4.2 Multistage Coupling & Negative Feedback

4.2 Multistage Coupling & Negative Feedback

A single transistor amplifier stage rarely provides sufficient voltage or power gain to process tiny sensor signals (such as thermocouple voltages, pitot-static pressure transducers, or antenna RF signals) up to levels required by aircraft instruments and flight control computers. Practical avionics systems therefore interconnect multiple amplifier stages in cascade, where the output of each preceding stage serves as the input to the succeeding stage.

Controlling the stability, frequency bandwidth, and distortion of such high-gain cascades requires interstage coupling networks and negative feedback loops.

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1. Multistage Cascade Analysis & Gain Mathematics

When $n$ amplifier stages are connected in cascade, the overall voltage gain ($A_v$) is the product of the individual stage voltage gains, accounting for the input loading that each stage imposes on the preceding stage:

$A_v = A_{v1} \cdot A_{v2} \cdot A_{v3} \cdots A_{vn}$

In avionics specifications and audio/RF telemetry calculations, gains are expressed in decibels (dB). When converted to logarithmic units, the multiplicative product transforms into a simple algebraic sum:

$A_{v(dB)} = 20 \log_{10} |A_v| = A_{v1(dB)} + A_{v2(dB)} + A_{v3(dB)} + \dots + A_{vn(dB)}$

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2. Interstage Coupling Methodologies

The method chosen to couple AC signals between cascaded stages dictates the amplifier's physical weight, manufacturing cost, and lower/upper cutoff frequencies.

A. Resistance-Capacitance (RC) Coupling

In RC coupling, a series coupling capacitor ($C_c$) connects the collector of the first transistor to the base of the second, with a collector resistor ($R_C$) providing DC supply current and base resistors ($R_1, R_2$) setting the second stage bias.

• DC Isolation: The coupling capacitor acts as an open circuit to DC ($X_C = \infty$ at $f = 0\text{ Hz}$), preventing the high positive DC collector voltage of Stage 1 from disrupting the delicate $0.7\text{ V}$ base bias of Stage 2.
• Frequency Limitations:
  • Low-Frequency Rolloff: As signal frequency decreases, capacitive reactance $X_C = \frac{1}{2\pi f C_c}$ increases, attenuating signal transmission and causing phase lead. The lower $-3\text{ dB}$ cutoff frequency is: $f_L = \frac{1}{2\pi (R_{out1} + R_{in2}) C_c}$
  • High-Frequency Rolloff: At high frequencies, transistor internal junction capacitances (depletion and diffusion capacitances) and stray wiring capacitances shunt signals to ground, causing gain rolloff ($f_H$).
• Avionics Application: Cockpit interphone audio amplifiers, cabin announcement systems, and general-purpose audio-frequency processing.

B. Transformer Coupling

In transformer coupling, the primary winding of an iron- or ferrite-core transformer serves as the collector load of Stage 1, while the secondary winding drives the base of Stage 2.

• DC Isolation & Efficiency: Complete physical and DC isolation between stages. Because the transformer primary exhibits very low DC winding resistance ($R_{copper} \approx 0\ \Omega$), there is negligible $I_C^2 R_C$ DC power loss, yielding high power efficiency.
• Impedance Matching: By selecting turns ratio $N_p/N_s$, the transformer transforms load impedance according to the square of the turns ratio: $\frac{Z_p}{Z_s} = \left(\frac{N_p}{N_s}\right)^2$ This enables matching a high-impedance collector output to a low-impedance base input or transmission line, maximizing power transfer.
• Drawbacks in Aviation: Transformers are bulky, heavy, expensive, vulnerable to electromagnetic hum pickup from aircraft 400 Hz power systems, and suffer poor frequency response at harmonic extremes due to core saturation and leakage inductance.

C. Direct Coupling

In direct coupling (DC), the collector of the driving transistor is wired directly to the base of the driven transistor without any series capacitor or transformer.

• Zero Frequency Response: Because there are no reactive components in the signal path, frequency response extends down to $0\text{ Hz}$ (true DC).
• Application: Essential in aircraft operational amplifiers (op-amps), servomechanism error detectors, thermocouple/strain gauge instrumentation, and analog flight control computers.
• The DC Drift Problem: The principal design challenge is temperature drift. BJT base-emitter voltage changes at approximately $-2\text{ mV}/^\circ\text{C}$ and leakage current ($I_{CBO}$) doubles every $10^\circ\text{C}$. In a direct-coupled cascade, a tiny temperature-induced DC voltage shift in Stage 1 is multiplied by the full DC gain of all subsequent stages, potentially driving the output stage into hard saturation or cutoff. Mitigating drift requires differential amplifier topologies.

D. Impedance Coupling (Inductive-Resistive / IR Coupling)

In impedance coupling - listed in the Part-66 syllabus as inductive-resistive (IR) coupling - the collector load resistor $R_C$ is replaced by an inductor (a choke) $L$, and the signal is still passed to the next stage through a series coupling capacitor. It is the halfway house between RC coupling and transformer coupling.

• DC Efficiency: The choke's copper resistance is only a few ohms, so almost none of the supply rail is wasted as a DC drop. The collector sits close to $V_{CC}$, which matters on a $28\text{ V}$ aircraft bus where an $R_C$ of several kilohms would throw away half the available swing as heat.
• AC Load and Gain: At signal frequencies the AC load is the inductive reactance $X_L = 2\pi f L$, so the stage gain becomes $A_v \approx X_L / r'_e$ and therefore rises with frequency.
• Frequency Response: This is exactly why impedance coupling is not used for wideband audio. At low frequencies $X_L$ collapses and gain disappears; at high frequencies the choke's own self-capacitance resonates with $L$ and the response peaks then falls away. The usable band is narrow.
• Weight and Interference: A choke is lighter and cheaper than a transformer but bulkier than two resistors, and its windings readily pick up stray magnetic fields from the aircraft $115\text{ V}$, $400\text{ Hz}$ distribution.
• Avionics Application: Narrowband and tuned work - radio receiver IF strips, $400\text{ Hz}$ servo-channel amplifiers, and RF stages where the choke is part of a tuned collector load.

Illustration of the frequency dependence. A $10\text{ mH}$ choke presents $X_L = 2\pi \times 400\text{ Hz} \times 0.010\text{ H} = 25.1\ \Omega$ at $400\text{ Hz}$ - a negligible AC load giving almost no gain. The same choke presents $X_L = 2\pi \times 100\text{ kHz} \times 0.010\text{ H} = 6.28\text{ k}\Omega$ at $100\text{ kHz}$, a perfectly usable collector load. One component, a gain ratio of 250:1 across the band.

Coupling Method Comparison

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3. Negative Feedback: Principles and Closed-Loop Gain

Negative (degenerative) feedback is the deliberate process of returning an anti-phase fraction of the amplifier's output signal back to the input, where it subtracts from the externally applied source signal.

Signal Mixing:   V_in' = V_s - V_f = V_s - (β_fb · V_out)

Derivation of Closed-Loop Gain ($A_f$)

Let $A$ represent the internal open-loop voltage gain of the forward amplifier, and $\beta_{fb}$ represent the feedback attenuation factor ($0 < \beta_{fb} < 1$):

1. The amplifier produces output voltage: $V_{out} = A \cdot V_{in}'$
2. The net differential input voltage is: $V_{in}' = V_s - \beta_{fb} V_{out}$
3. Substituting $V_{in}'$ into the output equation: $V_{out} = A (V_s - \beta_{fb} V_{out}) = A V_s - A \beta_{fb} V_{out}$
4. Regrouping terms: $V_{out} (1 + A \beta_{fb}) = A V_s$
5. Solving for the closed-loop voltage gain ($A_f = V_{out} / V_s$): $A_f = \frac{A}{1 + A \beta_{fb}}$

Here, $A\beta_{fb}$ is termed the loop gain, and $(1 + A\beta_{fb})$ is the feedback factor (or desensitization factor).

Gain Desensitization (Stability)

When loop gain is engineered to be large ($A\beta_{fb} \gg 1$), the closed-loop gain simplifies to:

$A_f \approx \frac{A}{A \beta_{fb}} = \frac{1}{\beta_{fb}}$

Under this condition, closed-loop gain becomes virtually independent of transistor $\beta$, supply rail fluctuations, and ambient temperature. Gain is governed exclusively by the passive, highly stable resistors in the feedback network $\beta_{fb}$, a vital requirement for flight-critical avionics.

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4. Key Engineering Benefits of Negative Feedback

Although negative feedback trades away raw gain, it dramatically enhances every other metric of amplifier performance:

1. Bandwidth Extension

Negative feedback widens the frequency bandwidth by the exact factor $(1 + A\beta_{fb})$. The upper $-3\text{ dB}$ cutoff frequency increases, while the lower cutoff frequency decreases:

$f_{Hf} = f_H \cdot (1 + A \beta_{fb}) \qquad \text{and} \qquad f_{Lf} = \frac{f_L}{1 + A \beta_{fb}}$

The Gain-Bandwidth Product (GBW) remains strictly constant: reducing gain by a factor of 10 increases bandwidth tenfold.

2. Harmonic Distortion and Internal Noise Reduction

Non-linear harmonic distortion generated within the amplifier's forward path is reduced by the feedback factor:

$D_f = \frac{D_{open}}{1 + A \beta_{fb}}$

If open-loop distortion is $5%$ and the feedback factor is $50$, closed-loop distortion drops to an imperceptible $0.1%$.

3. Impedance Tailoring

The impact of negative feedback on input ($Z_{in}$) and output ($Z_{out}$) impedance depends on the feedback topology:

• Voltage-Series Feedback (series input mixing, shunt output sampling):
  • Input Impedance Increases: $Z_{in,f} = Z_{in} \cdot (1 + A \beta_{fb})$. High input impedance prevents loading upstream avionics sensors.
  • Output Impedance Decreases: $Z_{out,f} = \frac{Z_{out}}{1 + A \beta_{fb}}$. Low output impedance allows driving low-resistance actuators and transmission cables with zero voltage sag.

5. Positive versus Negative Feedback: Advantages and Disadvantages

The syllabus asks specifically for the advantages and disadvantages of both feedback polarities. Positive feedback is not simply a fault condition - it is deliberately engineered into several circuits a Part-66 engineer will meet.

The practical lesson for line maintenance is that the two polarities are separated only by loop phase. A negative-feedback amplifier whose decoupling capacitor has dried out, or whose screening can has been left off after a repair, becomes a positive-feedback oscillator without a single component value having changed.

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[!WARNING] Instability and Oscillation Hazards: At high frequencies, parasitic capacitances and transformer leakage inductances introduce cumulative phase lag. If the internal phase lag reaches $180^\circ$, the total loop phase shift becomes $180^\circ + 180^\circ = 360^\circ$ ($0^\circ$). The degenerative negative feedback transforms into regenerative positive feedback. If the loop gain $|A\beta_{fb}| \ge 1$ at this crossover frequency, the amplifier transforms into an unstable, self-sustaining oscillator, a condition governed by the Nyquist Stability Criterion.