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

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Common-emitter versus common-source amplifier

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

For the following, T = 300K, VA = 100V, VGSVth = 500mV,  = 0.1V-1, CL = 10pF and IC = ID = 1mA.

1.  Calculate the DC voltage gain vout/vin for each structure. Determine the ratios gm/IC and gm/ID (transconductance efficiency).
2. For each structure, determine the small-signal transfer function vout/vin as a function of frequency. Plot the Bode magnitude and phase (by hand or using MATLAB/Python). For each, calculate the transit frequency fT, 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 gm arbitrarily large) as VGSVth approaches zero. For values of VGS smaller than Vth (subthreshold operation), the drain current is better described as

𝐼𝐷 = 𝐼𝑆𝑒𝑉𝐺𝑆/𝑛𝑉𝑇,

where IS and n are technology parameters related to the device structure. For n = 1.5, calculate the transconductance efficiency (gm/ID) 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 VBE of a BJT exhibits a slope of approximately −2mV/C (CTAT). Combining this with the difference of the VBE’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 (IS = 10-14A,  = 300, VA = 100) and the UniversalOpamp2 models in Ltspice. Use VCC = 5V for the supply voltage.

1.  Determine values for N and R1 such that IC1 = IC2 = 50A at room temperature (27C).
2.  Determine the temperature slope of VBE1 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 𝑉𝐵𝐸(𝑇) + 𝑀 × ∆𝑉𝐵𝐸(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 VBG 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 Vin = ain  sin(2f0t) + VDC, where VDC = Vth + 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 ain = 1mV and ain = 10mV.

• hw03-cvpc6x.zip