impedance

Electrically, impedance is the sum of a DC resistance and an AC reactance. The (DC) resistance is the real portion of the impedance, and the reactance is the imaginary (j) portion. Impedance is measured in the same units as resistance, Ohms.

Suppose one has the following circuit:

             i(t)
       ---------------->

       R      C     L
   +--\/\-----|(----@@@----+        R = 1 kΩ
   |                       |        C = 1 μF
   |  +                    |        L = 1 H
  (~) v(t)                 |
   |  -                    |        v(t) = 10 cos ωt V
   |                       |        ω = 2π*60 Hz = 120π  rad/s
   +-----------------------+

This circuit has a DC resistance contributed solely by the resistor:

R = R = 1kΩ

The capacitor and inductor both contribute to the net reactance:

X_C = 1/(jωC) = 1/j(120π*10^-6) = -j2652 Ω

X_L = jωL = j(120π) = j376 Ω

Reactances can be added when in series, just as resistance, so we obtain:

X_total = X_C+X_L = -j2276 Ω

Now, to obtain the total impedance, Z, of the circuit, add the reactance and the DC resistance, since they are in series:

Z = (1000-j2276) Ω

This can be expressed in exponential notation, and plotted on the polar plane:

Z = |Z|exp(j*arctan(X/R)) = 2485*exp(j(-66°))

         R                    Re           
      +----->------------------->          
      |\)Φ  |  
      | \   | X   
      |  \  |
      | Y \ |       
      |    \v
      |                      Y = |Z|
      |                      Φ = -66°
      |
      |
      |
      v Im

We can now calculate the the current, i(t), through the circuit, using Ohm's Law:

V = IZ => v(t)/Z = i(t)

i(t) = 10/Z cos ωt = (10/2485)*exp(j(66°)) cos ωt = 4.024 cos (ωt + 66°) (mA)

Note that the presence of a reactance introduces a phase shift between the current and the voltage souce.

---

One may note that the inverse of impedance is admittance, measured in the same units as conductance, Siemens (S).

Im*ped"ance (?), n. [Impede + -ance.] (Elec.)

The apparent resistance in an electric circuit to the flow of an alternating current, analogous to the actual electrical resistance to a direct current, being the ratio of electromotive force to the current. It is equal to R² + X², where R = ohmic resistance, X = reactance. For an inductive circuit, X = 2πfL, where f = frequency and L = self-inductance; for a circuit with capacity X = 1 ÷ 2πfC, where C = capacity.

 

© Webster 1913