Question

What is the formula for Z in a LR circuit?

Answer 1

Are you asking for the formula for impedance (Z) in a series LR circuit or a parallel LR circuit?

Answer 2

In an LR circuit, which consists of an inductor (L) and a resistor (R) in series, the impedance \( Z \) is a measure of how much the circuit resists the flow of alternating current (AC). The impedance combines the effects of both the resistance and the inductive reactance.

To find the impedance \( Z \) of an LR circuit, you can use the following formula:

\[ Z = \sqrt{R^2 + (X_L)^2} \]

where:
- \( R \) is the resistance of the resistor (in ohms, \(\Omega\)).
- \( X_L \) is the inductive reactance of the inductor (in ohms, \(\Omega\)).

The inductive reactance \( X_L \) is given by:

\[ X_L = \omega L \]

where:
- \( \omega \) is the angular frequency of the AC source, defined as \( \omega = 2\pi f \), with \( f \) being the frequency of the AC source (in hertz, Hz).
- \( L \) is the inductance of the inductor (in henrys, H).

So, the complete formula for impedance \( Z \) in an LR circuit is:

\[ Z = \sqrt{R^2 + (\omega L)^2} \]

### Example Calculation

Let's say you have an LR circuit with:
- Resistance \( R = 10 \, \Omega \)
- Inductance \( L = 0.1 \, \text{H} \)
- Frequency \( f = 50 \, \text{Hz} \)

First, calculate the angular frequency \( \omega \):

\[ \omega = 2\pi f = 2\pi \times 50 \approx 314.16 \, \text{rad/s} \]

Then, calculate the inductive reactance \( X_L \):

\[ X_L = \omega L = 314.16 \times 0.1 = 31.416 \, \Omega \]

Now, use the impedance formula:

\[ Z = \sqrt{R^2 + (X_L)^2} \]
\[ Z = \sqrt{10^2 + 31.416^2} \]
\[ Z = \sqrt{100 + 987.59} \]
\[ Z = \sqrt{1087.59} \approx 32.96 \, \Omega \]

Thus, the impedance of the LR circuit is approximately \( 32.96 \, \Omega \).