# 11. AC Circuit Exercises

by M. Bourne

The following exercises make use of what you learned in Definitions and Impedance and Phase Angle, as well as the Complex Number Basic Operations and Products and Quotients sections.

## Exercise 1

Given that the current in a given
circuit is 3.90 - 6.04*j* mA
and the impedance is 5.16 +
1.14*j* kΩ, find
the magnitude of the voltage.

Answer

Note the **units** in the question: `"mA"` and `"k"Omega`.

Applying the formula for voltage:

`|V|=|I|xx|Z|`

`=|(3.90-6.04j)xx10^-3|` `xx|(5.16+1.14j)xx10^3|`

We can move the powers of 10 outside the absolute value (magnitude) signs.

`=|(3.90-6.04j)|xx10^-3` `xx|(5.16+1.14j)|xx10^3`

The powers of 10 cancel out.

`=|3.90-6.04j|xx|5.16+1.14j|`

We now proceed to find the magnitude of each complex number. (See Impedance & Phase Angle for background.)

`=sqrt((3.90)^2+(-6.04)^2)` `xxsqrt((5.16)^2+(1.14)^2)`

`=7.190xx5.284`

`=38.0\ text(volts)`

So the voltage is `38\ "V"`.

## Exercise 2

A resistor, an inductor and a capacitor are connected in series across an ac voltage source. A voltmeter measures `12.0\ "V"`, `15.5\ "V"` and `10.5\ "V"` respectively, when placed across each element separately. What is the magnitude of the voltage of the source?

Answer

The total reactance across a circuit with a resistance, an inductance and a capacitance in series is given by:

V_{RLC}=V_{R}+j(V_{L}−V_{C}) (from the definitions)

Now

`|V_R| = 12\ "V"`

`|V_L| = 15.5\ "V"`

`|V_C| = 10.5\ "V"`

Now

*V*_{RLC} =* V*_{R} +* j*(*V*_{L} − *V*_{C})

= 12 +

j(15.5 − 10.5)= 12 + 5

jV

The **magnitude** of this voltage is:

|12 + 5*j*| = 13 V (from calculator)

Easy to understand math videos:

MathTutorDVD.com

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Algebra Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!