1. Resistive-Inductive (RL) Circuits
AC Circuits I

2. Series RL Circuits Series circuits: A comparison (Magnitude)
IT = IR1 = IR2
IT = IL1=IL2
IT = IL=IR
VT =VR1+VR2
VT = j(VL1 + VL2)
|VT| = sqrt(VR2 + VL2)
RT = R1+R2
XT = XL1 + XL2
|ZT| = sqrt(R2 + XL2)

3. Series RL Circuits
Series Voltages

4. Series RL Circuits Series Voltages (Continued)
Since VL leads VR by 90°, the values of the component voltages are usually represented using phasors
Phasor – a vector used to represent a value that constantly changes phase. Usually denoted in a polar form

5. Series RL circuits Series Impedance Polar form Polar form
Rectangular form
Rectangular form

6. Example
The component voltages shown below were measured using an ac voltmeter (HINT: rms values!!!). Calculate the source voltage for the circuit.

7. Example
Calculate the total impedance for the circuit shown (remember Z = R +jXL , where XL=2πfL )

8. Phase Reference
In a series RL circuit the current I and voltage across the resistor VR are considered to have 0o phase.
Sometimes however (especially in the lab) the source voltage Vs is assumed to have 0o phase.

9. Example
Calculate the circuit impedance, and current in the circuit shown below
I and Vs can be represented as either: (I lags Vs or Vs leads I)

10. Example
Determine the voltage, current and impedance values for the circuit shown below

11. Voltage Dividers RL Voltage Dividers
In a series RL circuit where I(VR )is used as the 0o phase reference, Vs and ZT always have the same phase angle!!!
where
Zn = magnitude of R or XL
Vn = voltage across the component

12. Example
Determine the voltages VL and VR in the circuit below

13. Example
Calculate ZT, VL1 VL2 and VR for the circuit shown below.

14. Series RL Circuit Frequency Response
Frequency Response – used to describe any changes that occur in a circuit as a result of a change in operating frequency
An increase in frequency causes XL to increase
An increase in XL causes ZT and  to increase
An increase in ZT causes IT to decrease
An increase in XL causes VL to increase
An increase in VL causes VR to decrease

15. Power
Power is also a complex quantity in AC circuits

16. Apparent, True and Reactive Power
Value
Definition
Resistive power (PR)
The power dissipated by the resistance in an RL circuit. Also known as true power.
Reactive power (PX)
The value found using P = I2XL. Also known as imaginary power. The energy stored by the inductor in its electromagnetic field. PX is measured in volt-amperes-reactive (VARs) to distinguish it from true power.
Apparent power (PAPP)
The combination of resistive (true) power and reactive (imaginary) power. Measured in volt-amperes (VAs).

17. Power Factor (PF)
The ratio of resistive power to apparent power

18. Example
Calculate PR, PX and PAPP for the following circuit - (see slide 13)

19. Parallel RL Circuits Note that VS, IR and VR are all in phase
IL is 90o out of phase with (lags) VS, IR and VR.
VS = VR1 = VR2
VS = VL1 = VL2
VT = VL= VR
IT =IR1+IR2
IT = j(IL1 + IL2)
|IT|= sqrt(IR2 + IL2)
RT = 1/[ (1/R1 )+(1/R2)]
XT = 1/[(1/XL1 ) + (1/XL2 )]
XT = 1/[(1/R ) + (1/jXL )]

20. Parallel RL Circuits
Branch Currents
IL lags IR , VR and VS by 90º

21. Parallel RL Circuits Parallel Circuit Impedance
Impedance phase angle is positive since current phase angle is negative
Angles of ZT and IT have the same magnitude but opposite signs
Calculating Parallel-Circuit Impedance (polar form)

22. Parallel RL Circuits Calculating Parallel-Circuit Impedance
rectangular form approach:

23. Parallel RL Circuits Parallel-Circuit Frequency Response
The increase in frequency causes XL to increase
The increase in XL causes:
IL to decrease
IT to increase
ZT to increase
The decrease in IL causes IT to decrease

24. Example 1
Calculate the total current for circuit (a) in both rectangular and polar forms
Calculate total impedance of circuit (a)
Repeat for circuit (b)

25. Example 2 (Rectangular)
Calculate the total impedance for circuit (a) in both rectangular and polar forms
Calculate total current of circuit (a)
Repeat for circuit (b)

26. Example 2 (Polar)
Calculate the total impedance for circuit (a) in both rectangular and polar forms
Calculate total current of circuit (a)
Repeat for circuit (b)

28. Series –Parallel RL Circuits
First collapse the parallel RL portion into a single impedance ZP
If this single impedance is in polar form, convert it to rectangular form
Add any resistances/ reactances in series with ZP to get ZT.
Remember you cannot add complex numbers in their polar form.
Always assume that IT is the zero Phase reference!! And that VS and ZT have the same phase i.e. Same assumption as for series RL circuit

29. Example
Calculate the Equivalent Impedance, the total current and the current in each branch of the parallel RL circuit shown below
Now convert to rect
and back to polar

30. Example Cont’d Knowing that
Remember in a series circuit, IT is the phase reference and VS and ZT always have the same phase!!!

31. Example Cont’d Alternatively using a current divider equation
Does IT = IL + IR2?

33. Remember!! For Series RL Circuits For Parallel RL Circuits
IT and VR are the 0o phase reference
VS and ZT have the same phase
For Parallel RL Circuits
VS is the 0o phase reference
IT and ZT have the same phase magnitude but opposite signs
For Series-Parallel RL Circuits
Use same rules as Series RL Circuits