9. Impedance and Phase Angle

Impedance

The impedance of a circuit is the total effective resistance to the flow of current by a combination of the elements of the circuit.

Symbol: Z

Units: `Ω`

The total voltage across all 3 elements (resistors, capacitors and inductors) is written

VRLC

To find this total voltage, we cannot just add the voltages VR, VL and VC.

Because VL and VC are considered to be imaginary quantities, we have:

Impedance VRLC = IZ

So `Z = R + j(X_L− X_C)`

Now, the magnitude (size, or absolute value) of Z is given by:

`|Z|=sqrt(R^2+(X_L-X_C)^2`

Phase angle

`tan\ theta=(X_L-X_C)/R`

Angle θ represents the phase angle between the current and the voltage.

Compare this to the Phase Angle that we met earlier in Graphs of y = a sin(bx + c).

Example 1

A circuit has a resistance of `5\ Ω` in series with a reactance across an inductor of `3\ Ω`. Represent the impedance by a complex number, in polar form.

Resistor and inductor in series

Example 2(a)

A particular ac circuit has a resistor of `4\ Ω`, a reactance across an inductor of `8\ Ω` and a reactance across a capacitor of `11\ Ω`. Express the impedance of the circuit as a complex number in polar form.

Interactive RLC graph

Below is an interactive graph to play with (it's not a static image). You can explore the effect of a resistor, capacitor and inductor on total impedance in an AC circuit.

Activities for this Interactive

  1. First, just play with the sliders. Drag the red dot left or right to vary the impedance due to the resistor, `R`, the blue dot up or down to vary the impedance due to the inductor, `X_L`, and the green dot up or down to vary the impedance due to the capacitor, `X_C`.
  2. Observe the effects of different impedances on the values of XLXC and Z.
  3. Observe the effects of different impedances on θ, the angle the black line makes with the horizontal (in radians).
  4. Consider the graphs of voltage and current on the right of the interactive. Observe the amount of lag or lead as you change the sliders.
  5. What have you learned from playing with this interactive?

Example 2(b)

Referring to Example 2 (a) above, suppose we have a current of 10 A in the circuit. Find the magnitude of the voltage across

i) the resistor (VR)

ii) the inductor (VL)

iii) the capacitor (VC)

iv) the combination (VRLC)

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