Inductors in AC Circuits

NCEA Level 3 Physics Electricity AS91526

1. Inductors in AC Circuits

2. Inductors An inductor affects a circuit whenever current (I) is changing. The magnetic field generated by the inductor acts to induce an opposing current (Lenz’s Law). The ideal inductor stores energy in its magnetic field which is then returned to the circuit as electrical energy, the only energy loss is from the resistance of the circuit.

3. Inductors in AC In an AC circuit current is constantly changing so inductors play an important role The current opposing ability of inductors is called reactance and given the symbol X L Like X C the units are Ohms

4. Voltage and Current Phase Differences In a circuit composed only of an inductor and an AC power source, there is a 90° phase difference between the voltage and the current in the inductor. For an inductor the current lags the voltage by 90°, so it reaches its peak ¼ cycle after the voltage peaks.

5. Relationship between V and I Because the inductor acts to oppose the change in current, as current increases a clear relationship with voltage can be seen ~ V L A 6V AC

6. Examples Find the inductor voltage of an AC circuit with a reactance of 2.4  and a current of 0.18A 0.43V An inductor has a voltage of 8.2V AC and a reactance of 54. Calculate the current of the circuit. 0.15A Calculate the reactance of a circuit with an inductor voltage of 16V and a current of 1.2A 13 

7. Factors Affecting Reactance ( X L ) Increasing the size of the inductor (L) will induce a higher opposing voltage and therefore increase X L Increasing frequency increases induced current (increasing reactance ). This is because more frequent creation and collapse of magnetic field produces greater opposing current The reactance of a capacitor with a supply frequency f;

8. Examples A 0.5H inductor is connected to a 6V 50Hz AC supply. Calculate the reactance of the inductor 157  The RMS current in the circuit 0.038A What size inductor is needed to give an reactance of 25 in a 18V 6 0Hz circuit? 66 mF

9. V L as ¼ cycle ahead of resistive voltage Because V L is maximum where V R is changing most (gradient steepest) Note: the value of V R and V L are not always equal as in this example Resistor and Inductor Phase Differences -1.5 -1 -0.5 0.5 1 1.5 200 400 600 800 Time (ms) Voltage (mV) Resistor Inductor Phase Differences in LR Circuits V L V R

10. The Effect of Phase Differences in LR Circuits In DC circuits the voltages across components in a circuit add up to the supply voltage In AC Inductor/Resistor (LR) circuits the same does not appear to apply (at first glance) just like RC circuits V S V C V R 0.50H 100  12V 6.4V 10V

11. The Effect of Phase Differences in LR Circuits However if we consider the phase differences, we see that this is a vector problem V L V R V S V L V R V S V S V C V R 0.50H 100  12V 6.4V 10V

12. In an LR circuit; At any instant Note the graph But when considering the rms voltages the phase differences are important The Effect of Phase Differences in LR Circuits

13. Exercises Find the AC supply voltage of an LR circuit where the resistor voltage is 3.4V and the inductor voltage is 1.5V 3.7V Calculate the voltage across the resistor in an AC circuit with a supply voltage of 8.5V and a inductor voltage of 2.4V 8.2V Calculate the voltage across the inductor in an 12V AC circuit with a voltage of 8.5V across the resistor. 8.5V Find the supply voltage of an 60Hz AC circuit with a 120V across a 2k  resistor and an inductor voltage of 0.80V 120V

14. Impedance As with LR circuits impedance relates supply voltage to current .   U sing Pythagoras from the addition of phasors

15. Examples Calculate the impedance of an L R circuit with a resistance of 75  and a reactance of 15  76  An LR circuit has an impedance of 65  and has a resistance of 24 . What is the reactance of the circuit? 60  Find the resistance of an LR circuit with 25  impedance and 12  reactance. 22 

16. Inductors in DC c.f. AC Both circuits have the same components but behave quite differently because of their power supplies; Find the resistance of the resistor What assumption did you make in 1? Calculate the reactance of the circuit What is the impedance of the circuit? Calculate the current in the AC circuit A A 400mH 400mH 0.15A 18V DC 18V AC 50Hz

17. The LCR Series Circuit The LCR circuit has some interesting and useful properties. T he current and voltage in the circuit vary considerably as frequency changes The voltage across each component will depend on the resistance or reactance of each component Variable Frequency AC A      

18. LRC Phase Differences Phase differences are the same as the individual RC and LR circuits combined Inductor voltage (V L ) leads resistor voltage (V R ) by 90  and V R leads capacitor voltage ( V C ) by 90  In LCR circuits inductor and capacitor voltages have an opposite phase, so fully or partially cancel each other V L V R V C

19. LCR Phasors In most cases the L, C and R phasors will be different lengths Most commonly voltage and reactance/resistor phasors are considered In either case remember to calculate the differences between the two opposite phasors before calculating V S or Z V L V R V C V L - V C V R X L R X C X L - X C R V S Z or;

20. Supply Voltage in LCR Circuits Calculations of the supply voltage must take the into account the differences of the components V L V R V C V L - V C V S  

21. Examples Calculate the supply voltage of an LCR circuit where the capacitor voltage is 12V , the resistor voltage is 18V  and the inductor voltage is 6V 19V Calculate the resistor voltage of an LCR circuit where the supply voltage 240V, the capacitor voltage is 85V and the inductor voltage is 220V 198  Find the inductor voltage of an LCR circuit where the supply voltage is 12V , the resistor voltage is 9.8V and the capacitor voltage is 4.5V 2.4V

22. Impedance in LCR Circuits Impedance is a measure of the combined opposition to alternating current of the components of a circuit. It describes not only the relative amplitudes of the voltage and current, but also the relative phases the components in the circuit. Impedance has the symbol Z and units Ohms X L R X C X L - X C R Z

23. Examples Calculate the impedance of an LCR circuit where the capacitor reactance is 25 , the resistance is 50  and the inductor reactance is 15  51  Calculate the resistance of an LCR circuit where the impedance 110  is capacitor reactance is 64  and the inductor reactance is 25  100  Find the inductor reactance of an LCR circuit where the impedance is 120 , the resistance is 110  and the capacitor reactance is 30  120 

24. Resonance Because reactance is dependant on supply frequency and directly proportional for inductors and inversely proportional for capacitors at a certain frequency (resonant frequency f O ) these reactances cancel each other out At this frequency current in the circuit reaches a maximum and the circuit is said to be tuned f o Resonant frequency Current (A)

25. Resonant Frequency Because at resonance; so;   Note that the resonant frequency is independent of the resistance

26. Examples Calculate the resonance frequency of an LRC circuit with a 200 F capacitor and a 0.5H inductor. Find the size of the capacitor needed for resonance in an LRC with a resonant frequency of 50Hz and an inductor of 0.20H

27. Voltage at Resonance At resonance; And because Z = R And cancel each other out

28. Examples

29. Exercises ESA Pg 282 Activity 16E, 16F, 16G, 16H ABA Pg 186-196