Before starting this module, you should be able to: When you complete this module, you should be able to: 
  • Apply Ohm's Law to determine the relationships between current, voltage, and reactance of an inductor.
  • Describe the phase angle for the current and voltage of an inductor in a series circuit.

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  • Define the impedance of an RL circuit.
  • Cite the equation for calculating the impedance of an RL circuit in terms of R and XL
  • Calculate the value of impedance for a series RL circuit, given the values of R and XL
  • Calculate the impedance of a series RL circuit, given the values of R, L, and f.
  • Cite the equation for determining the total phase angle for a series RL circuit in terms of voltage drops.
  • Given values of R and XL, determine the phase angles for the resistor, inductor, and total circuit.
  • Describe what is meant by a complete analysis of a series RL circuit.
  • Completely analyze a series RL circuit, given the values of R, L, VT, and f.

Topic 7-2.1 Impedance of a Series RL Circuit

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The impedance of an RL circuit is the total opposition to AC current flow caused by the resistor (R) and the reactance of the inductor (XL). 

The equation for the impedance of an RL circuit is: 

 

where: 

    Z = the total impedance in ohms 
    XL = the inductive reactance in ohms 
    R = the resistance in ohms

It is no accident that the equation for impedance looks like the equation for calculating the hypotenuse of a right triangle.

Impedance in series circuit is, in fact, often portrayed as a vector diagram where the horizontal side is the resistance, the vertical side is the reactance, and the hypotenuse is the resulting impedance. 

 

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In a series RL circuit, R = 100 W and XL = 150 W. What is the total impedance of this circuit? 

Ans: 180 W

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Topic 7-2.2 Voltages in a Series RC Circuit

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The total voltage in a series RL circuit is given by this equation: 

where: 

    VT = total voltage 
    VR = voltage across resistor R 
    VL = voltage across inductor L

It is very important to notice that the total voltage for a series RL circuit is NOT equal to the sum of the voltages across the resistor and inductor

The sum of voltages in a series RL circuit is always greater than the sum of the voltages across the resistive and inductive components.

 

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What is the total voltage applied to a series RL circuit when the voltage drop across the resistor is 12 V and the voltage across the inductor is 10 V? 

Ans: 15.6 V

 

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Reactance and Impedance Cannot be Directly Measured 

Although you can use an ordinary ohmmeter to measure resistance, there are no common lab instruments for directly measuring reactance and impedance. For all practical purposes, then, you must calculate reactance and impedance from other circuit values that are more readily available.  
 

In practical RL circuits, you can readily determine or directly measure the values of VT, f, R, and L. There is more than enough information among these items to calculate the values of XL and Z.
Procedure 

Step 1. Calculate the value of XL from the known values of f and L: 

XL  = 2pfL

Step 2. Calculate Z from the know value of R and the value of XL calculated in Step 1: 

 

 

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Given a series RL circuit where R = 20 kW, f = 220 kHz, and L = 10 mH, calculate (a) the inductive reactance and (b) the total impedance. 

Ans: (a) XL = 13.8 kW, (b) Z = 24.3 kW

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 Topic 7-2.3 Phase Angles of Series RL Circuits

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Phase Angles in Series RL Circuits 

It is a basic property of resistors and inductors that their phase angles in a series circuit are always give by: 

qR = 0  qL = 90

However, there are two equations you can use for defining the total phase angle for a series RL circuit.

The total phase angle can be determined by the equation:  qT = tan-1(XL/ R)  

  where: 

    qT = total phase angle in degrees or radians 
    XL= inductive reactance in ohms 
    R = resistance in ohms

The total phase angle is also determined by the equation:

qT = -tan-1(VL /  VR)

    where: 

    qT = total phase angle in degrees or radians 
    VL= AC voltage drop across the inductor in volts 
    VR = AC voltage drop across the resistor in volts

The total phase angle of a series RL circuit is always somewhere between 0 (purely resistive circuit) and 90 (purely inductive circuit). 

The  tan-1 expression is the inverse tangent which is used for calculating angle q for a right triangle, given the lengths of the two sides. 

 

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A series RL circuit has values of R = 120 W and XL = 150 W. Calculate the phase angles for R, L and the total circuit. 

Ans:  0, 90, 51.30 

 

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What is the phase angle for total voltage and current in a series RL circuit where R = 10 kW, L = 100 mH, and f = 30 MHz.  

Ans: 62

Step 1: Calculate the inductive reactance (18.8 kW).  

Step 2: Calculate the total phase shift.

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Topic 7-2.4 Analysis of Series RL Circuits

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Secondary Properties of Series RL Circuits 
The secondary properties of a series RL circuit are the: 

 

  •  Amount of applied AC voltage, VT in volts 
  • Amount of AC voltage drop across R and L, VR and VL in volts 
  • Total current through the circuit, IT in amperes 
  • Currents through R and L, IR and IL in amperes 
  • Total phase angle, qT in degrees or radians 
  • Phase angle for R and L, qR and qL in degrees or radians
A complete analysis of a series RL circuit usually proceeds from knowing the values for R, L, f, and VT. The analysis then amounts to determining the remaining secondary properties of the circuit. 

Some of these remaining properties are determined from the nature of the components, themselves, and do not have to be calculated. For example, the phase angle for R is always 0, and the phase angle for L in a series circuit is always 90. Other values must to be calculated by means of various equations.   

 

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Complete Analysis of a Series RL Circuit

Here is the procedure for doing a complete analysis of a series RL circuit, given the values of R, L, f, and VT

Step 1. Calculate the value of XLXL = 2pfL

Step 2. Calculate the total impedance: 

Step 3. Use Ohm's Law to calculate the total current:  IT = VT / Z

Step 4. Determine  the currents through R and L. Since this is a series circuit:  IR = IT  IL = IT

Step 5. Calculate the voltages across R and L. By Ohm's Law:  VR = RIR VL = XLIR

Step 6. Determine the phase angles for R and L. Phase angles for these components in a series circuit are always:  qR = 0  qL =  90

Step 7. Calculate the total phase angle for the circuit:  qT = tan-1(XL/ R)   

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