To solve the problem in the uploaded image, let's determine the resonant frequency of the circuit and how the frequency changes in part 2.

Problem 1:

The formula for the resonant frequency of a parallel LC circuit is:

$f = \frac{1}{2\pi\sqrt{L C_\text{total}}}$

Where:

• $L = 40 \, \mu\text{H} = 40 \times 10^{-6} \, \text{H}$
• $C_\text{total} = 330 \, \text{pF} + 50 \, \text{pF} = 380 \, \text{pF} = 380 \times 10^{-12} \, \text{F}$

Substitute the values:

$f = \frac{1}{2\pi \sqrt{(40 \times 10^{-6})(380 \times 10^{-12})}}$

Solve this to find the resonant frequency.

Problem 2:

If the varactor capacitance changes to $25 \, \text{pF}$, the new total capacitance becomes:

$C_\text{total} = 330 \, \text{pF} + 25 \, \text{pF} = 355 \, \text{pF} = 355 \times 10^{-12} \, \text{F}$

Substitute this new $C_\text{total}$ into the same resonant frequency formula to find the new frequency.

Let me calculate these frequencies for clarity.import math

Constants for Problem 1

L = 40e-6 # Inductance in Henries (H) C_total_1 = (330 + 50) * 1e-12 # Total capacitance in Farads (F)

Resonant frequency formula

def resonant_frequency(L, C): return 1 / (2 * math.pi * math.sqrt(L * C))

Frequency for Problem 1

f1 = resonant_frequency(L, C_total_1)

Constants for Problem 2

C_total_2 = (330 + 25) * 1e-12 # Total capacitance in Farads (F)

Frequency for Problem 2

f2 = resonant_frequency(L, C_total_2)

f1, f2(1290917.052417645, 1335598.6132348897)(1290917.052417645, 1335598.6132348897)successThe results are:

1. The resonant frequency for Problem 1 is approximately 1.29 MHz.
2. When the varactor capacitance decreases to 25 pF, the new resonant frequency is approximately 1.34 MHz.

Would you like further clarification or a step-by-step breakdown of these calculations?

Here are five related questions to explore:

1. How does increasing inductance affect the resonant frequency?
2. What happens to resonant frequency if capacitance doubles?
3. Can you calculate the quality factor of this circuit?
4. How do parasitic resistances influence the resonant frequency?
5. What applications rely on precise tuning of resonant circuits?

Tip: Always combine the inductance and capacitance properly to avoid miscalculations with parallel components.