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, VGS − Vth = 500mV, = 0.1V-1, CL = 10pF and IC = ID = 1mA.
- Calculate the DC voltage gain vout/vin for each structure. Determine the ratios gm/IC and gm/ID (transconductance efficiency).
- 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.
- The so-called “square-law” model of the FET incorrectly predicts that current becomes arbitrarily small (and gm arbitrarily large) as VGS – Vth 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.
- Determine values for N and R1 such that IC1 = IC2 = 50A at room temperature (27C).
- Determine the temperature slope of VBE1 via simulation and calculate the value of M that would satisfy the above equation.
- 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
- the value of VBG at room temperature, and
- 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
- (5 points) Assuming the amplifier is driven with a sinusoidal voltage Vin = ain sin(2f0t) + 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)]
- Calculate the ratio of a2/a1 for ain = 1mV and ain = 10mV.