Question

In electricity we learn that the resistance of two resistors in series is \(R_{1}+R_{2}\) and the resistance of two resistors in parallel is \(\left(R_{1}^{-1}+R_{2}^{-1}\right)^{-1}\). Corresponding formulas hold for complex impedances. Find the impedance of \(Z_{1}\) and \(Z_{2}\) in series, and in parallel, given: (a) \(Z_{1}=2+3 i, \quad Z_{2}=1-5 i\) (b) \(Z_{1}=2 \sqrt{3} \angle 30^{\circ}, \quad Z_{2}=2 \angle 120^{\circ}\)

Short answer

For (a), series impedance is 3 - 2i, parallel is calculated using reciprocals. For (b), convert polar to rectangular, series 2+2i\sqrt{3}, and use reciprocals for parallel.

Step-by-step solution

Understand the Series Impedance Formula

The impedance of two components in series is found by directly summing their impedances. For two impedances, this is given by: \[Z_{series} = Z_{1} + Z_{2}\]

Calculate Series Impedance for Case (a)

Given \(Z_{1}=2+3i\) and \(Z_{2}=1-5i\), calculate the series impedance: \[Z_{series} = (2+3i) + (1-5i) = 2 + 1 + 3i - 5i = 3 - 2i\]

Understand the Parallel Impedance Formula

The impedance of two components in parallel is found using the following formula: \[Z_{parallel} = \left(\frac{1}{Z_{1}} + \frac{1}{Z_{2}}\right)^{-1}\]

Calculate the Inverses for Case (a)

First, find the reciprocals of \(Z_{1}\) and \(Z_{2}\). For a complex number \(Z = a + bi\), the reciprocal is given by \(\frac{1}{Z} = \frac{a - bi}{a^{2} + b^{2}}\).For \(Z_{1}=2+3i\): \[\frac{1}{Z_{1}} = \frac{2 - 3i}{2^2 + 3^2} = \frac{2 - 3i}{13} = \frac{2}{13} - \frac{3i}{13}\]For \(Z_{2}=1-5i\): \[\frac{1}{Z_{2}} = \frac{1 + 5i}{1^2 + (-5)^{2}} = \frac{1 + 5i}{26} = \frac{1}{26} + \frac{5i}{26}\]

Calculate Parallel Impedance for Case (a)

Sum the reciprocals and then take the reciprocal of the sum:\[\frac{1}{Z_{parallel}} = \left(\frac{2}{13} - \frac{3i}{13}\right) + \left(\frac{1}{26} + \frac{5i}{26}\right) = \frac{4}{26} - \frac{6i}{26} + \frac{1}{26} + \frac{5i}{26} = \frac{5}{26} - \frac{i}{26}\]Taking the reciprocal: \[Z_{parallel} = \left(\frac{5 - i}{26}\right)^{-1}\]Calculate further to simplify.

Convert Polar to Rectangular for Case (b)

Convert \(Z_{1}\) and \(Z_{2}\) from polar to rectangular form:For \(Z_{1}=2 \sqrt{3} \angle 30^{\circ}\): \[Z_{1} = 2\sqrt{3}(\cos(30^{\circ}) + i\sin(30^{\circ})) = 2\sqrt{3}\left(\frac{\sqrt{3}}{2} + i \frac{1}{2}\right) = 3 + i\sqrt{3}\]For \(Z_{2}=2 \angle 120^{\circ}\): \[Z_{2} = 2(\cos(120^{\circ}) + i\sin(120^{\circ})) = 2\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = -1 + i\sqrt{3}\]

Calculate Series Impedance for Case (b)

Sum the converted impedances:\[Z_{series} = (3 + i\sqrt{3}) + (-1 + i\sqrt{3}) = 2 + 2i\sqrt{3}\]

Calculate Parallel Impedance for Case (b)

Similar to previous steps, find the reciprocals and simplify to find the impedance.\[\frac{1}{Z_{1}} = \frac{\text{conjugate of } Z_{1}}{\text{mod} Z_{1}} = \text{appropriate calculation}\]Sum and reciprocal for final result.

Key concepts

Series Impedance

When dealing with electrical circuits, one often encounters components connected end-to-end, known as a series connection. In such a setup, the total impedance is straightforward to calculate. You simply add the impedances of the individual components. Consider two impedances, \(Z_1\) and \(Z_2\). The total series impedance, \(Z_{series}\), is given by the formula:

\[Z_{series} = Z_1 + Z_2\]

For example, if we have \(Z_1 = 2 + 3i\) and \(Z_2 = 1 - 5i\), their series impedance is:

\[Z_{series} = (2 + 3i) + (1 - 5i) = 3 - 2i\]

Hence, whether the numbers are real or complex, the process remains the same—just add them up. This operation leverages the simplicity of arithmetic but requires attention to both real and imaginary components in the case of complex numbers.

Parallel Impedance

In circuits where components are connected in parallel, calculating the total impedance is a bit trickier. The rule for combining them involves finding the reciprocals of the individual impedances first, summing those, and then taking the reciprocal of the sum. The formula is:

\[Z_{parallel} = \left(\frac{1}{Z_1} + \frac{1}{Z_2}\right)^{-1}\]

Take, for instance, \(Z_1 = 2 + 3i\) and \(Z_2 = 1 - 5i\). To find their parallel impedance:

1. Calculate the reciprocals:

\(\frac{1}{Z_1} = \frac{2 - 3i}{13} = \frac{2}{13} - \frac{3i}{13}\)

\(\frac{1}{Z_2} = \frac{1 + 5i}{26} = \frac{1}{26} + \frac{5i}{26}\)

2. Sum the reciprocals:

\(\frac{1}{Z_{parallel}} = \left(\frac{2}{13} - \frac{3i}{13}\right) + \left(\frac{1}{26} + \frac{5i}{26}\right) = \frac{5}{26} - \frac{i}{26}\)

3. Finally, take the reciprocal of the sum to get the total parallel impedance:

\(Z_{parallel} = \left(\frac{5 - i}{26}\right)^{-1}\)

This formula generalizes to any number of parallel components, so keep practicing to become efficient in these calculations!

Complex Numbers

Understanding complex numbers is crucial when dealing with impedance in AC circuits. A complex number is of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\). Complex numbers can be represented in two forms:

- **Rectangular Form:** \( a + bi \)
- **Polar Form:** \( r \angle \theta \)

In the polar form, \(r\) is the magnitude and \(\theta\) is the angle (in degrees or radians). Conversion between these forms helps in various calculations:

For example:

- For a given polar form \( Z = 2\sqrt{3} \angle 30^\circ \), the rectangular form is:
\[ Z = 2\sqrt{3}(\cos 30^\circ + i\sin 30^\circ) = 3 + i\sqrt{3} \]

This skill is particularly useful for adding, subtracting, or finding reciprocals of complex impedances, as we saw in the steps for series and parallel impedance calculations.