# 9. Impedance and Phase Angle

### Don't miss...

Explore impedance, current and voltage in an RLC circuit in the applet later on this page.

## 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.

Answer

In this case, `X_L= 3\ Ω` and `X_C= 0` so `X_L- X_C= 3\ Ω`.

So in rectangular form, the impedance is written:

`Z = 5 + 3j\ Ω`

Using calculator, the **magnitude** of *Z* is given
by: `5.83`, and the angle `θ` (the phase difference) is given by:
`30.96^@`.

So the voltage leads the current by `30.96^@`, as shown in the diagram.

Presenting *Z* as a complex number (in polar form), we
have:

`Z = 5.83 ∠ 30.96^@\ Ω`.

### 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.

Answer

In this case, we have: `X_L- X_C= 8 - 11 = -3\ Ω`

So `Z = 4 - 3j\ Ω` in rectangular form.

Now to express it in polar form:

Using calculator, we find `r = 5` and ` θ = -36.87^@`.

[NOTE: We usually express the phase angle (when voltage lags
the current) using a **negative value**, rather than the
equivalent positive value `323.13^@`.]

So `Z = 5 ∠ -36.87^@\ Ω`

## 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. You can:

Drag the**top slider**left or right to vary the impedance due to the resistor, `R`,

Drag the**X**up or down to vary the impedance due to the inductor, `X_L`, and_{L}slider

Drag the**X**up or down to vary the impedance due to the capacitor, `X_C`._{C}slider - 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 red "result" line makes with the horizontal (in radians).
- Consider the graphs of voltage and current in 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})

Answer

i) | *V*_{R} | = | *IR* | = 10 × 4 = 40 V

ii) | *V*_{L} | = | *IX*_{L} | = 10 × 8 = 80 V

iii) | *V*_{C} | = | *IX*_{C} | = 10 × 11 = 110 V

iv) | *V*_{RLC} | = | *IZ* | = 10 × 5 = 50 V