Chapter 3 AC RL Circuits

3-1 A Look at AC Circuit Analysis
3-2 Series RL Circuit Analysis
3-3 Parallel RL Circuit Analysis
3-4 Combination RL Circuit Analysis
3-5 Applications of RL Circuits

 

 

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You should know by now that there are two kinds forces that oppose current in a circuit:  resistance and reactance. Resistance is a fixed opposition to current that is created by the molecular structure of the resistor material. Reactance--both inductive and capacitive--is caused by dynamic forces that oppose current flow and are sensitive to the operating frequency of their circuits.

In this lesson, you will begin working with AC circuits that combine resistance and reactance. So what do you call  this combination of resistive and reactive opposition?  It's called impedance.

Definition

Impedance

Impedance is the opposition to AC current by a combination of resistance and  reactance.

The unit of measure for impedance is Ohms (W)
The math symbol for impedance is Z.

 

 

 

 

3-2 Series R-L Circuit Analysis

 

Z = R + j0

Z = R

Series RL Impedance

 


An ideal series RL circuit.


Phasor diagram for series RL circuits.

 

 

Equation

The impedance of a series RL circuit is:

Z = R2 + XL2

Where:

Z = RL circuit impedance in Ohms
R = Resistor value in Ohms
XL = Inductive reactance in Ohms

If you know the values for R and XL, you can solve the equation directly. In most practical situations, however, you know the values of L  and f, and must calculate XL in order to use the series-Z equation.

Since this is a series circuit, the magnitude of the current is the same at all point. It's the voltage that is subject to phase shifting. Recall there is no phase shift for a resistor (qR = 0). And for the inductor, the voltage is always leading the current by 90 ( qL = 90).  It is the phase of the total circuit impedance ( q ) that varies between 0 and 90.

 

Equation

The phase angle for a series RL circuit is:

q = tan-1 XL
R

Where:

q = Phase angle in degrees or radians*
XL = Inductive reactance
R = Resistor value
 
*Unless clearly stated otherwise, angles in this course are specified in degrees

Examples

Endless Examples & Exercises

Work these problems until you are confident you have mastered the procedures.
  • All angles are expressed in degrees.
  • Round answers to the nearest tenth.

Voltages in Series RL Circuits

A series RL circuit is a simple-looking circuit. In most  respects, it looks similar to a series resistor circuit that is operating from an ac source. But something very interesting happens when you measure and compare the rms voltages across the  resistor and inductorthe sum of  those  two  voltages is greater than the source voltage!

Suppose you measure the ac source voltage (VT) and find it to be 18 V. Then you measure the voltages for VR and VL, and find them to be 6 V and 17 V, respectively. How can that be? 6V + 17V is a long  way from 18V

This apparent discrepancy is caused by the fact that the ac waveforms across the resistor and inductor are out of phase. The phase diagram shows that the voltages across the two components  are 90 out of phase, with the inductor voltage leading the resistor voltage. The total voltage is not the sum of the resistive and inductive voltage, but rather a level determined by the length of the VT vector:

VT = VR2 + VL2

And, yes, that  is a variation of the Pythagorean Theorem

Equation

The total voltage of a series RL circuit is:

VT = VR2 + VL2

Where:

VT = Source voltage in Volts
VR = Resistor voltage in Volts
VL = Inductor voltage in Volts

In actual practice, it makes more sense to measure VT with a meter or oscilloscope than to calculate it. The real value of the equation is that it explains how the total voltage in a series RL circuit is not the sum of the voltages across the components.

Secondly, the equation can be rearranged to deal with calculations where you know the total voltage and the voltage across one of the other components:

For the resistor voltage: VR = VT2 VL2
For the inductor voltage: VL = VT2 VR2

Equation

The phase angle for a series RL circuit is:

q = tan-1 VL
VR

Where:

q = Phase angle in degrees or radians
VL = Inductor voltage
VR = Resistor voltage

 

 

 

 

 

 

Exercise

VR = 12.6V

VL = 8V

VT = VR2 + VL2
VT = 12.62 + 82

 VT = 14.9V

q = Tan-1 VL
VR

 

q = Tan-1 8
12.6

q = 32.4

 

Currents in Series RL Circuits

 

 

Analysis of Series RL Circuits

 

A typical analysis of a series RL circuit begins with known values for:
Total rms voltage applied to the circuit (VT)
Applied frequency (f)
Value of the resistor (R)
Value of the inductor (L)

The objective, then, is to determine all other relevant circuit values:

Inductive reactance (XL)
Impedance (Z)
Total current (IT)
Resistor current (IR)
Inductor current (IL)
Voltage across the resistor (VR)
Voltage across the inductor (VL)
Phase angle (q)

Procedure

1. Calculate XL

XL = 2pfL

2. Calculate Z

Z = R2 + XL2

3. Calculate IT

IT =

VT  
Z

4. Specify the values for IR and IL

For a series circuit, IR = IL = IT

 5. Calculate VR

VR = IR x R

6. Calculate VL

VL = IR x XL

7. Calculate the phase angle

q = tan-1 XL
R

Students who are new to the procedures for analyzing circuit often ask, "How do you know where to start? How do you know what to do next?"  The answer is simple and so very true:  Do whatever you can do with the data that is available.

Endless Examples & Exercises

Work these problems until you are confident you have mastered the procedures.
  • All angles are expressed in degrees.
  • Round answers to the nearest tenth.

 

I and V Phase Relationships

 

 

 

Variation of Z with Frequency

 

 

 

Variation of Phase Angle with Frequency

 

 

 

fra03 Parallel RL Circuit Analysis

 

 

 

Currents in Parallel RL Circuits

 

 


Currents in parallel RL circuits


Phasor diagram for parallel RL circuits.

 

 

Equation

The total current of a series RL circuit is:

IT =  IR2 + IL2

Where:

IT = Total current
IR = Resistor current
IL = Inductor current

In most practical situations, however, the resistor and inductor currents are not known. So  the application of the equation for total current is  preceded by determining IR and IL:

IR VT
R
 and
IL = VT
XL

If XL is not directly know, you also have to calculate XL = 2pfL.

The phase angle for a parallel RL circuit is usually determined from the resistive and reactive currents.


Currents in a parallel RL circuit.

The phasor diagram shows that there is no phase shift for the resistor current (IR) and there is a phase of -90 for the inductor current. The phase shift for the total  circuit is thus somewhere between 0 and -90.

Equation

The phase shift for a parallel RL circuit is:

q = tan-1 IL
IR

Where:

q = Phase angle*

IL = Inductor current

IR = Resistor current

 

 

Procedure

Given: VT, f, R, and L

Determine: IT and q

1. Calculate XL

XL = 2pfl

2. Calculate IL

IL = VT
XL

3. Calculate IR

IR VT
R

At this point, you know IL and IR

 

4. Calculate IT

IT =  IR2 + IL2

5. Calculate q

q = tan-1 IL
IR

Steps 4 and 5 are the values to be determined.

 

Examples

Impedance in Parallel RL Circuits

Equation

The impedance of a parallel RL circuit is:

Z = VT
IT

Where:

Z = Circuit impedance
VT= Total voltage
IT = Total current

 

Examples

 

 

Analyzing Parallel RL Circuits

A typical analysis of a parallel RL circuit begins with known values for:
Total rms voltage applied to the circuit (VT)
Applied frequency (f)
Value of the resistor (R)
Value of the inductor (L)

The objective, then, is to determine all other relevant circuit values:

Voltage across the resistor (VR)
Voltage across the inductor (VL)
Inductive reactance (XL)
Resistor current (IR)
Inductor current (IL)
Total current (IT)
Impedance (Z)
Phase angle (q)

General Procedure

1. Determine VR and VL from  VT

2. Calculate XL from f and L

3. Calculate IR by applying Ohm's law to VR and R

4. Calculate IL by applying Ohm's Law to VL  and XL

5. Calculate IT from IR and IL

6. Calculate Z by applying Ohm's law to VT and  IT

7. Calculate q from IL and IR

 

Examples

Endless Examples & Exercises

Work these problems until you are confident you have mastered the procedures.
  • All angles are expressed in degrees.
  • Round answers to the nearest tenth.

 

 

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Analysis of Combination RL Circuits

 

 

 

 

 

 

 

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Endless Examples & Exercises

 

 

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Examples

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Definition

 

 

 

Procedure

 

 

 

Note

 

 

Equation

 

 

     
 

 

         
   

Series RL Circuits

VT =  VR2 + VL2

 

IT = IR = IL = VT
Z

 

ZT =  R2 + XL2

 

q = tan-1 XL
R

 

q = tan-1 VL
VR

 

XL = 2pfL

 

XL = VL
IL

 

 

 

 

 

 

Parallel RL Circuits

 

VT = ITZ

IT =  IR2 + IL2

 

ZT = VT
IT

 

q = tan-1 IL
IR

 

XL = 2pfL

 

XL = VL
IL