Q1:
Q2:
1k
470 ohms
2.2k
Parameter
R=470 Ω
R=1.0 kΩ
R=2.2 kΩ
Resonant Frequency
(𝑓0)
799. 8𝐻𝑧
799. 8𝐻𝑧
799. 8𝐻𝑧
Lower Cutoff (𝑓1)
625. 0𝐻𝑧
605. 0𝐻𝑧
256. 7𝐻𝑧
Upper Cutoff (𝑓2)
1037. 0𝐻𝑧
1350. 0𝐻𝑧
2200. 0𝐻𝑧
Bandwidth (𝐵𝑊)
412. 0𝐻𝑧
745. 0𝐻𝑧
1943. 3𝐻𝑧
Simulated Quality
Factor (𝑄)
1. 94
1. 07
0. 41
Analytical Quality
Factor (𝑄)
1. 92
0. 91
0. 41
The resonant frequency f0​remains strictly constant at 799.8 Hz regardless of
resistance because it depends only on L (180 mH) and C (220 nF). As R increases
from 470 Ω to 2.2 kΩ, the lower cutoff frequency f1​decreases (625 Hz→256.7 Hz)
while the upper cutoff frequency f2​increases (1.037 kHz→2.2 kHz). This expands
the total bandwidth from 412 Hz to 1.943 kHz, causing the Quality Factor Q to drop
from 1.94 down to 0.41, verifying that Q is inversely proportional to resistance R.
Q3:
4.3k
Q4:
Parallel:
For an ideal parallel 𝑅𝐿𝐶 circuit driven by a current source (or evaluated via input admittance
𝑌(𝑠)), the admittance is:
1
1
𝑌(𝑠) = 𝑅 + 𝑠𝐿 + 𝑠𝐶
Setting 𝑠 = 𝑗ω:
1
(
1
𝑌(𝑗ω) = 𝑅 + 𝑗 ω𝐶 − ω𝐿
)
1. Resonant Frequency (ω0)
Resonance occurs when the imaginary part of the admittance (the total susceptance) equals
zero:
$$\omega_0 C - \frac{1}{\omega_0 L} = 0 \implies \omega_0^2 = \frac{1}{LC} \implies \omega_0
= \frac{1}{\sqrt{LC}}$$
5
Given ω0 = 100𝑘𝑟𝑎𝑑/𝑠 = 10 𝑟𝑎𝑑/𝑠:
𝐿𝐶 =
1
2
ω0
=
1
5 2
(10 )
−10 2
= 10
𝑠
2. Bandwidth (β) and Quality Factor (𝑄)
The bandwidth β is defined as the frequency range between the upper and lower half-power (
− 3𝑑𝐵) cutoff frequencies ω1 and ω2:
1
β = ω2 − ω1 = 𝑅𝐶 (𝑟𝑎𝑑/𝑠)
Given β = 5𝑘𝑟𝑎𝑑/𝑠 = 5000𝑟𝑎𝑑/𝑠:
1
1
−4
𝑅𝐶 = β = 5000 = 2 × 10 𝑠
The Quality Factor 𝑄 is:
𝑄=
ω0
β
=
100𝑘𝑟𝑎𝑑/𝑠
5𝑘𝑟𝑎𝑑/𝑠
= 20
3. Component Selection
Since 𝑅𝐿𝐶 circuit parameters leave one degree of freedom, selecting a standard baseline
resistance 𝑅 yields the inductance and capacitance. Choosing 𝑅 = 1𝑘Ω:
●​ Capacitance (𝐶):​
$$C = \frac{1}{\beta \cdot R} = \frac{1}{(5000\text{ rad/s})(1000\ \Omega)} = 0.2\
\mu\text{F} \quad (200\text{ nF})$$
●​ Inductance (𝐿):​
$$L = \frac{1}{\omega_0^2 C} = \frac{1}{(10^5)^2 (0.2 \times 10^{-6})} = 0.5\text{ mH}
\quad (500\ \mu\text{H})$$
Verification Summary Table
Parameter
Theoretical Expression
Resonant Frequency (
ω0)
1
Bandwidth (β)
1
𝑅𝐶
Target / Calculated Value
100𝑘𝑟𝑎𝑑/𝑠 (≈ 15. 915𝑘𝐻𝑧)
𝐿𝐶
5𝑘𝑟𝑎𝑑/𝑠 (≈ 795. 77𝐻𝑧)
Quality Factor (𝑄)
𝐶
𝐿
𝑅
=
ω0
20
β
Resistor (𝑅)
Chosen
1𝑘Ω
Capacitor (𝐶)
1
β𝑅
200𝑛𝐹
Inductor (𝐿)
1
0. 5𝑚𝐻
2
ω0𝐶
Q5:
Nmos transf ch
Op ch
Pmos op
Pmos tf ch