Numerical Example 4.1 (RL Series Circuit)
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to RL Series Circuits
π Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let's begin our analysis of an RL series circuit. To start, can anyone tell me what an RL circuit consists of?
An RL circuit consists of a resistor and an inductor connected in series.
Exactly! The resistor dissipates energy, while the inductor stores energy. Now, can someone explain how we calculate the total impedance?
We add the resistance and the inductive reactance as complex numbers.
Right! Remember, the total impedance, Z, is given by Z = R + jXL. We'll compute the total impedance in our example.
How do we find the phase angle?
Good question! The phase angle can be found using the arctan function: ΞΈ = arctan(XL/R).
To summarize, to find the total impedance in an RL circuit, we use Z = R + jXL and find the phase angle with ΞΈ = arctan(XL/R).
Calculating Total Current and Voltage
π Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Now let's find the total current flowing through our circuit. Does anyone remember how to apply Ohm's Law in AC circuits?
We use I = V / Z, where V is the source voltage and Z is the total impedance.
Correct! In our example with a 120V source, we will calculate the total current I. What will we find?
We'd need to divide 120V by the total impedance we calculated.
Exactly! After calculating, weβll express the current in phasor notation. What do we do next?
We calculate the voltages across the resistor and inductor using VR = I * ZR and VL = I * ZL.
Very good! This shows how we can determine the voltage drops across each component using the total current. Summarizing: we use I=V/Z for total current and VR, VL for component voltages.
Verifying and Summarizing Results
π Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
After performing all these calculations, what should we do to ensure our results are reasonable?
We can verify the sum of voltages across the components against the source voltage.
Exactly! This is crucial for ensuring valid calculations in scenarios like these. Can anyone summarize what we learned today in one sentence?
We learned how to calculate the total impedance, current, and voltage in an RL series circuit connected to an AC supply.
Well done! Remember these key points β they will be essential for more complex circuit analyses.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section provides a detailed numerical example involving a series circuit comprising a resistor and inductor connected to an AC supply. Key calculations include the total impedance, total current, and the voltages across the resistor and inductor, demonstrating practical applications of AC circuit analysis.
Detailed
Numerical Example 4.1 (RL Series Circuit)
In this section, we analyze a series RL (Resistor-Inductor) circuit subjected to an alternating current (AC) supply. The primary goal is to compute the total impedance of the circuit, the total current flowing through it, and the voltage drop across each componentβthe resistor and the inductor.
Circuit Description
We have a circuit containing:
- A resistor (R) with a resistance of 15Ξ©.
- An inductor (L) with an inductive reactance (XL) of 20Ξ©.
- The circuit is connected to a 120V, 60Hz AC supply.
Key Calculations
- Total Impedance (Ztotal):
- The total impedance of a series circuit is the sum of the resistive and reactive components. Thus:
-
Calculation:
- ZR = 15β 0Β° = 15 + j0 Ξ©
- ZL = 20β 90Β° = 0 + j20 Ξ©
- Ztotal = ZR + ZL = (15 + j0) + (0 + j20) = 15 + j20 Ξ©.
- Magnitude:
- |Ztotal| = β(15Β² + 20Β²) = β(225 + 400) = 25Ξ©
- Phase Angle:
- ΞΈ = arctan(20/15) β 53.13Β°
-
Polar Form:
- Ztotal = 25β 53.13Β° Ξ©
- Total Current (I):
- Using Ohm's Law: I = Vsource / Ztotal
-
Calculation:
- I = (120β 0Β°) / (25β 53.13Β°) = 4.8β -53.13Β° A (indicating the current lags the voltage)
- Voltage across Resistor (VR):
- From Ohm's Law: VR = I * ZR
-
Calculation:
- VR = (4.8β -53.13Β°) * (15β 0Β°) = 72β -53.13Β° V
- Voltage across Inductor (VL):
- From Ohm's Law: VL = I * ZL
-
Calculation:
- VL = (4.8β -53.13Β°) * (20β 90Β°) = 96β 36.87Β° V
- Verification:
- Using Kirchhoff's Voltage Law, we check that the sum of voltages across the components equals the source voltage, confirming our calculations.
Significance
This example demonstrates the practical application of AC circuit principles, notably how to systematically analyze and calculate circuit parameters using complex impedance and phasors, essential for understanding more complex AC systems in various engineering contexts.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Total Impedance Calculation
Chapter 1 of 4
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Total Impedance (Ztotal ): ZR =15β 0β=15+j0 Ξ© ZL =20β 90β=0+j20 Ξ© Ztotal =ZR +ZL =(15+j0)+(0+j20)=15+j20 Ξ©
β In polar form: β£Ztotalβ£ =15Β²+20Β² =225+400 =625 =25 Ξ©
β ΞΈ=arctan(20/15)=arctan(1.333)β53.13β
β So, Ztotal =25β 53.13β Ξ©.
Detailed Explanation
To find the total impedance in a series circuit that includes a resistor (ZR) and an inductor (ZL), we start by determining their individual impedances. The resistor has an impedance of ZR = 15β 0Β°, while the inductor has an impedance represented as ZL = 20β 90Β°. In rectangular form, this translates to ZR = 15 + j0 and ZL = 0 + j20. The total impedance (Ztotal) is simply the sum of these two: Ztotal = ZR + ZL = (15 + j0) + (0 + j20) = 15 + j20.
Next, we convert this rectangular form into polar form. To do this, we first calculate the magnitude of Ztotal, given by Ztotal = β(ZRΒ² + ZLΒ²) = β(15Β² + 20Β²) = β625 = 25. The angle ΞΈ, which indicates the phase, is found using the arctangent function: ΞΈ = arctan(20/15) β 53.13Β°. Thus, the final total impedance is represented as Ztotal = 25β 53.13Β° Ξ©.
Examples & Analogies
Think of the total impedance as finding the effective route in a journey where you face a highway (the resistor) and a side road (the inductor). The highway allows for direct, straight travel with no detours, whereas the side road might have twists and turns. By understanding the length and angle of these paths (impedances), you can effectively calculate the combined journey's complexity (total impedance) and determine the best approach for a smooth ride.
Total Current Calculation
Chapter 2 of 4
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Total Current (I): Assume the supply voltage is the reference: Vsource =120β 0β V.
I=Vsource /Ztotal =(120β 0β)/(25β 53.13β)=(120/25)β (0ββ53.13β)=4.8β β53.13β A. (The current lags the voltage by 53.13β, as expected for an inductive circuit).
Detailed Explanation
To calculate the total current (I) flowing through the circuit, we use Ohm's Law for AC circuits, which states that current is equal to voltage divided by impedance (I = V/Z). Here, we have a supply voltage (Vsource) of 120β 0Β°, meaning the voltage is at its maximum. The total impedance we've calculated earlier is Ztotal = 25β 53.13Β°. Therefore, the calculation of current becomes I = 120β 0Β° / 25β 53.13Β°.
When we perform the division, we first divide the magnitudes: 120/25 = 4.8. For the angle, we subtract the impedance angle from the voltage angle: 0Β° - 53.13Β° = -53.13Β°. Thus, the current in phasor form is I = 4.8β -53.13Β° A. This negative angle indicates that the current lags behind the voltage, which is typical for inductive circuits where current responds more slowly than the applied voltage.
Examples & Analogies
Imagine trying to synchronize two people starting a race: one directly follows a whistle (voltage), while the other is a runner who takes a moment to react due to a small delay (inductance). The runner's reaction time is like the lag in current, represented by the angle in the calculation. By knowing their respective starting positions (voltage) and the runner's adjustment time (impedance), you can determine how far behind the runner is expected to be at any point in the race (current).
Voltage Across Components Calculation
Chapter 3 of 4
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Voltage across Resistor (VR ): VR =IΓZR =(4.8β β53.13β)Γ(15β 0β)=(4.8Γ15)β (β53.13β+0β)=72β β53.13β V.
Voltage across Inductor (VL ): VL =IΓZL =(4.8β β53.13β)Γ(20β 90β)=(4.8Γ20)β (β53.13β+90β)=96β 36.87β V.
Detailed Explanation
To find the voltage across each component in a series circuit, we apply the formula VR = I Γ Z, where VR is the voltage across the resistor and I is the total current we computed earlier.
For the resistor, we calculate: VR = (4.8β -53.13Β°) Γ (15β 0Β°). This means we multiply the magnitudes: 4.8 Γ 15 = 72, and for the angle, we add the respective angles: -53.13Β° + 0Β° = -53.13Β°. Hence, the voltage across the resistor (VR) is 72β -53.13Β° V.
For the inductor, we similarly use the total current: VL = (4.8β -53.13Β°) Γ (20β 90Β°). The magnitude is calculated as 4.8 Γ 20 = 96. Now we add the angles: -53.13Β° + 90Β° = 36.87Β°. Therefore, the voltage across the inductor (VL) is 96β 36.87Β° V.
Examples & Analogies
Visualize a team of workers pulling a heavy object up a slope. Each worker represents a different electrical component; in this case, the resistor and the inductor are working against the load (current). The total effort each worker contributes (voltage) depends on their strength (impedance) and the total weight of the load (current). As the workers pull the object, the effectiveness of their individual contributions (voltage across each component) can be calculated just as we computed the voltage across the resistor and inductor here.
Verification Using Kirchhoff's Voltage Law
Chapter 4 of 4
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Verification (KVL): VR +VL =(72cos(β53.13β)+j72sin(β53.13β))+(96cos(36.87β)+j96sin(36.87β))
=(43.2βj57.6)+(76.8+j57.6)=120+j0=120β 0β V (matches source voltage).
Detailed Explanation
After calculating the voltage across each component, we will verify the results using Kirchhoff's Voltage Law (KVL), which asserts that the sum of the voltages in a closed loop must equal the source voltage. In this case, this involves adding the calculated voltages (VR and VL) together and checking whether their total equals the source voltage.
When applying the cosine and sine functions based on the angles calculated earlier, we find the real and imaginary parts of the voltages: VR and VL. Adding these together gives a resultant voltage of 120 + j0, meaning the total is 120β 0Β° V. As the results match the source voltage of 120V, we've correctly described the circuit performance.
Examples & Analogies
Imagine youβre at an amusement park and decide to check if all the rides add up to the same amount of fun you paid for. Each ride contributes differently based on your enjoyment (voltages across components). By totaling the experiences and ensuring they sum to your expected enjoyment (source voltage), you verify that the fun was worth the ticket price, just like we verified that the voltage across our components equaled the supply voltage.
Key Concepts
-
Impedance: The total opposition to current flow in an AC circuit, represented as a complex number.
-
Inductive Reactance: The opposition to current flow due to inductance, increasing with frequency.
-
Ohm's Law in AC: Current is calculated using the ratio of voltage to impedance.
-
Voltage Drop: The difference in voltage across circuit components caused by current flow.
Examples & Applications
Example of calculating total impedance and current in a series RL circuit connected to an AC supply.
Illustration of voltage drops across individual components in an RL series circuit under AC conditions.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In an RL circuit, Z adds up straight, Reactance and Resistance pave the fate.
Stories
Imagine R and L on a journey. R walks straight, but L lags behind, always needing more time, represents phase difference.
Memory Tools
To remember impedance: 'Zap Resistance + Inductive reactance = Impedance!'
Acronyms
RIP - Resistance, Inductive reactance, Phase angle.
Flash Cards
Glossary
- Impedance (Z)
The total opposition to current flow in an AC circuit, combining resistive and reactive components.
- Inductive Reactance (XL)
The opposition to current flow caused by an inductor; calculated as XL = ΟL.
- Phasor
A complex number used to represent a sinusoidal function in terms of magnitude and phase.
- Ohm's Law (AC)
A relationship stating that current (I) equals voltage (V) divided by impedance (Z) in an AC circuit: I = V/Z.
- Voltage Drop
The reduction in voltage across a component in a circuit due to a current passing through it.
Reference links
Supplementary resources to enhance your learning experience.