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In this assignment, you will design a differential amplifier as shown in the schematics below.

Your design should satisfy the required differential gain, input impedance, single-ended common-mode
gain, and the power dissipation specifications provided in Section A below. Then in Sections B and C, you
will simulate your circuit on LTSpice to compare the simulation results with hand calculations.

A. Hand design: Design the bipolar differential amplifier and the current source and bias network
(𝑅1, 𝑄3, 𝑎𝑛𝑑 𝑄4) above such that:

(i) Differential gain: 𝐴𝑑 ≥ 200 𝑉,
𝑉

(ii) Input differential resistance: 𝑅𝑖𝑑 ≥ 50 𝑘Ω,
(iii) 𝐴𝑐𝑚 < 0.1 where 𝐴𝑐𝑚 is the single-ended common-mode gain, defined as the gain for a common-
mode input signal when the output is measured from one of the outputs with respect to ground, rather
than differentially.
(iv) To maximize battery life, the power consumption of the differential amplifier, 𝑃𝑇𝑂𝑇𝐴𝐿, must be less
than 2 milliwatts.

Design Suggestion for Part A.
1. Derive the expression for 𝑅 , ignoring 𝑟 of Q1 and Q2.). Replace small signal parameters with

𝑖𝑑 𝑜

“DC currents”, “resistors”, “𝑉 “, “𝛽”, “𝑉 “, etc., use specific values (e.g., 𝑉 = 100 𝑉, 𝛽 = 200,
𝑡h𝐴 𝐴

etc.), and simplify the expressions. The 𝑅𝑖𝑑 design specification will help determine an upper limit
for the DC collector current of Q1 (and Q2 since 𝐼𝐶1 = 𝐼𝐶2).

2. The total power dissipation is 𝑃
𝑇𝑂𝑇𝐴𝐿

=𝑃 +𝑃 =9𝑉×(𝐼 +𝐼 ). Here, 𝐼 is the
𝑉𝐷𝐷 𝑉𝑆𝑆 𝑉𝐷𝐷 𝑉𝑆𝑆 𝑉𝐷𝐷

sum of all DC currents leaving the +𝑉
𝐷𝐷

and 𝐼 is the sum of all dc currents entering the −𝑉 .
𝑉𝑆𝑆 𝑆𝑆

Express 𝐼𝑉𝐷𝐷 and 𝐼𝑉𝑆𝑆 in terms of the collector currents of Q1 (or Q2), ignoring the current in the

reference current generation branch formed by 𝑅1 and 𝑄4. The 𝑃𝑇𝑂𝑇𝐴𝐿 design specification will
help determine another upper limit for 𝐼𝐶1 = 𝐼𝐶2.

1. Derive expressions for 𝐴 and 𝐴 . When deriving 𝐴 , include 𝑟 of Q1 or Q2. When deriving 𝐴
   𝑑𝑐𝑚𝑑𝑜 𝑐𝑚
   ignore 𝑟 of Q1 and Q2 but include 𝑟 of Q3. Replace small signal parameters with “DC currents”,
   𝑜𝑜
   “resistors”, “𝑉 “, “𝛽”, “𝑉 “, etc., use specific values (e.g., 𝑉 = 100 𝑉, 𝛽 = 200, etc.), and
   𝑡h𝐴 𝐴
   simplify the expressions. The 𝐴𝑑 and 𝐴𝑐𝑚 design specifications will respectively provide lower and
   upper limits for the product 𝑅𝐶 × 𝐼𝐶1.
2. Consider the forward-active region requirement of Q1 (or Q2). For 𝑉 = 0 𝑉, find another upper
   𝐶𝑀
   limit for the product 𝑅𝐶 × 𝐼𝐶1.
3. Pick an 𝐼𝐶1 satisfying all upper limits for 𝐼𝐶1 found in steps 1 and 2. Find 𝑅1 based on the value you
   pick for 𝐼𝐶1. Pick an 𝑅𝐶 satisfying the upper and lower limits for 𝑅𝐶 ∗ 𝐼𝐶1 found in steps 3 and 4.

Inyoursimulations,usetheBJTmodel2N2222ofNXP,whichhasaSPICEmodelasbelowwith𝑉 and𝛽

highlighted:

𝐴

B. DC Analysis: In LTSpice do a DC operating point simulation (.op) with both inputs connected to ground
(i.e.,𝑉 =0𝑉).DeterminethesimulatedDCvaluesfor𝐼 ,𝐼 ,𝐼 ,𝐼 ,𝐼 ,𝑉 ,𝑉 ,𝑉 ,𝑉 .Compare

𝐶𝑀 𝑅1𝐶3𝐶4𝐶1𝐶2 𝐵3 𝐸1,2 𝑂1 𝑂2
these results with your hand calculations. Additionally, comment on the matching between 𝐼𝑅1 𝑎𝑛𝑑 𝐼𝐶3
and comment on the calculated vs. simulated match between 𝐼𝑅1 𝑎𝑛𝑑 𝐼𝐶3.

C. Transient Analysis: In LTSpice do a transient simulation (.tran) for 100 ms.
For differential small-signal input simulations:

Apply 𝑣 =1𝑚𝑉 𝑠𝑖𝑛𝑢𝑠𝑜𝑖𝑑𝑎𝑙𝑠𝑖𝑔𝑛𝑎𝑙𝑎𝑡100𝐻𝑧. [i.e., 𝑣 =+𝑣𝑖𝑑 =0.5𝑚𝑉sin(2∗𝜋∗100𝐻𝑧∗𝑡)

𝑖𝑑 𝑝 𝑖𝑑1

2

and𝑣𝑖𝑑2 =−𝑣𝑖𝑑 =0.5𝑚𝑉sin((2∗𝜋∗100𝐻𝑧∗𝑡)+𝜋)withDCoffset=0V.]
2

𝑣 =𝑣 =𝑣
𝑖𝑐𝑚1 𝑖𝑐𝑚2 𝑐𝑚

For common-mode small-signal input simulations:
Apply 𝑣 =1𝑚𝑉 𝑠𝑖𝑛𝑢𝑠𝑜𝑖𝑑𝑎𝑙𝑠𝑖𝑔𝑛𝑎𝑙𝑎𝑡100𝐻𝑧. [i.e.,

=1𝑚𝑉sin(2∗𝜋∗

𝑐𝑚 𝑝
100𝐻𝑧 ∗ 𝑡) with DC offset = 0V.]

1. For the differential small-signal input, what is the expected emitter voltage of Q1 and Q2,
𝑣𝑒1(= 𝑣𝑒2)? Plot the simulated waveform. What is the simulated value of 𝑣𝑒1(= 𝑣𝑒2)?

2. Plot 𝑣𝑖𝑑, 𝑣𝑜𝑑(𝑣𝑜𝑑 = 𝑣𝑜2 − 𝑣𝑜1), 𝑎𝑛𝑑 𝑖𝑖𝑑. Note that 𝑖𝑖𝑑 is the base current of Q1 (𝑖𝑖𝑑 = 𝑖𝑏1).
Calculate the simulated 𝐴𝑑 = 𝑣𝑜𝑑 /𝑣𝑖𝑑 and 𝑅𝑖𝑑 = 𝑣𝑖𝑑 /𝑖𝑖𝑑 . Compare the values with your
design targets.

3. If the simulation results do not match the design constraints, revisit your design to achieve
the targets.

4. For the common-mode small-signal input, plot 𝑣𝑐𝑚 and 𝑣𝑜𝑐𝑚 . (𝑣𝑜𝑐𝑚 = 𝑣𝑜𝑐𝑚2 =
𝑣𝑜𝑐𝑚1 𝑤h𝑒𝑛 𝑡h𝑒 𝑖𝑛𝑝𝑢𝑡 𝑖𝑠 𝑎 𝑐𝑜𝑚𝑚𝑜𝑛 − 𝑚𝑜𝑑𝑒 𝑠𝑖𝑔𝑛𝑎𝑙)

Report Requirements
A. Your hand calculations for part A must be detailed and sequential. Show every step of the derivations
clearly, and explain how and why you select each parameter, including any approximations made.
B and C. For parts B and C, besides answering the questions and plotting the requested simulation results,
complete the table below. Provide explanations for any discrepancies exceeding 10%.

𝐼𝑅1 𝐼𝐶4 𝐼𝐶3

𝐼𝐶1 𝐼𝐶2

Hand
calculations
Simulated

Percent
discrepancy

Hand
calculations
Simulated

Percent
discrepancy

𝑉𝑉𝑉𝑉𝐴𝑅𝐴
𝐵3 𝐸1,2 𝑂1 𝑂2 𝑑 𝑖𝑛 𝑐𝑚