In the circuit shown, V_AB = 48.3<30° V. Find V_in.

Steps to Solve

1. Circuit Analysis

Identify the components of the circuit. The circuit includes a voltage source $V_{in}$, a 4Ω resistor, a 5Ω resistor, and a load with impedance $Z_L = 38.66Ω$. The voltage $V_{AB}$ across points A and B is given as $48.3∠30° V$.

2. Using Ohm's Law

The voltage across the load $Z_L$ can be expressed in terms of the current $I$ flowing through it:

$$ V_{AB} = I \cdot Z_L $$

Where $Z_L = 38.66Ω$.

3. Calculate the Current

From the above equation, we can rearrange to find the current $I$:

$$ I = \frac{V_{AB}}{Z_L} $$

Substituting $V_{AB}$ into the equation:

$$ I = \frac{48.3∠30°}{38.66} $$

Calculate this value in polar form.

4. Calculate the Total Impedance

Now find the total impedance of the series circuit formed by the 4Ω and 5Ω resistors:

$$ Z_{total} = 4Ω + 5Ω = 9Ω $$

5. Find the Input Voltage $V_{in}$

Apply Ohm's Law again to find $V_{in}$:

$$ V_{in} = I \cdot Z_{total} $$

Substitute $I$ and the total impedance to find $V_{in}$.

6. Determine the Range for $V_{in}$

Calculate the expected range of $V_{in}$ based on the values obtained, and compare it with the choices given (a-e).