The Total Resistance $Z_T$ In A Circuit Is Given By $Z_T=\frac{Z_1 Z_2}{Z_1+Z_2}$, Where \$Z_1=50+60i$[/tex\] And $Z_2=40-25i$. What Is The Total Resistance In Standard Form? Round To Three Decimal

Introduction

In electrical engineering, understanding the total resistance in a circuit is crucial for designing and analyzing complex electrical systems. The total resistance is a measure of the opposition to the flow of electric current in a circuit. In this article, we will explore the concept of total resistance and derive a formula to calculate it. We will then apply this formula to a specific circuit with complex impedances and determine the total resistance in standard form.

The Formula for Total Resistance

The total resistance in a circuit is given by the formula:

$$Z_T=\frac{Z_1 Z_2}{Z_1+Z_2}$$

where $Z_1$ and $Z_2$ are the individual impedances in the circuit.

Complex Impedances

In electrical engineering, impedances are complex quantities that represent the opposition to the flow of electric current in a circuit. They are typically represented in the form:

$$Z=a+bi$$

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

In this article, we will consider two complex impedances:

$$Z_1=50+60i$$

$$Z_2=40-25i$$

Calculating the Total Resistance

To calculate the total resistance, we need to substitute the values of $Z_1$ and $Z_2$ into the formula:

$$Z_T=\frac{(50+60i)(40-25i)}{(50+60i)+(40-25i)}$$

To simplify this expression, we can use the distributive property of multiplication over addition:

$$Z_T=\frac{2000-1250i+2400i-1500i^2}{90+35i}$$

Since $i^2=-1$, we can substitute this value into the expression:

$$Z_T=\frac{2000-1250i+2400i+1500}{90+35i}$$

$$Z_T=\frac{3500+150i}{90+35i}$$

To simplify this expression further, we can multiply the numerator and denominator by the conjugate of the denominator:

$$Z_T=\frac{(3500+150i)(90-35i)}{(90+35i)(90-35i)}$$

Expanding the numerator and denominator, we get:

$$Z_T=\frac{315000-122500i+13500i-5250i^2}{8100-1225i^2}$$

Substituting $i^2=-1$ into the expression, we get:

$$Z_T=\frac{315000-122500i+13500i+5250}{8100+1225}$$

$$Z_T=\frac{320250-109000i}{9325}$$

Simplifying this expression, we get:

$$Z_T=34.38-11.65i$$

Conclusion

In this article, we derived a formula for calculating the total resistance in a circuit with complex impedances. We then applied this formula to a specific circuit with complex impedances and determined the total resistance in standard form. The total resistance was found to be $34.38-11.65i$.

Discussion

The total resistance in a circuit is a critical parameter in electrical engineering. It determines the opposition to the flow of electric current in a circuit and affects the overall performance of the circuit. In this article, we demonstrated how to calculate the total resistance in a circuit with complex impedances. This knowledge is essential for designing and analyzing complex electrical systems.

References

• [1] "Electrical Engineering: Principles and Applications" by Allan R. Hambley
• [2] "Circuit Analysis: Theory and Practice" by Robert L. Boylestad

Appendix

The following is a summary of the calculations performed in this article:

Expression	Value
$Z_1$	$50+60i$
$Z_2$	$40-25i$
$Z_T$	$\frac{(50+60i)(40-25i)}{(50+60i)+(40-25i)}$
$Z_T$	$\frac{3500+150i}{90+35i}$
$Z_T$	$\frac{(3500+150i)(90-35i)}{(90+35i)(90-35i)}$
$Z_T$	$\frac{320250-109000i}{9325}$
$Z_T$	$34.38-11.65i$

Introduction

In our previous article, we explored the concept of total resistance in a circuit and derived a formula to calculate it. We then applied this formula to a specific circuit with complex impedances and determined the total resistance in standard form. In this article, we will answer some frequently asked questions related to total resistance in a circuit.

Q&A

Q: What is the total resistance in a circuit?

A: The total resistance in a circuit is a measure of the opposition to the flow of electric current in a circuit. It is typically represented by the symbol $Z_T$ and is calculated using the formula:

$$Z_T=\frac{Z_1 Z_2}{Z_1+Z_2}$$

where $Z_1$ and $Z_2$ are the individual impedances in the circuit.

Q: What are complex impedances?

A: Complex impedances are quantities that represent the opposition to the flow of electric current in a circuit. They are typically represented in the form:

$$Z=a+bi$$

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: How do I calculate the total resistance in a circuit with complex impedances?

A: To calculate the total resistance in a circuit with complex impedances, you can use the formula:

$$Z_T=\frac{Z_1 Z_2}{Z_1+Z_2}$$

where $Z_1$ and $Z_2$ are the individual impedances in the circuit.

Q: What is the significance of the total resistance in a circuit?

A: The total resistance in a circuit is a critical parameter in electrical engineering. It determines the opposition to the flow of electric current in a circuit and affects the overall performance of the circuit.

Q: Can I use the total resistance formula for circuits with more than two impedances?

A: Yes, you can use the total resistance formula for circuits with more than two impedances. However, you will need to substitute the values of all the impedances into the formula and simplify the expression.

Q: How do I simplify complex expressions involving complex impedances?

A: To simplify complex expressions involving complex impedances, you can use the following techniques:

• Multiply the numerator and denominator by the conjugate of the denominator
• Use the distributive property of multiplication over addition
• Substitute $i^2=-1$ into the expression

Q: What is the standard form of the total resistance in a circuit?

A: The standard form of the total resistance in a circuit is typically represented as:

$$Z_T=a+bi$$

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: Can I use the total resistance formula for AC circuits?

A: Yes, you can use the total resistance formula for AC circuits. However, you will need to use complex impedances and the formula:

$$Z_T=\frac{Z_1 Z_2}{Z_1+Z_2}$$

Q: How do I determine the total resistance in a circuit with multiple branches?

A: To determine the total resistance in a circuit with multiple branches, you can use the following techniques:

• Use the formula for total resistance in a series circuit
• Use the formula for total resistance in a parallel circuit
• Use the formula for total resistance in a combination of series and parallel circuits

Conclusion

In this article, we answered some frequently asked questions related to total resistance in a circuit. We hope that this article has provided you with a better understanding of the concept of total resistance in a circuit and how to calculate it.

Discussion

The total resistance in a circuit is a critical parameter in electrical engineering. It determines the opposition to the flow of electric current in a circuit and affects the overall performance of the circuit. In this article, we demonstrated how to calculate the total resistance in a circuit with complex impedances and answered some frequently asked questions related to total resistance in a circuit.

References

• [1] "Electrical Engineering: Principles and Applications" by Allan R. Hambley
• [2] "Circuit Analysis: Theory and Practice" by Robert L. Boylestad

Appendix

The following is a summary of the calculations performed in this article:

Expression	Value
$Z_1$	$50+60i$
$Z_2$	$40-25i$
$Z_T$	$\frac{(50+60i)(40-25i)}{(50+60i)+(40-25i)}$
$Z_T$	$\frac{3500+150i}{90+35i}$
$Z_T$	$\frac{(3500+150i)(90-35i)}{(90+35i)(90-35i)}$
$Z_T$	$\frac{320250-109000i}{9325}$
$Z_T$	$34.38-11.65i$