# 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

V_{RLC}

To find this total voltage, we cannot just **add** the
voltages *V*_{R},
*V*_{L} and *V*_{C}.

Because *V*_{L} and
*V*_{C} are considered to be imaginary
quantities, we have:

Impedance

V_{RLC}=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.

### 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

- 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`. - Observe the effects of different impedances on the values of
*X*_{L}−*X*_{C}and*Z*. - Observe the effects of different impedances on θ, the angle the black line makes with the horizontal (in radians).
- 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. - What have you learned from playing with this interactive?

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### 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 (

V_{R})ii) the inductor (

V_{L})iii) the capacitor (

V_{C})iv) the combination (

V_{RLC})

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