Impedance and admittance sound strange for people who do not study electric circuits in advanced level.

Impedance is often used in ac electric circuit analysis as well as resistance in dc electric circuit.

*Make sure to read what is ac circuit first.*

## Impedance and Admittance

From the previous post about sinusoidal and phasor, we knew the voltage-current relations for three passive elements of R, L, and C as

(1) |

These equations may be written in terms of the ratio of the phasor voltage to the phasor current as

(3) |

where **Z** is a frequency-dependent quantity known as *impedance*, measured in Ω.

The

impedance Zof a circuit is the ratio of the phasor voltageVto the phasor currentI, measured in ohms (Ω)

The impedance shows the opposition that the circuit exhibits to the flow of sinusoidal current.

Although the impedance is the ratio of two phasors, it is not a phasor, because it does not correspond to a sinusoidally varying quantity.

The impedance of resistors, inductors, and capacitors can be readily obtained from Equation.(2). Table.1 gives a summary of their impedances.

Table.1 |

Looking from the table, we notice that **Z**_{L} = *jωL* and **Z**_{C} = *-j/ωC*. Consider two extreme cases of angular frequency.

When *ω* = 0 (i.e., for dc sources), **Z**_{L} = 0 and **Z**_{C} → ∞, confirming what we already know – that the inductor acts like a short circuit, while the capacitor acts like an open circuit.

When *ω*→ ∞ (i.e., for high frequencies), **Z**_{L} → ∞ and **Z**_{C} = 0, showing that the inductor is an open circuit, while the capacitor is a short circuit as can be seen in Figure.(1).

Figure 1. Equivalent circuits at dc and high frequencies: (a) inductor, (b) capacitor |

As a complex quantity, the impedance may be written in rectangular form as

(4) |

where *R* = Re **Z** is the **resistance** and *X* = Im **Z** is the *reactance*. The reactance X may be negative or positive.

We may say that the impedance is inductive when *X* is positive or capacitive when *X* is negative.

Hence, impedance **Z** = *R + jX* is said to be *inductive* or lagging since current lags voltage, while impedance **Z** = *R – jX* is *capacitive* or leading since current leads voltage.

All three impedance, resistance, and reactance are measured in ohms. The impedance may also be expressed in polar form as

(8) |

It is sometimes convenient to work with the reciprocal of impedance, known as *admittance*.

The

admittanceY is the reciprocal of impedance, measured in siemens (S).

The admittance Y of an element (or a circuit) is the ratio of the phasor current through it to the phasor voltage across it, or

(9) |

The admittance of resistors, inductors, and capacitors can be obtained from Equation.(2). They also summarized in Table.(1).

As a complex quantity, we may write Y as

(10) |

where *G* = Re **Y** is called the *conductance* and *B* = Im **Y** is called the *susceptance*. Admittance, conductance, and susceptance are all expressed in the unit of siemens (or mhos). From Equations.(4) and (10),

(13) |

showing that *G* ≠ 1/*R* as it is in resistive circuits. Of course, if *X* = 0, then *G* = 1/*R*.

We will move on to the:

- Equivalent impedance
- What is phasor
- Kirchhoff’s laws for ac circuit
- Power calculation in ac circuit
- Three phase ac circuit

And its applications:

## Impedance and Admittance Example

For better understanding let us review the example below :

**1. Find v(t) and i(t) in the circuit in Figure.(2)**

*t*,

*ω*= 4,

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