UWEE538 Homework 3-Common-emitter versus common-source amplifier Solved

Description

Common-emitter versus common-source amplifier

 Figure 1a. Common-emitter (CE) amplifier                   Figure 1b. Common-source (CS) amplifier

For the following, T = 300K, V_A = 100V, V_GS − V_th = 500mV,  = 0.1V^-1, C_L = 10pF and I_C = I_D = 1mA.

1.  Calculate the DC voltage gain v_out/v_in for each structure. Determine the ratios g_m/I_C and g_m/I_D (transconductance efficiency).
2. For each structure, determine the small-signal transfer function v_out/v_in as a function of frequency. Plot the Bode magnitude and phase (by hand or using MATLAB/Python). For each, calculate the transit frequency f_T, the frequency at which the magnitude of the transfer function is equal to 1V/V.
3. The so-called “square-law” model of the FET incorrectly predicts that current becomes arbitrarily small (and g_m arbitrarily large) as V_GS – V_th approaches zero. For values of V_GS smaller than V_th (subthreshold operation), the drain current is better described as

𝐼𝐷 = 𝐼𝑆𝑒𝑉𝐺𝑆/𝑛𝑉𝑇,

where I_S and n are technology parameters related to the device structure. For n = 1.5, calculate the transconductance efficiency (g_m/I_D) of the FET assuming subthreshold operation. How does it compare to your answers in Part a)?

Problem 2: Temperature-independent voltage reference (BJT DC analysis)

Figure 2. PTAT Voltage Generator

Temperature-insensitive voltage and current references are critical components of precision sensor systems. A temperature-independent reference is created by combining something (e.g. a voltage) that has a positive temperature coefficient (proportional-to-absolute-temperature, PTAT) with something that has a negative temperature coefficient (complementary-to-absolute-temperature, CTAT). When biased with a constant current, the V_BE of a BJT exhibits a slope of approximately −2mV/C (CTAT). Combining this with the difference of the V_BE’s of two BJT’s biased with different current densities (which is PTAT), properly scaled, will yield a voltage that is (approximately) independent of temperature:

𝑉𝐵𝐺 = 𝑉𝐶𝑇𝐴𝑇 + 𝑉𝑃𝑇𝐴𝑇 = 𝑉𝐵𝐸(𝑇) + 𝑀 × ∆𝑉𝐵𝐸(𝑇)

Note that different current densities for Q₁ and Q₂ are achieved by connecting N transistors in parallel for Q₂.

For the following, use the 2N3904 npn transistor (I_S = 10^-14A,  = 300, V_A = 100) and the UniversalOpamp2 models in Ltspice. Use V_CC = 5V for the supply voltage.

1.  Determine values for N and R₁ such that I_C1 = I_C2 = 50A at room temperature (27C).
2.  Determine the temperature slope of V_BE1 via simulation and calculate the value of M that would satisfy the above equation.
3. Verify the design of the PTAT generator in Ltspice, plotting the expression 𝑉_𝐵𝐸(𝑇) + 𝑀 × ∆𝑉_𝐵𝐸(𝑇) as a function of temperature. Include your schematic in your submission, showing all relevant voltages and currents at room temperature. Evaluate
   1. the value of V_BG at room temperature, and
   2. the maximum deviation from this value over the temperature range −40C to 125C.

Bonus (: Complete the design of the Brokaw bandgap circuit.

                 Problem 3: Nonlinear distortion in a common-source amplifier

The output voltage of a resistor-loaded common-source amplifier is expressed (neglecting ) as

𝑉𝑜𝑢𝑡 = 𝑉𝐷𝐷 −(𝑉𝑖𝑛 − 𝑉𝑡ℎ)2𝑅𝐷

1. (5 points) Assuming the amplifier is driven with a sinusoidal voltage V_in = a_in  sin(2f₀t) + V_DC, where V_DC = V_th + 500mV, determine expressions for the amplitudes of the fundamental (sinusoid at f₀ with amplitude a₁) and second harmonic (sinusoid at 2f₀ with amplitude a₂) using the trigonometric relationship

sin²(𝑥) =  [1 − cos(2𝑥)]

1.  Calculate the ratio of a₂/a₁ for a_in = 1mV and a_in = 10mV.