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12.5 Exponential Form of a Complex Number

Exponential Form $r e^{j θ}$ $r e^{j θ}$ • Expressing Numbers in Exponential Form • Summary of Important Forms

Another important form of a complex number is the exponential form. It is commonly used in electronics, engineering, and physics applications. As we will see in the next section, it is also convenient for multiplication and division of complex numbers, as the rectangular form is for addition and subtraction.

The exponential form of a complex number is written as $r e^{j θ},$ $r e^{j θ},$ where r and $θ$ $θ$ have the same meanings as given in the previous section, although $θ$ $θ$ is expressed in radians. The number e is a special irrational number and has an approximate value

e = 2.7182818284590452

$e = 2.7182818284590452$

This number e is very important in mathematics, and we will see it again in the next chapter. For now, it is necessary to accept the value for e, although in calculus its meaning is shown along with the reason it has the above value. We can find its value on a calculator by using the key, with $x = 1,$ $x = 1,$ or by using the key.

In advanced mathematics, it is shown that

r e^{j θ} = r (\cos θ + j \sin θ)

$r e^{j θ} = r (\cos θ + j \sin θ)$ (12.11)

By expressing $θ$ $θ$ in radians, the expression $j θ$ $j θ$ is an exponent, and it can be shown to obey all the laws of exponents as discussed in Chapter 11.

NOTE

[Therefore, we will always express $θ$ $θ$ in radians when using exponential form.]

EXAMPLE 1 Polar form to exponential form

Express the number $8.50 ∠ \underline{136.3 °}$ $8.50 ∠ \underline{136.3 °}$ in exponential form.

Because this complex number is in polar form, we note that $r = 8.50$ $r = 8.50$ and that we must express $136.3 °$ $136.3 °$ in radians. Changing $136.3 °$ $136.3 °$ to radians, we have

\frac{136.3 π}{180} = 2.38 rad

$\frac{136.3 π}{180} = 2.38 rad$

Therefore, the required exponential form is $8.50 e^{2.38 j} .$ $8.50 e^{2.38 j} .$ This means that

8.50 angle 136.3 degrees = 8.50 e raised to the power of, start expression 2.38 j end expression. where 8.50 is the value of r and 136.3 degree in the L H S is converted to 2.38 radiant in the R H S.

We see that the principal step in changing from polar form to exponential form is to change $θ$ $θ$ from degrees to radians.

Complex numbers are often used to represent vectors, where r is the magnitude and $θ$ $θ$ is the direction. The following example illustrates this.

EXAMPLE 2 Rectangular to exponential form—robotic displacement

The displacement d of a welding point from the end of a robot arm can be expressed as $2.00 - 4.00 j ft .$ $2.00 - 4.00 j ft .$ Express the displacement in exponential form and find its magnitude.

From the rectangular form, we have $x = 2.00$ $x = 2.00$ and $y = - 4.00.$ $y = - 4.00.$ Therefore,

\begin{array}{rcl} d & = & \sqrt{{(2.00)}^{2} + {(- 4.00)}^{2}} = 4.47 ft \\ θ & = & \tan^{- 1} \frac{- 4.00}{2.00} = - 63.4 ° \end{array}

$\begin{array}{rcl} d & = & \sqrt{{(2.00)}^{2} + {(- 4.00)}^{2}} = 4.47 ft \\ θ & = & \tan^{- 1} \frac{- 4.00}{2.00} = - 63.4 ° \end{array}$

It is common to express the direction in terms of negative angles when the imaginary part is negative. Since $63.4 ° = 1.11$ $63.4 ° = 1.11$ rad, the exponential form is $4.47 e^{- 1.11 j} .$ $4.47 e^{- 1.11 j} .$ The modulus of 4.47 means the magnitude of the displacement is 4.47 ft. Figure 12.15 shows the calculator features for directly converting between rectangular and polar forms. The calculator treats exponential and polar forms as being the same since each uses r and $θ .$ $θ .$

A calculator screen with input 2 minus 4 i right arrow polar, and output 4.472 e raised to negative 1.107 i; input answer right arrow rectangle, output 2.000 minus 4.000 i. — Fig. 12.15

EXAMPLE 3 Exponential form to other forms

Express the complex number $2.00 e^{4.80 j}$ $2.00 e^{4.80 j}$ in polar and rectangular forms.

We first express 4.80 rad as $275.0 °$ $275.0 °$ . From the exponential form, we know that $r = 2.00.$ $r = 2.00.$ Thus, the polar form is

2.00 (\cos 275.0 ° + j \sin 275.0 °)

$2.00 (\cos 275.0 ° + j \sin 275.0 °)$

Using the distributive law, we rewrite the polar form and then evaluate. Thus,

\begin{array}{rcl} 2.00 e^{4.80 j} & = & 2.00 (\cos 275.0 ° + j \sin 275.0 °) \\ = & 2.00 \cos 275.0 ° + (2.00 \sin 275.0 °) j \\ = & 0.174 - 1.99 j \end{array}

$\begin{array}{rcl} 2.00 e^{4.80 j} & = & 2.00 (\cos 275.0 ° + j \sin 275.0 °) \\ = & 2.00 \cos 275.0 ° + (2.00 \sin 275.0 °) j \\ = & 0.174 - 1.99 j \end{array}$

We now summarize the three important forms of a complex number. See Fig. 12.16.

A position vector of length r points to x plus y j at angle theta to the positive real axis. Horizontal length is x = r times cosine of theta, and vertical length is y = r times sine of theta. — Fig. 12.16

\begin{array}{l} Rectangular : & x + y j \\ Polar : & r (\cos θ + j \sin θ) = r ∠ \underline{θ} \\ Exponential : & {r e}^{j θ} \end{array}

$\begin{array}{l} Rectangular : & x + y j \\ Polar : & r (\cos θ + j \sin θ) = r ∠ \underline{θ} \\ Exponential : & {r e}^{j θ} \end{array}$

It follows that

x + y j = r (\cos θ + j \sin θ) = r ∠ \underline{θ} = r e^{j θ}

$x + y j = r (\cos θ + j \sin θ) = r ∠ \underline{θ} = r e^{j θ}$ (12.12)

where

r^{2} = x^{2} + y^{2} \tan θ = \frac{y}{x}

$r^{2} = x^{2} + y^{2} \tan θ = \frac{y}{x}$ (12.8)

In Eq. (12.12), the argument $θ$ $θ$ is the same for exponential and polar forms. It is usually expressed in radians in exponential form and in degrees in polar form.

Exercises 12.5

In Exercises 1 and 2, perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.

In Example 1, change $136.3 °$ $136.3 °$ to $226.3 °$ $226.3 °$ and then find the exponential form.
In Example 3, change the exponent to 3.80j and then find the polar and rectangular forms.

In Exercises 3–22, express the given numbers in exponential form.

$3.00 (\cos 60.0 ° + j \sin 60.0 °)$ $3.00 (\cos 60.0 ° + j \sin 60.0 °)$
$575 (\cos 135.0 ° + j \sin 135 °)$ $575 (\cos 135.0 ° + j \sin 135 °)$
$0.450 (\cos 282.3 ° + j \sin 282.3 °)$ $0.450 (\cos 282.3 ° + j \sin 282.3 °)$
$2.10 (\cos 588.7 ° + j \sin 588.7 °)$ $2.10 (\cos 588.7 ° + j \sin 588.7 °)$
$375.5 [\cos (- 95.46 °) + j \sin (- 95.46 °)]$ $375.5 [\cos (- 95.46 °) + j \sin (- 95.46 °)]$
$1672 [\cos (- 7.14 °) + j \sin (- 7.14 °)]$ $1672 [\cos (- 7.14 °) + j \sin (- 7.14 °)]$
$0.515 ∠ \underline{198.3 °}$ $0.515 ∠ \underline{198.3 °}$
$4650 ∠ \underline{326.5 °}$ $4650 ∠ \underline{326.5 °}$
$4.06 ∠ \underline{- 61.4 °}$ $4.06 ∠ \underline{- 61.4 °}$
$0.0192 ∠ \underline{76.7 °}$ $0.0192 ∠ \underline{76.7 °}$
$9245 ∠ \underline{296.32 °}$ $9245 ∠ \underline{296.32 °}$
$82.76 ∠ \underline{470.09 °}$ $82.76 ∠ \underline{470.09 °}$
$3 - 4 j$ $3 - 4 j$
$- 1 - 5 j$ $- 1 - 5 j$
$- 30 + 20 j$ $- 30 + 20 j$
$100 j + 600$ $100 j + 600$
$5.90 + 2.40 j$ $5.90 + 2.40 j$
$47.3 - 10.9 j$ $47.3 - 10.9 j$
$- 634.6 - 528.2 j$ $- 634.6 - 528.2 j$
$5477 j - 8573$ $5477 j - 8573$

In Exercises 23–30, express the given complex numbers in polar and rectangular forms.

$3.00 e^{0.500 j}$ $3.00 e^{0.500 j}$
$20.0 e^{1.00 j}$ $20.0 e^{1.00 j}$
$464 e^{1.85 j}$ $464 e^{1.85 j}$
$2.50 e^{3.84 j}$ $2.50 e^{3.84 j}$
$3.20 e^{- 5.41 j}$ $3.20 e^{- 5.41 j}$
$0.800 e^{3.00 j}$ $0.800 e^{3.00 j}$
$1724 e^{2.391 j}$ $1724 e^{2.391 j}$
$820.7 e^{- 3.492 j}$ $820.7 e^{- 3.492 j}$

In Exercises 31–34, perform the indicated operations by using properties of exponents and express results in rectangular and polar forms.

${(4.55 e^{1.32 j})}^{2}$ ${(4.55 e^{1.32 j})}^{2}$
${(0.926 e^{0.253 j})}^{3}$ ${(0.926 e^{0.253 j})}^{3}$
$(625 e^{3.46 j}) (4.40 e^{1.22 j})$ $(625 e^{3.46 j}) (4.40 e^{1.22 j})$
$(18.0 e^{5.13 j}) (25.5 e^{0.77 j})$ $(18.0 e^{5.13 j}) (25.5 e^{0.77 j})$

In Exercises 35–40, perform the indicated operations.

Using a calculator, express $3.73 + 5.24 j$ $3.73 + 5.24 j$ in exponential form. See Fig. 12.15.
Using a calculator, express $75.6 e^{1.25 j}$ $75.6 e^{1.25 j}$ in rectangular form. See Fig. 12.15.
The impedance in an antenna circuit is $3.75 + 1.10 j$ $3.75 + 1.10 j$ ohms. Write this in exponential form and find the magnitude of the impedance.
The intensity of the signal from a radar microwave signal is $37.0 [\cos (- 65.3 °) + j \sin (- 65.3 °)] V / m .$ $37.0 [\cos (- 65.3 °) + j \sin (- 65.3 °)] V / m .$ Write this in exponential form.
In an electric circuit, the admittance is the reciprocal of the impedance. If the impedance is $2800 - 1450 j$ $2800 - 1450 j$ ohms in a certain circuit, find the exponential form of the admittance.
If $E = 115 e^{0.315 j} V$ $E = 115 e^{0.315 j} V$ and $I = 28.6 e^{- 0.723 j} A,$ $I = 28.6 e^{- 0.723 j} A,$ find the exponential form of Z given that $E = I Z .$ $E = I Z .$

Answers to Practice Exercises

$25.0 e^{2.22 j}$ $25.0 e^{2.22 j}$
$26.5 e^{3.83 j}$ $26.5 e^{3.83 j}$
$- 2.66 + 1.39 j$ $- 2.66 + 1.39 j$

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Table of Contents for 12.5 Exponential Form of a Complex Number

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Table of Contents for
12.5 Exponential Form of a Complex Number