options.

Amplitude modulation (AM) and frequency modulation (FM) refer to the modulation of an oscillating signal $$s{\left( t \right) =}A\cos\left( \omega t + \varphi \right)$$, the so-called carrier. A and $$\omega t + \varphi$$ are the amplitude and the phase of the signal, respectively. Figure 1 depicts the phasor representation of s(t). The phasor follows a circle with radius A, and the phase wraps around after a full revolution of 360°. The signal s(t) is the projection of the phasor on the abscissa.

In the case of AM signals, the amplitude A, i.e. the phasor length, is time dependent, as in Figure 1(b). In the case of FM signals, the phase offset $$\varphi$$ is time dependent and the phasor has a constant amplitude, see Figure 1(c).

Amplitude and frequency modulation, best known from radio transmission, are also common lock-in detection techniques.

Figure 1. A sinusoidal signal represented as a phasor: the signal corresponds to the projection on the x axis. Amplitude (b) and frequency (c) modulated signals affect the amplitude of the phasor or its phase

## What is Amplitude Modulation?

In the time domain, amplitude modulation of the carrier signal produces a variation of the carrier amplitude proportional to the amplitude of the modulating signal. For example, when the amplitude of a carrier with a frequency $$f_c=\omega_c / 2\pi$$ is modulated by a signal with frequency $$f_m=\omega_m / 2\pi$$ (where $$f_m<f_c$$), the resulting signal has the form

\tag{1} \begin{aligned} s(t)&=[A+M sin(\omega_m t)]sin(\omega_c t + \varphi) \\ &= A sin(\omega_c t + \varphi) + \frac{M}/{2} cos [(\omega_c - \omega_m)t+ \varphi] - \frac{M}/{2} cos[(\omega_c + \omega_m)t+ \varphi] \end{aligned}

where A and M are the amplitudes of the fast and slow modulations respectively and $$\varphi$$ the phase offset. There is no restriction on the magnitude of $$M$$ compared to $$A$$. The information of interest is encoded in these three parameters, A, M and $$\varphi$$.

In the frequency domain, the AM signal s(t) is composed of three frequencies: the carrier at $$f_{c}$$ and two additional sidebands at $$f_{c} - f_{m}$$ and $$f_{c} + f_{m}$$, as shown in Equation 1. The two sidebands have equal amplitude M/2, half of the modulating signal, and the carrier amplitude is independent on the modulation amplitude.

The traditional way of measuring an AM signal is called double (or tandem) demodulation and requires two lock-in amplifiers: the first one demodulates the signal at $$f_{c}$$ with a bandwidth that is at least as large as $$f_{m}$$ (but smaller than $$f_{c} - f_{m}$$). This is to ensure that the full amplitude of the modulation signal is retained. The demodulated signal after the first lock-in becomes

$$$\tag{2} s(t) \cdot cos(\omega_c t) \xrightarrow[\text{filtering}]{\text{after}} d_1(t)=\frac{A+M sin(\omega_m t)}{2}cos(\varphi)$$$

In $$d_{1}\left( t \right)$$, the two sidebands are now located at the same frequency $$f_{m}$$, while the carrier appears as a DC component. When the demodulated signal $$d_{1}\left( t \right)$$ is fed to a second lock-in amplifier, the result of the second demodulation at $$f_{m}$$ is proportional to $$M\cos\varphi$$.

In order to recover the amplitude $$M$$ it is necessary to measure the $$\varphi$$ so one can divide the result by the factor $$\cos\varphi$$. To measure the phase one can use a third lock-in to demodulate s(t) at the carrier frequency $$f_{c}$$ with a bandwidth smaller than $$f_{m}$$ as shown in Figure 2.

Instead of using a tandem configuration, the HF2-MOD option allows the user to demodulate directly at the three frequencies $$f_{c}$$ and $$f_{c} \pm f_{m}$$ simultaneously. The three parameters A, M and $$\varphi$$ can be measured and displayed with a single instrument.

Internally, the HF2LI generates the phases $$\omega_{c}t$$ and $$\omega_{m}t$$ from which it produces $$\left( \omega_{c} - \omega_{m} \right)t$$, $$\omega_{c}t$$ and $$\left( \omega_{c} + \omega_{m} \right)t$$. This ensures the correct phase relationship for the demodulations of the sidebands.

Figure 2. Comparison between tandem demodulation and the HF2-MOD option of an AM modulated signal

## Generate the Test Signal

In this tutorial, you are going to generate an AM signal with a carrier frequency of 1 MHz and a modulation frequency of 100 kHz. The signal is generated at Signal Output 2 and is demodulated by the first lock-in unit by feeding it into Signal Input 1. The HF2-MOD option requires the HF2-MF Multi-frequency option because each modulated signal requires at least two oscillators. Note that changing the Modulation tab settings will modify some of the settings found in the Lock-in MF tab. The reader is kindly referred to MOD Tab .

Start by enabling the Signal Output 2 in the Lock-in MF tab and disabling all demodulator Output Amplitudes. This will ensure that only the desired components of the amplitude-modulated signal appear on the output.

 Signal Output 2 Enable ON Signal Output Amplitudes Demodulators 1-8 OFF

In the Modulation tab, in the MOD 2 section, select the following parameters:

 Carrier Oscillator (Osc)/Frequency Osc 1 / 1 MHz Sideband 1 Oscillator (Osc)/Frequency Osc 2 / 100 kHz Carrier Mode AM Generation Carrier/Modulation Amplitude 200 mV / 100 mV Generation Carrier/Modulation Enable ON / ON MOD 2 Enable ON

This generates an AM signal with two sidebands of equal amplitude. To look at this signal, connect Signal Output 2 to Signal Input 1 of the HF2LI. Select the correct input parameters: in the Lock-in tab, for Signal Input 1, make sure Differential mode and 50 Ω are disabled. Then click on the auto range button In the Scope tab, select Source to be Signal Input 1, Trigger to be Signal Output 2 and click on Run to activate the Scope. Observe how the carrier amplitude is modulated at 100 kHz as seen in . In Frequency Domain FFT mode, the plot shows three peaks: the carrier at 1 MHz and two sidebands at 0.9 and 1.1 MHz (see the cursors in the frequency domain representation in Figure 4).

Figure 3. Time domain representation of the AM signal generated by MOD2 measured with the LabOne Scope
Figure 4. Frequency domain representation of the AM signal generated by MOD2 measured with the LabOne Scope

## Measure the Test Signal

In the Modulation tab, in the MOD 1 section, select the following parameters:

 Carrier oscillator (Osc) 1 Sideband 1 oscillator (Osc) 2 Carrier Mode AM Low-pass Filter BW (Carrier) 10 Hz Low-pass Filter BW (Sideband 1) 10 Hz MOD 1 Enable ON Demod 1, 2, 3 Data Transfer Enable (Lock-in tab) ON

This sets the correct demodulation of the AM signal with the two sidebands. In the Numeric tab, look at the amplitude of the carrier, 142 mVRMS and of the two sidebands, 35 mVRMS each, one quarter of the carrier amplitude: this corresponds to a modulation index of 50%.