Question

An ac voltage $V = 280 \sin(100\pi t)$ volt is connected across a series LCR circuit in which $R = 400 \Omega$, $L = 5/\pi \text{ H}$ and $C = 50/\pi \mu\text{F}$. Taking $\sqrt{2} = 1.4$, calculate impedance of the circuit.

Hint:

Continuously watch actively for familiar Pythagorean triplets like 3-4-5, 5-12-13, or 8-15-17 during complex impedance calculations to cleanly and rapidly bypass tedious arithmetic.
Always meticulously convert microfarads ($\mu\text{F}$) entirely into standard farads by aggressively applying the critical $10^{-6}$ multiplier before inserting them deeply into denominator fractions.

Answer 1

Concept:
• A classic series LCR circuit possesses a unified total impedance constructed physically from three key components: ohmic resistance, inductive reactance, and capacitive reactance.

• The driving AC voltage elegantly follows a standard sinusoidal mathematical waveform, where the coefficient of time $t$ uniquely represents the angular frequency.

• These distinct reactances strictly operate entirely out of phase with the standard resistance, necessitating a careful vector-like Pythagorean addition to solidly find total impedance.

Step 1: Extract Fundamental AC Parameters
From the rigorously standard mathematical voltage form $V = V_m \sin(\omega t)$, we intelligently compare it directly with the heavily given equation $V = 280 \sin(100\pi t)$.
This comparison decisively reveals two critical physical parameters:
The absolute peak driving voltage is $V_m = 280 \text{ V}$.
The operating angular frequency is rigidly $\omega = 100\pi \text{ rad/s}$.

Step 2: Calculate Individual Component Reactances
We logically compute the explicit inductive reactance $X_L$ exactly using its defining formula:
\[ X_L = \omega L = (100\pi) \times \left(\frac{5}{\pi}\right) \]
The $\pi$ cleanly cancels out, yielding:
\[ X_L = 500 \Omega \]
Next, we strictly compute the explicit capacitive reactance $X_C$, carefully incorporating the crucial microfarad ($10^{-6}$) conversion:
\[ X_C = \frac{1}{\omega C} = \frac{1}{(100\pi) \times \left(\frac{50}{\pi} \times 10^{-6}\right)} \]
The $\pi$ again cleanly cancels out from the denominator:
\[ X_C = \frac{1}{5000 \times 10^{-6}} = \frac{10^6}{5000} \]
Simplify the fraction completely:
\[ X_C = \frac{1000}{5} = 200 \Omega \]

Step 3: Calculate the Total Circuit Impedance
The overarching comprehensive impedance $Z$ for a fully functioning series LCR circuit is mathematically formalized rigorously as:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
We systematically substitute the confirmed physical resistance and calculated reactance values directly into the square root:
\[ Z = \sqrt{400^2 + (500 - 200)^2} \]
Perform the strict internal subtraction first:
\[ Z = \sqrt{400^2 + 300^2} \]
Recognizing the ubiquitous 3-4-5 mathematical Pythagorean triplet drastically and beautifully simplifies the heavy arithmetic square root calculation without pain:
\[ Z = \sqrt{160000 + 90000} = \sqrt{250000} \]
\[ Z = 500 \Omega \]
The total effective impedance strictly opposing current flow in this complex circuit is 500 Ohms.