Express The Impedance Of A 52 Mh Inductor

Express The Impedance Of A 52mh Induc

Express the impedance of a 5.2 mH inductor at 60 Hz in polar form. Determine the impedance of a capacitor with a magnitude of 240 Ω at a frequency of 1.8 kHz. Find the equivalent impedance (Z_eq) in various circuit cases, including resistors, capacitors, and inductors at specified frequencies. Calculate the current I(jω) in these circuits. Find the total impedance Z, and analyze phase relationships between voltage and current. Examine the circuit with voltage and current phasors to determine the magnitude of impedance, resistor value, reactive component, and phase angles. Compute the AC current source I_in in polar form, and analyze other circuit impedances and phase relationships for various combinations of resistors, capacitors, and inductors as presented. Provide all results with proper calculations and explanations necessary for understanding in an academic context.

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The task of expressing the impedance of a 5.2 millihenry (mH) inductor at 60 Hz in polar form is fundamental in AC circuit analysis. Impedance (Z) for an inductor is calculated using the formula Z = jωL, where ω = 2πf. At 60 Hz, ω = 2π × 60 ≈ 377 rad/sec. Therefore, Z = j(377 × 0.0052) Ω ≈ j1.96 Ω. Converting to polar form, the magnitude is approximately 1.96 Ω, and the phase angle is +90°, resulting in Z ≈ 1.96∠90° Ω. This indicates that the inductor's impedance is purely reactive and leads the current by 90°.

Similarly, the impedance of a capacitor can be found knowing its magnitude at a specific frequency. For a capacitor with impedance magnitude 240 Ω at 1.8 kHz, we use Z_C = 1 / (jωC). Rearranged, C = 1 / (ω|Z|). With ω = 2π × 1800 ≈ 11,309 rad/sec, the capacitor value is C ≈ 1 / (11,309 × 240) ≈ 3.68 × 10^-6 F, or approximately 0.368 μF.

When analyzing circuits, the total impedance (Z_eq) depends on whether resistors, capacitors, or inductors are connected in series or parallel at specified frequencies. For example, in a series R-L circuit with R = 0.330 Ω and L = 100 mH at 1 kHz, the inductor's impedance Z_L = jωL = j(2π × 1000 × 0.1) ≈ j628.3 Ω. The total impedance becomes Z_total = R + Z_L ≈ 0.33 + j628.3 Ω, which in polar form is approximately 628.7∠89.97°, emphasizing the dominant inductive reactance.

The current I(jω) in such a circuit is obtained by dividing the applied voltage by the total impedance, using I = V / Z. For an applied voltage V = 10∠0° V, the current would be I ≈ 10∠0° / 628.7∠89.97° ≈ 0.0159∠-89.97°, or approximately 15.9 mA lagging by nearly 90°, consistent with the mostly reactive impedance.

In phase relationship analysis, the load impedance magnitude and phase angle influence whether the current leads or lags the voltage. For instance, a circuit with resistance R and inductance L leads to a phase angle θ = arctan(X_L/R). If R is small relative to X_L, the current lags significantly behind voltage, confirming the inductive nature of the load.

When considering complex impedance in parallel or series, such as in the phasor diagram with voltage V_s = 60∠0° V and current I in the circuit, the impedance magnitude can be computed from the ratio V / I. If, for example, I = 3.4∠-30° A, then |Z| = 60 V / 3.4 A ≈ 17.6 Ω, and the phase angle of Z is 30°, indicating the load mainly behaves as an inductor with a lagging current.

Calculations involving voltage, current, and impedance in combined networks of resistors, inductors, and capacitors demand careful attention to phase relationships. For instance, in a simple series circuit with a resistor R = 180 Ω and a reactive component, the net impedance Z can be found by vector addition, and the phase angle derived from the tangent of the reactance over resistance.

In more advanced circuit analysis, the use of AC current sources and impedance models such as Z₁ and Z₂ are common. Given Z₁ = 120∠90° Ω and Z₂ = 75∠30° Ω, the total impedance can be computed by vector addition. The resultant impedance influences the AC source current I_in, which can be expressed in polar form and indicates whether the source leads or lags the applied voltage, depending on the phase relationships established in the circuit.

Finally, detailed calculations suggest that the total impedance Z from combined circuits can have a magnitude and phase that dictate the power factor and energy transfer efficiency. For example, Z = 100∠53.4° Ω reflects a circuit with both resistive and reactive components in a certain phase relationship, guiding the solution to the phase angle and real/reactive power distribution.

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