Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• Convert polar to rectangular using calculator
• 5. Exponential Form
• 6. Products and Quotients
• 7. Powers and Roots
• 8. AC Circuit Definitions
• 9. Impedance and Phase Angle
• 10. Reactance and Angular Velocity
• 11. AC Circuit Exercises
• 12. Parallel AC Circuits
• Vocabulary
• Fractals
• Complex Numbers Problem Solver

• Comments, Questions?

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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.

Answer

Example 2(a)

A particular ac circuit has a resistor of 4Ω, 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

Flash Interactive

Here is a Flash example to play with. Play with the sliders to see the effect of each component. Also, you can enter your own values.

Activities for this Flash Interactive

1. First, just play with the sliders and observe the effects on the values of X_L - X_C and Z. Also, consider the graphs of voltage and current in the bottom right corner.
2. Set each of the sliders to 0. Why does the graph of current disappear?
3. On paper, calculate Z and θ if R = 60 Ω, X_L = 50 Ω and X_C = 90 Ω.
4. Now click where it says "Own values" and enter the values that you found in Activity 3 above (R = 60 Ω, X_L = 50 Ω and X_C = 90 Ω) to check your answer. Observe the amount of lag or lead in the bottom right corner.
5. What have you learned from playing with this Flash movie?

Click where it says "Start"

Example 2(b)

In 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