# Why do capacitors and inductors have imaginary impedance?

• Hybird
In summary, the impedance of capacitors and inductors are imaginary numbers because the voltage and current in these devices are 90 degrees out of phase. The phrase "ELI the ICE man" can help remember which is which. In the case of an inductor, its impedance is represented by the complex number j in polar form, which is equal to j times the inductance multiplied by the angular frequency.
Hybird
Consider an AC source with series elements (resistor,inductor, capacitor). In order to determine information from the circuit it is useful to consider impedance. So here's the question, total impedance is a complex number, and that is because the impedance of the cap and inductor are imaginary numbers. I'm tryin to find out why they are imaginary.

Any help?

Hybird said:
Consider an AC source with series elements (resistor,inductor, capacitor). In order to determine information from the circuit it is useful to consider impedance. So here's the question, total impedance is a complex number, and that is because the impedance of the cap and inductor are imaginary numbers. I'm tryin to find out why they are imaginary.

Any help?

In a complex number representation, sinusoidal functions are written as complex exponentials. The imaginary numbers for the impedence of capacitors and inductors indicates that the voltage and current in these devices is 90 degrees out of phase. Voltage leads the current in an inductor and lags the current in a capicitor. The catchy little phrase "ELI the ICE man" can help you remember which is which.

Last edited:
Well let's just talk about the impedance of an induction. The impedance is defined as followed

$$Z_L \equiv \frac {\mathbf{V}_L}{\mathbf{I}_L}$$

where $V_L$ and $I_L$ are phasors. Consider $v_s(t) = A \cos (\omega t)$, then $\mathbf{V}_L = A \angle 0$. The current through an inductor is given by

$$i_L(t) = \frac {1}{L} \int_{t_0}^t v_L ~dt + I_0$$

If you work that integral out will you get $i_L(t) = \frac {A}{\omega L} \sin(\omega t) = \frac {A}{\omega L} \cos(\omega t - \frac {\pi}{2})$. So $I_L = \frac {A}{\omega L} \angle -\frac {\pi}{2}$

Then

$$Z_L \equiv \frac {\mathbf{V}_L}{\mathbf{I}_L} = \frac {A \angle 0}{\frac {A}{\omega L} \angle - \frac {\pi}{2}} = \omega L \angle \frac {\pi}{2} = j \omega L$$

The complex number $j$ appears in the impedance of an inductor when converting from polar form to rectangular form. Hope that answered your question.

## What is an AC circuit?

An AC circuit is a type of electrical circuit that uses alternating current (AC) to power devices. This means that the direction of the electrical current changes periodically, which is different from a direct current (DC) circuit where the current flows in one direction.

## What is the difference between AC and DC circuits?

The main difference between AC and DC circuits is the direction of the current. In AC circuits, the current changes direction periodically, while in DC circuits, the current flows in one direction. Additionally, AC circuits use transformers to change the voltage level, while DC circuits use converters.

## How is current measured in an AC circuit?

In an AC circuit, current is measured using an ammeter. This device measures the flow of electrons through a conductor and is typically connected in series with the circuit. The unit for measuring current is amperes (A).

## What factors can affect the current in an AC circuit?

The current in an AC circuit can be affected by several factors, including the voltage of the power source, the resistance of the circuit, and the frequency of the alternating current. Changes in any of these factors can result in a change in the current flowing through the circuit.

## How does current change in an AC circuit?

In an AC circuit, the current changes direction periodically, meaning it alternates between positive and negative values. This change in direction is represented by a sinusoidal wave, with the peak value representing the maximum current and the zero crossing representing the current's direction change.

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