8.4 Small-signal models and parameters

Small-signal models are crucial for understanding how semiconductor devices behave under small input signals. These models simplify complex device physics into manageable circuit elements, allowing engineers to analyze and design analog circuits effectively.

The key parameters in small-signal models include transconductance, input and output resistances, and capacitances. These parameters determine a device's gain, impedance, and frequency response, shaping its performance in various applications like amplifiers, oscillators, and mixers.

Small-signal equivalent circuit

• Small-signal equivalent circuits model the behavior of semiconductor devices under small-signal conditions, where the input signal is much smaller than the DC bias
• These models are essential for analyzing and designing analog circuits such as amplifiers, oscillators, and mixers in the context of semiconductor device physics

Hybrid-pi model

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• Represents the small-signal behavior of a bipolar junction transistor (BJT)
• Consists of a voltage-controlled current source $(g_m v_{\pi})$, input resistance $(r_{\pi})$, and output resistance $(r_o)$
• The transconductance $(g_m)$ relates the output current to the input voltage, while the resistances model the input and output impedances
• Widely used in the analysis and design of BJT-based analog circuits (common-emitter amplifier)

T-model

• Another small-signal model for BJTs, similar to the hybrid-pi model
• Represents the transistor using a voltage-controlled current source $(g_m v_{be})$ and two resistances $(r_e$ and $r_c)$
• The resistances model the emitter and collector impedances, respectively
• Useful for analyzing circuits with multiple transistors (cascode amplifier)

Y-parameters

• A matrix representation of the small-signal behavior of a two-port network, such as a transistor
• Relates the input and output currents to the input and output voltages through admittance parameters $(y_{11}, y_{12}, y_{21}, y_{22})$
• Allows for easy analysis of the small-signal performance of a device or circuit
• Commonly used in microwave and RF circuit design (low-noise amplifiers)

Transconductance

• Transconductance is a key parameter in small-signal models, describing the relationship between the input voltage and output current of a device

Definition of transconductance

• Transconductance $(g_m)$ is defined as the ratio of the small-signal output current change to the small-signal input voltage change, measured in siemens (S) or mhos
• Mathematically, $g_m = \frac{\partial I_c}{\partial V_{be}}$ for BJTs and $g_m = \frac{\partial I_d}{\partial V_{gs}}$ for field-effect transistors (FETs)
• Represents the gain of the device, with higher transconductance indicating higher gain

Transconductance vs bias current

• Transconductance is proportional to the DC bias current in both BJTs and FETs
• For BJTs, $g_m = \frac{I_c}{V_T}$, where $I_c$ is the collector current and $V_T$ is the thermal voltage (approximately 26 mV at room temperature)
• For FETs, $g_m = \frac{2I_d}{V_{gs} - V_{th}}$ in the saturation region, where $I_d$ is the drain current, $V_{gs}$ is the gate-source voltage, and $V_{th}$ is the threshold voltage
• Increasing the bias current leads to higher transconductance and gain

Measurement of transconductance

• Transconductance can be measured using a small-signal AC test setup
• Apply a small AC voltage to the input (base-emitter for BJTs or gate-source for FETs) and measure the resulting AC output current (collector for BJTs or drain for FETs)
• The ratio of the output current to the input voltage gives the transconductance
• Automated measurement techniques (network analyzers) are commonly used for accurate and efficient characterization

Input resistance

• Input resistance is another important parameter in small-signal models, representing the impedance seen looking into the input of a device

Definition of input resistance

• Input resistance $(r_{\pi}$ for BJTs or $r_{gs}$ for FETs) is defined as the ratio of the small-signal input voltage change to the small-signal input current change
• Mathematically, $r_{\pi} = \frac{\partial V_{be}}{\partial I_b}$ for BJTs and $r_{gs} = \frac{\partial V_{gs}}{\partial I_g}$ for FETs
• A high input resistance is desirable for minimizing loading effects on the input signal source

Input resistance vs bias current

• Input resistance is inversely proportional to the DC bias current in both BJTs and FETs
• For BJTs, $r_{\pi} = \frac{\beta V_T}{I_c}$, where $\beta$ is the current gain, $V_T$ is the thermal voltage, and $I_c$ is the collector current
• For FETs, $r_{gs}$ is typically very high (on the order of megaohms) due to the insulated gate structure
• Increasing the bias current leads to lower input resistance

Measurement of input resistance

• Input resistance can be measured using a small-signal AC test setup
• Apply a small AC voltage to the input and measure the resulting AC input current
• The ratio of the input voltage to the input current gives the input resistance
• Accurate measurement requires proper de-embedding of parasitic impedances (probe pads, interconnects)

Output resistance

• Output resistance is the third key parameter in small-signal models, representing the impedance seen looking into the output of a device

Definition of output resistance

• Output resistance $(r_o)$ is defined as the ratio of the small-signal output voltage change to the small-signal output current change
• Mathematically, $r_o = \frac{\partial V_{ce}}{\partial I_c}$ for BJTs and $r_o = \frac{\partial V_{ds}}{\partial I_d}$ for FETs
• A high output resistance is desirable for maximizing the voltage gain and minimizing loading effects on the output load

Output resistance vs bias current

• Output resistance is inversely proportional to the DC bias current in both BJTs and FETs
• For BJTs, $r_o = \frac{V_A}{I_c}$, where $V_A$ is the Early voltage and $I_c$ is the collector current
• For FETs, $r_o = \frac{1}{\lambda I_d}$ in the saturation region, where $\lambda$ is the channel-length modulation parameter and $I_d$ is the drain current
• Increasing the bias current leads to lower output resistance

Measurement of output resistance

• Output resistance can be measured using a small-signal AC test setup
• Apply a small AC current to the output and measure the resulting AC output voltage
• The ratio of the output voltage to the output current gives the output resistance
• Accurate measurement requires proper de-embedding of parasitic impedances and consideration of device nonlinearities

Capacitances

• Capacitances in semiconductor devices play a crucial role in determining the high-frequency performance and transient response of small-signal circuits

Depletion layer capacitance

• Depletion layer capacitance $(C_j)$ arises from the voltage-dependent variation of the depletion region width in p-n junctions
• For a p-n junction, $C_j = \frac{C_{j0}}{\sqrt{1 - \frac{V}{V_0}}}$, where $C_{j0}$ is the zero-bias capacitance, $V$ is the applied voltage, and $V_0$ is the built-in potential
• Depletion layer capacitance is present in the base-collector and base-emitter junctions of BJTs and the drain-substrate and source-substrate junctions of FETs

Diffusion capacitance

• Diffusion capacitance $(C_d)$ is associated with the charge storage in the neutral regions of a forward-biased p-n junction
• For a p-n junction, $C_d = \frac{\tau I}{V_T}$, where $\tau$ is the carrier lifetime, $I$ is the forward bias current, and $V_T$ is the thermal voltage
• Diffusion capacitance is present in the base-emitter junction of BJTs and can limit the high-frequency performance

Parasitic capacitances

• Parasitic capacitances arise from the device geometry and interconnect layout
• Examples include the base-collector capacitance $(C_{bc})$ in BJTs and the gate-drain $(C_{gd})$, gate-source $(C_{gs})$, and drain-source $(C_{ds})$ capacitances in FETs
• These capacitances can cause high-frequency roll-off, oscillations, and crosstalk in small-signal circuits
• Careful device and layout design is necessary to minimize parasitic capacitances

Frequency response

• The frequency response of semiconductor devices and small-signal circuits determines their behavior and performance at different operating frequencies

Cutoff frequency

• The cutoff frequency $(f_T)$ is the frequency at which the small-signal current gain of a device falls to unity
• For BJTs, $f_T = \frac{g_m}{2\pi(C_{\pi} + C_{\mu})}$, where $g_m$ is the transconductance, $C_{\pi}$ is the base-emitter capacitance, and $C_{\mu}$ is the base-collector capacitance
• For FETs, $f_T = \frac{g_m}{2\pi(C_{gs} + C_{gd})}$, where $C_{gs}$ and $C_{gd}$ are the gate-source and gate-drain capacitances, respectively
• A higher cutoff frequency indicates better high-frequency performance

Gain-bandwidth product

• The gain-bandwidth product (GBW) is the product of the low-frequency voltage gain $(A_v)$ and the cutoff frequency $(f_T)$
• For a single-pole system, GBW is constant and equal to $\frac{g_m}{2\pi C}$, where $C$ is the dominant capacitance
• The GBW is a figure of merit for the high-frequency performance of amplifiers and other small-signal circuits
• Tradeoffs between gain and bandwidth can be made based on the GBW

Miller effect

• The Miller effect refers to the increase in the apparent input capacitance of an inverting amplifier due to the amplification of the feedback capacitance
• The Miller capacitance is given by $C_M = (1 - A_v)C_f$, where $A_v$ is the voltage gain and $C_f$ is the feedback capacitance
• The Miller effect can significantly impact the high-frequency performance of amplifiers and should be considered in the design process
• Techniques such as cascoding and neutralization can be used to mitigate the Miller effect

Small-signal analysis

• Small-signal analysis is the process of analyzing the behavior of semiconductor devices and circuits under small-signal conditions

AC equivalent circuit

• The AC equivalent circuit is a simplified representation of a device or circuit that models its small-signal behavior
• It consists of resistances, capacitances, and voltage- or current-controlled sources that represent the device parameters
• The AC equivalent circuit is derived by linearizing the device equations around the DC operating point
• It forms the basis for small-signal analysis techniques such as the hybrid-pi and Y-parameter methods

Hybrid-pi model analysis

• The hybrid-pi model is a widely used small-signal model for BJTs
• It represents the transistor using a voltage-controlled current source $(g_m v_{\pi})$, input resistance $(r_{\pi})$, and output resistance $(r_o)$
• The hybrid-pi model can be used to analyze the small-signal behavior of BJT-based circuits such as amplifiers and oscillators
• Analysis involves applying Kirchhoff's laws and solving for the desired voltages, currents, and gains

Y-parameter analysis

• Y-parameter analysis is a matrix-based approach to small-signal modeling and analysis
• The Y-parameters relate the input and output currents to the input and output voltages of a two-port network
• The Y-parameter matrix can be derived from the device equations or measured using a network analyzer
• Y-parameter analysis allows for the determination of small-signal parameters such as input and output impedances, voltage and current gains, and stability factors

Noise in small-signal models

• Noise is an important consideration in small-signal models, as it can limit the sensitivity and dynamic range of semiconductor devices and circuits

Thermal noise

• Thermal noise, also known as Johnson-Nyquist noise, is caused by the random motion of charge carriers in a conductor due to thermal agitation
• The thermal noise voltage across a resistance $R$ is given by $v_n = \sqrt{4kTRB}$, where $k$ is Boltzmann's constant, $T$ is the absolute temperature, and $B$ is the bandwidth
• Thermal noise is present in the resistive elements of small-signal models, such as the base and emitter resistances of BJTs and the channel resistance of FETs

Shot noise

• Shot noise is caused by the discrete nature of charge carriers and the randomness of their emission and collection processes
• The shot noise current in a p-n junction is given by $i_n = \sqrt{2qIB}$, where $q$ is the electron charge, $I$ is the DC current, and $B$ is the bandwidth
• Shot noise is present in the p-n junctions of BJTs and the channel of FETs, particularly at high bias currents

Flicker noise

• Flicker noise, also known as 1/f noise, is a low-frequency noise whose power spectral density is inversely proportional to frequency
• The origins of flicker noise are not fully understood, but it is believed to be related to surface and interface states, defects, and traps in semiconductor devices
• Flicker noise is more prominent in FETs than in BJTs due to the larger surface-to-volume ratio of the channel
• Flicker noise can be minimized through proper device design, fabrication, and biasing techniques

Applications of small-signal models

• Small-signal models are essential tools for the analysis, design, and optimization of various analog and RF circuits

Amplifier design

• Small-signal models are used to design and analyze amplifiers, such as common-emitter, common-source, and differential amplifiers
• The models allow for the determination of key performance parameters, such as voltage and current gains, input and output impedances, and frequency response
• Design considerations include biasing, device sizing, and component selection to achieve the desired gain, bandwidth, noise, and linearity specifications
• Techniques such as cascoding, feedback, and neutralization can be applied to improve the performance of amplifiers

Oscillator design

• Small-signal models are used in the design and analysis of oscillators, which generate periodic signals at a desired frequency
• The models help in determining the oscillation condition, which requires a loop gain of unity and a total phase shift of 360 degrees
• Common oscillator topologies include LC, crystal, and ring oscillators, each with its own small-signal model and design considerations
• Oscillator design involves selecting the device biasing, passive components, and feedback network to achieve the desired frequency, power, and stability

Mixer design

• Mixers are used in RF and microwave systems to perform frequency translation by multiplying two signals
• Small-signal models are used to analyze the performance of mixers, including conversion gain, noise figure, and linearity
• Mixer topologies include passive (diode) and active (transistor) mixers, each with its own small-signal model and design trade-offs
• Mixer design involves optimizing the device biasing, matching networks, and filtering to achieve the desired frequency translation, isolation, and spurious response suppression