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AC Components and Circuits
AC RL Circuits

Section 3-2 Series RL Circuit Analysis

Series RL Impedance

 An ideal series RL circuit. Phasor diagram for series RL circuits.

Equation

The impedance of a series RL circuit is:

 Z = Ö 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 = 0º). And for the inductor, the voltage is always leading the current by 90º ( qL = 90º).  It is the phase of the total circuit impedance ( q ) that varies between 0º and 90º.

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 = Ö VR2 + VL2

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

Equation

The total voltage of a series RL circuit is:

 VT = Ö 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 = Ö VT2 – VL2
 For the inductor voltage: VL = Ö 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 = Ö VR2 + VL2
 VT = Ö 12.62 + 82

VT = 14.9V

 q = Tan-1 VL VR

 q = Tan-1 8 12.6

q = 32.4º

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 = Ö 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

 David L. Heiserman, Editor Copyright ©  SweetHaven Publishing Services All Rights Reserved

Revised: June 06, 2015