Simple Difference of Impedance and Admittance

impedance and admittance

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.

The difference between these two is, impedance has magnitude and phase, while resistance only has magnitude.

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

impedance admittance
(1)

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

impedance admittance
(2)

From these three expressions, we obtain Ohm’s law in phasor form for any type of element as

impedance admittance
(3)

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

The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, 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.

impedance and admittance
Table.1

Looking from the table, we notice that ZL = jωL and ZC = -j/ωC. Consider two extreme cases of angular frequency.

When ω = 0 (i.e., for dc sources), ZL = 0 and ZC → ∞, 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), ZL → ∞ and ZC = 0, showing that the inductor is an open circuit, while the capacitor is a short circuit as can be seen in Figure.(1).

impedance admittance
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

impedance admittance
(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

impedance admittance
(5)

Comparing Equations.(4) and (5), we conclude that

impedance admittance
(6)

where

impedance admittance
(7)

and

impedance admittance
(8)

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

The admittance Y 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

impedance admittance
(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

impedance admittance
(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),

impedance admittance
(11)

By rationalization,

impedance admittance
(12)

Equating the real and imaginary parts give

impedance admittance
(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:

  1. Equivalent impedance
  2. What is phasor
  3. Kirchhoff’s laws for ac circuit
  4. Power calculation in ac circuit
  5. Three phase ac circuit

And its applications:

  1. Phase shifter circuit and formula
  2. AC bridge
  3. AC op-amp
  4. Capacitance multiplier circuit
  5. Wien bridge oscillator

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)

impedance admittance
Figure 2

Solution :

From the voltage source 10 cos 4t, ω = 4,
impedance admittance

The impedance is

impedance admittance

Thus the current

impedance admittance
(1.1)

The voltage across the capacitor is

impedance admittance
(1.2)

Converting I and V in Equations.(1.1) and (1.2) to the time domain results

impedance admittance

Notice that i(t) leads v(t) by 90o as expected.