Alternating Current Electricity NCEA A.S 3.6 Text Chapters 18-19

Why AC? It can be produced directly from generators It can be controlled by a wide range of components eg resistors,capacitors and inductors. The max voltage can be changed easily using a transformer The frequency of the AC can be used for timing

AC Current

AC Voltage

AC Power P=VxI Multiplying the graphs gives us a graph where the power is always positive

AC Power The average voltage in ac is zero since there is an equal amount of positive and negative voltage. Same for current The average value of the power used in ac is half that of the peak power

RMS Values Since voltage and current are always changing we need some way of averaging out their effect. We use r.m.s values (root-mean-square) The r.m.s values are the DC values which give the same average power output

RMS Values AC VoltageDC Voltage (with same power output) V rms V max

RMS Values (See text pg for derivations of these formulae)

AC in Capacitors In a DC circuit, the current flows until the cap is fully charged and then stops. In an AC circuit, the current can continue to flow, as the plates become alternately charged positively and negatively ~

Reactance For both AC and DC circuits, the voltage across the resistor is related to the current by V=IR A similar relationship exists for a capacitor: Where X c is the reactance of the capacitor ~

Reactance Reactance is a measure of how a capacitor can limit alternating current Unit: Ohms It is similar to resistance but differs in that it is dependent on the frequency of the ac supply. It also depends on the size of the capacitor.

Reactance Explanations: Higher f means cap never gets full before current direction changes, so never limits current, so low X Higher C means that it takes more charge to fill it, so never fills before current direction changes, so never limits current, so low X

Phase Relationship In a DC circuit the voltage across components connected in series will add up to the supply voltage In AC circuits this does not happen Eg. ~ VSVS VCVC VRVR

Phase Relationship Reasons: The meters used to measure the voltage will give rms values, not actual voltages at a point in time The meters used to measure the voltage will give rms values, not actual voltages at a point in time The voltages across the resistor and capacitor are out of phase with each other ie they do not both reach maxs and mins at the same time. The voltages across the resistor and capacitor are out of phase with each other ie they do not both reach maxs and mins at the same time.

Phase Relationship The current in the circuit will always be in phase with V R (Reason: because R is constant so bigger V gives bigger I) This can be shown on a phasor diagram: VRVR VRVR I t I ω VRVR

Phase Relationship V C will lag 90° behind I (and therefore V R ) because the max current flows when the voltage across it’s plates is zero, ie uncharged, and zero current flows when voltage is max ie cap is fully charged The phasor diagram will look like:

Phase Relationship The voltage phasors are not necessarily the same size, but are always 90°out of phase VRVR I t I ω VCVC VRVR VCVC

RC Circuits The total voltage in the circuit can be found by adding the V R and V C phasors together VRVR t ω VCVC VRVR VCVC VsVs VSVS

Impedance The current is the same everywhere in the circuit so V R and V C are proportional to R and X C This combination of resistance and reactance which both act to limit the current is called impedance Z V R =IR V C =IX C V S =IZ R XCXC Z

AC in Inductors In a DC circuit an inductor produces an opposing voltage whenever the current changes. In an AC circuit, the current is always changing so the inductor is always producing an opposing voltage so is always limiting the amount of current that can flow ~

Reactance For both AC and DC circuits, the voltage across the resistor is related to the current by V=IR A similar relationship exists for an inductor: Where X L is the reactance of the inductor ~

Reactance It measures how well an inductor can limit alternating current It depends on the frequency of the ac supply. It depends on the size of the inductor.

Reactance Explanations: Higher f means faster rate of change of current, so more back e.m.f, so less current, so higher X L Higher L means more back e.m.f, so less current, so higher X L

Phase Relationship V L will lead I (and therefore V R ) by 90° because the greatest back e.m.f occurs when the current is changing most rapidly, which is when it is passing through zero. When the current has reached it’s max, it is not changing as rapidly so there is no back e.m.f The phasor diagram will look like:

Phase Relationship Again the voltages may be different sizes but will always be 90° out of phase VRVR I t I ω VLVL VRVR VLVL

LR Circuits The total voltage in the circuit can be found by adding the V R and V L phasors together VRVR t ω VLVL VRVR VLVL VsVs VSVS

Impedance The impedance Z is found by adding R and X L V R =IR V L =IX L V S =IZ R XLXL Z

LCR Circuits This can be an extremely useful circuit set- up, as the current and voltages can change considerably as the frequency is changed ~

LCR Circuits The combined phasor diagram now looks like: t VRVR ω VLVL VRVR VLVL VsVs VSVS VCVC VCVC

Supply Voltage The supply voltage is now found by adding all 3 phasors together (V L and V C are combined into one first) V R =IR V L =IX L V S =IZ V C =IX C V L -V C

Impedance The impedance of an LCR circuit is a combination of both the resistance and the reactance. It is found by adding phasors: R XLXL Z XCXC X L -X C

Resonance At low f, V C >V L so V R (and therefore I) is small. ie. Capacitors limit the current better at low frequencies VRVR VLVL VSVS VCVC

Resonance At high f, V L >V C so V R (and therefore I) is small. ie. Inductors limit the current better at high frequencies VRVR VLVL VSVS VCVC

Resonance At resonance, V L =V C and they cancel each other out. So V S =V R and if V R is at max then I is at max. VRVR VLVL VSVS VCVC

Resonance At resonance, a circuit has the maximum possible current for a given supply voltage V S. At resonance:

Resonant Frequency A circuit will have a resonant frequency f 0 which depends on L and C:

Rectifying AC Rectifying – turning AC into DC Putting a diode into the circuit will do this: t

Rectifying AC A bridge rectifier will do this: t

Rectifying AC A bridge rectifier circuit looks like this: 240V AC in 12V AC out 12V DC (smoothing cap)

Rectifying AC A bridge rectifier with a capacitor in parallel with it will do this: (the bigger the cap the smoother the DC) t