Understanding Slope Detectors for FM Demodulation

In the previous article, we learned how a differentiator followed by an envelope detector can be used to demodulate FM waves. Figure 1 shows the simplified block diagram of this FM demodulator.

Figure 1. Simplified block diagram of an FM demodulator employing a differentiator.

The differentiator converts the FM signal to a conventional AM wave, allowing the envelope detector to retrieve the original message. In this article, we'll explore two ways of implementing this concept using slope detector circuits for the differentiator.

Differentiation Using an RL Circuit

As we learned in the preceding article, the frequency response of an ideal differentiator can be described by:

$$H( \omega)~=~j \omega K$$

Equation 1.

where K is a constant. Therefore, for an ideal differentiator, the magnitude response increases linearly with frequency, while the phase response remains constant at 90 degrees.

Obviously, we won't see an ideal response in the real world. However, the differentiator can be approximated by any circuit whose transfer function shows a linear slope near the carrier frequency. The RL discriminator in Figure 2(a) is one example.

Figure 2. Circuit schematic for an RL circuit used to approximate a differentiator (a) and voltage-frequency characteristic of the RL discriminator (b).

As shown in Figure 2(b), the output of the discriminator (v_o1) increases almost linearly with frequency. Due to its linear frequency response, this type of circuit is sometimes known as a slope detector.

Differentiation Using a Tuned LC Circuit: The Single-Tuned Discriminator

Instead of using a highpass circuit like the discriminator in Figure 2, we can approximate a differentiator over a narrow frequency range with a bandpass filter. This is illustrated in Figure 3(a), where the input-tuned circuit represents the output stage of the preceding IF amplifier.

Figure 3. Using an LC circuit to approximate a differentiator: the circuit schematic (a) and voltage-frequency characteristic of the tuned circuit (b).

As shown in Figure 3(b), the magnitude response varies with frequency, causing the voltage across the tuned circuit to change with the instantaneous frequency of the input FM wave.

Both the input and output LC circuits are tuned to a frequency (f_r) above the signal's IF carrier frequency. Since both LC circuits are tuned to the same frequency, we sometimes refer to this circuit as the single-tuned discriminator.

Analyzing the Frequency Response of a Tuned RLC Circuit

For a parallel RLC circuit tuned to frequency ω_r, it can be easily shown that the magnitude of the impedance of the tuned circuit is given by:

$$|Z(\omega)| ~=~ \frac{R}{\sqrt{1~+~Q^2 \big( \frac{\omega}{\omega_r} ~-~ \frac{\omega_r}{\omega} \big)^2}}$$

Equation 2.

where Q, the quality factor of the tuned circuit, is given by:

$$Q ~=~ RC \omega_r$$

Equation 3.

Away from the passband, Equation 2 shows that the magnitude response of the tuned circuit is proportional to ω below the resonant frequency and to 1/ω above the resonant frequency. The tuned circuit can thus perform FM-to-AM conversion.

For instance, if the carrier frequency is f_c = 10 MHz, we might choose a resonant frequency of f_r = 12 MHz. If we assume that the frequency deviation is 75 kHz, the instantaneous frequency applied to the resonant circuit would change between 9.925 MHz and 10.075 MHz.

Figure 4 plots the impedance magnitude for example values of R = 10 kΩ, Q = 20, and f_r = 12 MHz. The yellow-shaded section represents the frequency variation range for this example.

Figure 4. The magnitude of the impedance of an RLC circuit with R = 10 kΩ, Q = 20, and f_r = 12 MHz. The yellow-colored area indicates the range of frequency variation.

Within the frequency variation range, the response should be linear and the output amplitude should be directly proportional to the carrier frequency deviation. However, we know that the magnitude response is only approximately proportional to frequency. We can use the total harmonic distortion (THD) concept to examine the tuned circuit's linearity.

Example: Linearity of a Single-Tuned Discriminator

Figure 5 shows an example circuit from "Analog Integrated Circuits for Communication" by D. O. Pederson and K. Mayaram. This FM demodulator employs a single-tuned circuit with an emitter-coupled pair serving as the driver stage.

Figure 5. A single-tuned discriminator driven by an emitter-coupled pair.

The book's authors simulate the above circuit for an input FM wave with a carrier frequency of 10 MHz, frequency deviation of ±75 kHz, and a single-tone message signal at 10 kHz. Simulation results show that the second-harmonic distortion is HD₂ = 2.3%, the third-harmonic distortion is HD₃ = 4.2%, and the total harmonic distortion is THD = 7.9%.

These results show that the single-tuned discriminator has reasonable performance, albeit not high fidelity. Lowpass filtering in the subsequent audio amplifiers would attenuate the higher-order harmonics of this circuit.

Before proceeding, keep in mind that FM detection happens after the IF amplifiers, meaning the frequency deviation (±75 kHz in FM broadcast) is preserved, but the carrier frequency translation (usually to 10.7 MHz) has occurred.

The Balanced Discriminator

To achieve a better linear range of operation, we can employ the circuit in Figure 6. This is what's known as a balanced discriminator.

Figure 6. Schematic of the balanced discriminator.

A balanced discriminator incorporates two resonant circuits. One is tuned to a frequency above the carrier frequency (f_c); the other is tuned to a frequency below it. The slope detectors are configured back-to-back, with the midpoint of the envelope detectors connected to the transformer's center tap. In this circuit, the output (v_out) is taken as the difference between the voltages of the two envelope detectors (v_out = v_o1 – v_o2).

Understanding the Output Response

It can be shown that subtracting the outputs of two staggered tuned circuits produces a more linear response. To understand this, let's assume that the magnitude response of each of the slope detectors is proportional to the impedance of its corresponding tuned circuit, which can be described by Equation 2. Using this equation, the overall output can be described by:

$$v_{out} ~=~ v_{o1} ~-~ v_{o2} ~\propto~ \frac{1}{\sqrt{1~+~Q_{1}^2 \big( \frac{\omega}{\omega_{r1}} ~-~ \frac{\omega_{r1}}{\omega} \big)^2}} ~-~ \frac{1}{\sqrt{1~+~Q_{2}^2 \big( \frac{\omega}{\omega_{r2}} ~-~ \frac{\omega_{r2}}{\omega} \big)^2}}$$

Equation 4.

where:

Q₁ is the quality factor of the upper slope detector

ω_r1 is the resonant frequency of the upper slope detector

Q₂ is the quality factor of the lower slope detector

ω_r2 is the resonant frequency of the lower slope detector.

We could mathematically analyze the above equation to verify the improved linearity. However, I've instead chosen to plot the equation with some example values to get an idea of the circuit's operation.

Simulating the Output Response

Let's say the carrier frequency is 10.7 MHz and the frequency deviation is 75 kHz. The upper slope detector is tuned to f_r1 = 10.8 MHz, while the lower slope detector is tuned to f_r2 = 10.6 MHz. Both values are 100 kHz from the carrier frequency. For simplicity, assume both resonant circuits have identical Q-factors of 20. The calculated response of the circuit is shown in Figure 7.

Figure 7. The frequency-to-voltage characteristic of the upper path (red), lower path (blue), and the overall output (green) for the balanced discriminator in Figure 6.

The overall output is equal to H₁ – H₂. The upper path (H₁) produces its maximum value at f_r1 = 10.8 MHz, resulting in the peak near this frequency. Similarly, the lower path produces its maximum value at f_r2 = 10.6 MHz, resulting in a trough in the overall output characteristic. When the input frequency is equal to the carrier frequency (f_c = 10.7 MHz), both the H₁ and lower H₂ paths produce the same voltage, leading to zero overall output.

A visual inspection of the above curves shows that the overall output is more linear than the individual outputs. For instance, at 10.7 MHz, H₁ has a positive slope, but it flattens out to a zero slope at 10.8 MHz. However, the slope of H₁ – H₂ is still a positive value at 10.8 MHz. This implies that the overall output has smaller slope variations, which means it provides a broader linear range of operation.

Wrapping Up

While the rising half of the frequency characteristic of a tuned circuit can be used to perform FM-to-AM conversion, the linear region of such a circuit may not be wide enough. To achieve a linear characteristic over a broader frequency range, we can use a balanced discriminator that includes two resonant circuits: one tuned to a frequency above the carrier frequency and the other below it.

One advantage of the balanced configuration is that it doesn't need a DC block. This is because the constant terms in the individual envelopes cancel out when we subtract them to produce the overall output. The absence of a DC block stage allows the circuit to perform effectively at low modulating frequencies. The balanced configuration is easily adaptable to the microwave band by using resonant cavities for the tuned circuits and crystal diodes as envelope detectors.

All images used courtesy of Steve Arar