Signals & Amplifiers Comprehensive Study Notes

Fundamentals of Signals, Transducers, and Signal Sources

• Signals Definition: Signals are time-varying physical quantities that carry information about processes and events in the real world.

  • Weather Signals: Air temperature, barometric pressure, atmospheric humidity, wind speed, and wind direction.
  • Radio Signals: Audio signals generated by microphones, modulated transmitted radio-frequency signals.
  • Television Signals: Video camera image signals, transmitted TV broadcast signals.
  • Data Signals: Binary digital signals used in computation and digital communications.

• Signal Processing: Performed by electronic systems to extract meaningful information from signals, allowing interpretation by humans or machines (e.g., inside radio receivers, TV sets, computers).

• Transducers: Interface devices that convert non-electrical physical quantities into electrical signals or vice versa:

  • Physical-to-Electrical Transducers: Convert physical parameters into electrical voltages or currents (e.g., microphones, video image sensors).
  • Electrical-to-Physical Transducers: Convert electrical voltages or currents into physical actions or visual displays (e.g., loudspeakers, LCD displays).

• Equivalent Signal Source Models:

  • Signal sources are commonly modeled using linear equivalent circuits:
  • Thévenin Equivalent Source: A time-varying voltage source $v_s(t)$ in series with an internal source resistance $R_s$.
  • Norton Equivalent Source: A time-varying current source $i_s(t)$ in parallel with an internal source resistance $R_s$

• Signal Waveforms:
  • Information is represented by variations in amplitude over time.
  • An arbitrary continuous-time signal waveform is plotted as amplitude versus time $t$

Frequency Spectrum of Signals

• Mathematical Representation:

  • Any arbitrary signal $v_s(t)$ or $i_s(t)$ can be expressed as a sum of sinusoidal components of varying amplitudes, frequencies, and phases using the Fourier series (for periodic signals) or Fourier transform (for non-periodic signals).
  • Time Domain vs. Frequency Domain:
  • Waveform: Plots signal amplitude against time $t$
  • Frequency Spectrum: Plots component amplitudes against frequency $\omega$

• Periodic Signals: Sine Wave:

  • The simplest periodic signal is the pure sine wave, completely specified by peak amplitude $V_a$, frequency, and phase relative to a time origin.
  • Root Mean Square (RMS) Value: Represents the equivalent DC energy value, defined as:     $\text{RMS Value} = \frac{V_a}{\sqrt{2}}$
  • Fundamental Angular Frequency:     $\omega_0 = 2\pi f_0\,\text{rad/s}$     where $f_0 = \frac{1}{T}\,\text{Hz}$ and $T$ is the period in seconds.

• Periodic Signals: Square Wave:
  • A symmetrical square wave of period $T$, peak value $V_o$, and DC offset $V_{DC}$ is composed of a discrete line spectrum with harmonically related frequencies (odd integer multiples of fundamental frequency $\omega_0$):     $v(t) = V_{DC} + \frac{4V_o}{\pi} \left( \sin(\omega_0 t) + \frac{1}{3}\sin(3\omega_0 t) + \frac{1}{5}\sin(5\omega_0 t) + \frac{1}{7}\sin(7\omega_0 t) + \frac{1}{9}\sin(9\omega_0 t) + \dots \right)$
  • The spectral lines occur at discrete frequencies $\omega_0, 3\omega_0, 5\omega_0, 7\omega_0, 9\omega_0, \dots$
  • The fundamental component has an amplitude of $\frac{4V_o}{\pi}$, while the $n\text{-th}$ harmonic has an amplitude scaled by $\frac{1}{n}$.

• Non-Periodic Signals:
  • Non-periodic signals (such as human speech or musical waveforms) are represented in the frequency domain via the Fourier transform.
  • As period $T \rightarrow \infty$, spacing between spectral lines $\omega_0 = \frac{2\pi}{T} \rightarrow 0$, turning discrete spectral lines into a continuous frequency spectrum containing all frequencies.
  • Audible Frequency Band: Although continuous signals mathematically span infinite frequencies, practical signals concentrate energy within specific bands. For example, audible speech and music occupy approximately $20\,\text{Hz}$ to $20\,\text{kHz}$.

Analogue and Digital Signals

• Analogue Signals:

  • Continuous in time and amplitude, taking any value within a continuous range.
  • Analogous directly to the physical phenomena they represent.

• Discrete-Time Signals:

  • Formed by sampling continuous-time signals at discrete time instants $t_0, t_1, t_2, t_3, \dots$
  • Continuous in amplitude (sample values can take any real number), but discrete in time.

• Digital Signals:
  • Created when sample magnitudes are quantized into discrete levels represented by finite digits.
  • Binary Number System: Uses two discrete voltage levels (logic 0 and logic 1, typically $0\,\text{V}$ and $+5\,\text{V}$ in practical electronic circuits).
  • $N$--Bit Binary Representation Formula:     $D = b_0 2^0 + b_1 2^1 + b_2 2^2 + \dots + b_{N-1} 2^{N-1}$     where $b_0$ is the Least Significant Bit (LSB) and $b_{N-1}$ is the Most Significant Bit (MSB), written as word $b_{N-1}b_{N-2}\dots b_0$
  • Quantization & Resolution: An $N$-bit binary digit quantizes the signal into $2^N$ discrete amplitude levels. Higher $N$ values decrease quantization error and improve analog-to-digital resolution.
  • Worked Example: For an 8-bit binary word 10110100 ($b_7=1, b_6=0, b_5=1, b_4=1, b_3=0, b_2=1, b_1=0, b_0=0$):     $D = 0 \cdot 2^0 + 0 \cdot 2^1 + 1 \cdot 2^2 + 0 \cdot 2^3 + 1 \cdot 2^4 + 1 \cdot 2^5 + 0 \cdot 2^6 + 1 \cdot 2^7 = 180$

Amplifiers and Signal Amplification

• Need for Amplification:

  • Most transducers produce weak electrical signals containing minimal energy, typically in the microvolt ($\mu\text{V}$) to millivolt ($\text{mV}$) range. Amplification boosts these signal levels for reliable processing.

• Linear Amplifiers:

  • An ideal amplifier increases signal amplitude without changing the waveform shape (preventing distortion):     $v_o(t) = A \cdot v_i(t)$     where $v_i(t)$ is input signal, $v_o(t)$ is output signal, and $A$ is constant amplifier gain.

• Gain Definitions:

  • Voltage Gain ($A_v$): Ratio of output voltage to input voltage:     $A_v \equiv \frac{v_O}{v_I}\,\text{[V/V]}$
  • Current Gain ($A_i$): Ratio of output current to input current:     $A_i \equiv \frac{i_O}{i_I}\,\text{[A/A]}$
  • Power Gain ($A_p$): Ratio of load power to input power:     $A_p \equiv \frac{\text{load power } (P_L)}{\text{input power } (P_I)} = \frac{v_O i_O}{v_I i_I} = A_v A_i\,\text{[W/W]}$

• Gain in Decibels (dB):

  • Logarithmic representations reflect human sensory response and facilitate system-level calculations:     $\text{Power gain in decibels} = 10 \log(A_p)\,\text{dB}$$\text{Voltage gain in decibels} = 20 \log(|A_v|)\,\text{dB}$$\text{Current gain in decibels} = 20 \log(|A_i|)\,\text{dB}$

• Power Gain vs. Decibel Ratios:

  • $A_p = 0.1 \rightarrow -10\,\text{dB}$
  • $A_p = 0.5 \rightarrow -3\,\text{dB}$
  • $A_p = 1.0 \rightarrow 0\,\text{dB}$
  • $A_p = 2.0 \rightarrow 3\,\text{dB}$
  • $A_p = 10 \rightarrow 10\,\text{dB}$
  • $A_p = 100 \rightarrow 20\,\text{dB}$

• Voltage/Current Gain vs. Decibel Ratios:

  • $A_v\text{ or }A_i = 0.1 \rightarrow -20\,\text{dB}$
  • $A_v\text{ or }A_i = 0.707 \left(\frac{1}{\sqrt{2}}\right) \rightarrow -3\,\text{dB}$
  • $A_v\text{ or }A_i = 1.0 \rightarrow 0\,\text{dB}$
  • $A_v\text{ or }A_i = 1.414 \left(\sqrt{2}\right) \rightarrow 3\,\text{dB}$
  • $A_v\text{ or }A_i = 2.0 \rightarrow 6\,\text{dB}$
  • $A_v\text{ or }A_i = 10 \rightarrow 20\,\text{dB}$
  • $A_v\text{ or }A_i = 100 \rightarrow 40\,\text{dB}$
  • Half-Power Point: The $-3\,\text{dB}$ attenuation point corresponds to a voltage/current gain magnitude of $\frac{1}{\sqrt{2}} \approx 0.707$

Amplifier Power Supplies, Efficiency, and Saturation

• Power Supplies:
  • Power delivered to the load is drawn from DC power supplies (dual-rail/split-rail supplies $V^+$ and $V^-$, or single supply).

• DC Supply Power:   $P_{dc} = V_1 I_1 + V_2 I_2$

• Power Balance Equation:   $P_{dc} + P_I = P_L + P_{diss}$   where $P_I$ is input signal power, $P_L$ is load power, and $P_{diss}$ is internal heat dissipation.

• Amplifier Efficiency ($\eta$):   $\eta \equiv \frac{P_L}{P_{dc}} \times 100\,\%$

• Amplifier Saturation:

  • Practical amplifiers are constrained by finite power supply voltage limits $L^+$ and $L^-$.
  • Linear Range Limits:     $\frac{L^-}{A_v} \le v_I \le \frac{L^+}{A_v}$
  • Input signals within this window produce undistorted output signals. Exceeding this range causes output clipping due to saturation.

Non-Linear Transfer Characteristics and Biasing

• Non-Linear Amplifiers:

  • Single-supply amplifiers typically display non-linear curvature and off-origin transfer characteristics.

• Biasing Technique:

  • A constant DC bias voltage $V_B$ is applied to set the quiescent point (Q point, operating point) at DC output voltage $VO$
  • The small AC input signal $v_i(t)$ is superimposed on $V_B$.
  • Confining signal excursion to a narrow region around Q point ensures approximately linear operation (small-signal operation).

• Small-Signal Voltage Gain:

  • Evaluated as the derivative (slope) at the Q point:     $A_v = \left. \frac{d v_O}{d v_I} \right|_{\text{Q pt}}$

Two-Port Circuit Models for Amplifiers

• Voltage Amplifier Model:
  • Represents input as voltage $v_i$ and output as dependent voltage source $A_{vo} v_i$ in series with $R_o$
  • Open-Circuit Voltage Gain: $A_{vo} \equiv \left.\frac{v_o}{v_i}\right|_{R_L=\infty}$
  • Ideal Parameters: $R_i = \infty$, $R_o = 0$
  • Overall Voltage Gain:     $\frac{v_o}{v_s} = \frac{R_i}{R_s + R_i} \cdot A_{vo} \cdot \frac{R_L}{R_o + R_L}$

• Current Amplifier Model:
  • Represents input as current $i_i$ and output as dependent current source $A_{is} i_i$ in parallel with $R_o$
  • Short-Circuit Current Gain: $A_{is} \equiv \left.\frac{i_o}{i_i}\right|_{R_L=0}$
  • Ideal Parameters: $R_i = 0$, $R_o = \infty$
  • Overall Current Gain:     $\frac{i_o}{i_s} = \frac{R_s}{R_s + R_i} \cdot A_{is} \cdot \frac{R_o}{R_o + R_L}$

• Transconductance Amplifier Model:
  • Represents input as voltage $v_i$ and output as dependent current source $G_m v_i$ in parallel with $R_o$
  • Short-Circuit Transconductance: $G_m \equiv \left.\frac{i_o}{v_i}\right|_{R_L=0}$
  • Ideal Parameters: $R_i = \infty$, $R_o = \infty$
  • Overall Gain:     $\frac{i_o}{v_s} = \frac{R_i}{R_s + R_i} \cdot G_m \cdot \frac{R_o}{R_o + R_L}$

• Transresistance Amplifier Model:
  • Represents input as current $i_i$ and output as dependent voltage source $R_m i_i$ in series with $R_o$
  • Open-Circuit Transresistance: $R_m \equiv \left.\frac{v_o}{i_i}\right|_{R_L=\infty}$
  • Ideal Parameters: $R_i = 0$, $R_o = 0$
  • Overall Gain:     $\frac{v_o}{i_s} = \frac{R_s}{R_s + R_i} \cdot R_m \cdot \frac{R_L}{R_o + R_L}$

Frequency Response of Amplifiers

• Measurement Method:
  • Injected sine wave: $v_i(t) = V_i \sin(\omega t)$
  • Resulting linear output: $v_o(t) = V_o \sin(\omega t + \varphi)$
  • Complex Transfer Function:     $T(\omega) = \frac{V_o}{V_i} \angle \varphi$     where magnitude is $|T(\omega)| = \frac{V_o}{V_i}$ and phase shift is $\angle T(\omega) = \varphi$

• Amplifier Bandwidth:
  • Defined by lower cutoff frequency $\omega_L$ and upper cutoff frequency $\omega_H$ where gain drops by $3\,\text{dB}$ (half-power points):     $\text{Bandwidth} = \omega_H - \omega_L$

Single-Time-Constant (STC) Networks

• Definition: First-order circuits containing one reactive element ($C$ or $L$) and one resistance ($R$

• Time Constant & Corner Frequency:   $\tau = RC\,\text{[s]}$$\omega_0 = \frac{1}{\tau} = \frac{1}{RC}\,\text{[rad/s]}$$f_0 = \frac{1}{2\pi \tau} = \frac{1}{2\pi RC}\,\text{[Hz]}$

• Low Pass (LP) STC Network:

  • Transfer Function:     $T(j\omega) = \frac{V_o}{V_i} = \frac{Z_C}{Z_C + R} = \frac{1 / (j\omega C)}{1 / (j\omega C) + R} = \frac{1}{1 + j\omega CR} = \frac{1}{1 + j\omega \tau} = \frac{1}{1 + j\frac{\omega}{\omega_0}} = \frac{1}{1 + j\frac{f}{f_0}}$
  • Magnitude Response:     $|T(j\omega)| = \frac{1}{\sqrt{1 + \left(\frac{\omega}{\omega_0}\right)^2}} = \frac{1}{\sqrt{1 + \left(\frac{f}{f_0}\right)^2}}$
  • Phase Response:     $\angle T(j\omega) = -\tan^{-1}\left(\frac{\omega}{\omega_0}\right) = -\tan^{-1}\left(\frac{f}{f_0}\right)$

• High Pass (HP) STC Network:
  • Transfer Function:     $T(j\omega) = \frac{V_o}{V_i} = \frac{R}{R + Z_C} = \frac{R}{R + 1 / (j\omega C)} = \frac{1}{1 + \frac{1}{j\omega CR}} = \frac{1}{1 + \frac{1}{j\omega \tau}} = \frac{1}{1 - j\frac{\omega_0}{\omega}} = \frac{1}{1 - j\frac{f_0}{f}}$
  • Magnitude Response:     $|T(j\omega)| = \frac{1}{\sqrt{1 + \left(\frac{\omega_0}{\omega}\right)^2}} = \frac{1}{\sqrt{1 + \left(\frac{f_0}{f}\right)^2}}$
  • Phase Response:     $\angle T(j\omega) = \tan^{-1}\left(\frac{\omega_0}{\omega}\right) = \tan^{-1}\left(\frac{f_0}{f}\right)$

Asymptote Bode Plots for STC Networks

• Bode Format: Uses logarithmic frequency horizontal axes and magnitude in decibels ($20 \log |T(j\omega)|\,\text{dB}$.

• Low Pass Bode Plot:

  • Magnitude: Constant at $0\,\text{dB}$ for $\omega \ll \omega_0$; rolls off at $-20\,\text{dB/dec}$ for $\omega \gg \omega_0$; drops to $-3\,\text{dB}$ at corner frequency $\omega_0$
  • Phase: $0^\circ$ at $0.1\omega_0$; $-45^\circ$ at $\omega_0$; $-90^\circ$ at $10\omega_0$

• High Pass Bode Plot:
  • Magnitude: Rises at $+20\,\text{dB/dec}$ for $\omega \ll \omega_0$; flat at $0\,\text{dB}$ for $\omega \gg \omega_0$; drops to $-3\,\text{dB}$ at corner frequency $\omega_0$
  • Phase: $+90^\circ$ at $0.1\omega_0$; $+45^\circ$ at $\omega_0$; $0^\circ$ at $10\omega_0$