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Alternating current (A.C) is the current that changes in magnitude direction continuously with respect to time.
It can be represented as,
The currents and voltages in a.c circuits can be expressed by the following terms:-
(i) Instantaneous (I or V) :- It is the current or voltage that in the circuit at any instant. (ii) Peak (Im or Vm ):- It is the maximum available voltage or current in the circuit. (iii) Average (Iav or Vav):- It is the arithmetic mean of the instantaneous values of voltages or currents in the circuit. (iv) Room Mean Square (R.M.S) or Effective :- It is the square root of the average of the squares of the instantaneous values of the voltages or currents in the circuit.
Vrms = √
R.M.S value is the d.c equivalent of a.c
Relation between Peak and R.M.S values
=∫ =∫
= (^) ∫ ( ) where = 2/T
But (^) ∫ = [ = = = 0
Thus equation (1) becomes, = [ T – 0 ] So, = (^) √
Similarly, it can also be shown that , = (^) √
Note: the measurement of ac is done by the comparison of the heating effect produced by ac with that by dc. Hence ac measuring instruments are also called ‘hot-wire instruments’.
Show that the average current in the complete cycle of a.c is zero.
Iav = (^) ∫ = (^) ∫ = = ⁄ (^) ∫ ⁄ [ ]
= ⁄ [^ ] = ⁄ = 0 (Hence proved)
Note:- But in half a cycle, Iav = (^) ∫ = (^) ∫ = = ⁄ (^) ∫ ⁄ (^) [ ]
= ⁄ [^ ] = ⁄ = ⁄
This gives Iav = ⁄ ^ Similary, it can be shown that Vav = ⁄
Phasor diagrams
These are the vector diagrams in which the magnitudes represent the peak values and directions represent the phase differences between voltages and currents in the a.c circuits.
A.C Circuits
Fundamental components in a.c circuits.
- Resistor
- Inductor
- Capacitor 1. Circuit containing only resistor
Circuit diagram: Let V = Vm sint Current (I) = V/R = (Vm sint)/R
i.e., I = Im sint. Where Im = Vm/R. R = Vm/Im
V Hence the phase difference between voltage and current is, = 0.
So the phasor diagram can be as shown I V
The graph showing the variation of current and voltage with time can be represented as:
Power associated with the circuit:
= ∫^ ^ = ∫ - ∫^ ^ ]
But (^) ∫^ ^ = 0 Thus we get, PR = (Vm Im) /2 Hence PR = Vrms. Irms
2. Circuit containing only inductor
Circuit diagram: Let V = Vm sint
The induced voltage in the inductor is Vind = - L(dI/dt)
V By loop rule, Vm sint - L(dI/dt) = 0
i.e., Vm sint = L(dI/dt) this gives, dI = (Vm/L) sint dt
On integrating the above equation we get, Current (I) = )
I = Hence I = Im sin ( t - /2) where Im = ( Vm/L)
Here ‘L’ represents the non-resistive opposition offered by the inductor to the flow of current in an ac circuit. It is called inductive reactance (XL) i.e ., XL = L
Hence the phase difference between voltage and current is, = /2.
By parallelogram law, VL - VC V I
The resultant voltage is given VR
by, V =√ But VR = IR, VL = IXL and VC = IXC
Thus V = √ Hence V/I = √
This gives the total opposition to the flow of current in an a.c circuit, including resistance and reactances. It is called the impedance (Z) of the circuit.
Thus the impedance is given by Z = √
Also the phase difference between the voltage and current is given by
tan = i.e. tan = or tan =
Hence, current in the circuit can be expressed as I = Imsin( t- ), the negative sign indicates that the inductive reactance dominates capacitive. i.e. current lags behind voltage.
(Refer page 246 of Text book for ‘Analytical solution’)
Power associated with the circuit:
∫ =^ ∫^ ^ ^ ^ =^ ∫^ ^ ^ ^ ^
= ∫ (^ ^ ) ∫^
This gives P = Vrms. Irms.cos . ( Refer above for the integration results)
It shows that power in the circuit depends not only the values of current and voltage, but on the value of ‘cos’ also. Hence it is called the power factor of the circuit.
i.e. power factor, cos = (VR/V) = (IR/IZ) cos =
Thus power can also be expressed as P = Vrms. Irms.
Wattless current :- It is the current in the a.c circuit in which there is a phase difference of ‘/2’ between current and voltage.
We have P = Vrms. Irms.cos Here cos = 0
It can be represented as
Irms Thus by resolving the current into the respective components, we get the
Vrms Wattless current = Irms.sin
RESONANCE:- It is the condition at which the inductive and capacitive reactances are equal so that the current in the circuit is maximum.
We know Im = V/Z Current is maximum, when impedance is minimum.
But Z = √ So ‘Z’ is minimum when
Hence Zmin = R L = 1/C or ^2 = 1/LC
Thus (Im)max = Vm/R Resonant frequency r =√ and (^) √
Frequency response curves with current
The variation of peak current
with different frequencies
can be represented for different values
of resistances as shown beside:
VARIATION OF RESISTANCE, REACTANCE AND IMPEDANCE WITH FREQUENCY
Resistance is independent of the frequency.
Xc = So, Xc α
XL = L XL α
r
POWER AT RESONANCE
We have Power P = Vrms. Irms. But Vrms = (^) √ , Irms = (^) √ and Im = Vm / Z
So P = ------------- (1)
At resonance, Z is minimum. Thus power also is maximum at resonance.
Hence Pmax = Eqn(1) becomes P = Pmax
The frequency response curve with power can be drawn as:
The frequencies at which the power in the circuit is half of
Its maximum value are called ‘ half power points’.
The difference between half power points is called
Bandwidth (2 )
2 = 2 - 1
[For the derivation of the expression for band width, 2 = R/L, refer the note book]
Q-FACTOR (Quality factor)
It measures the sharpness of the resonance curve. It is defined as the ratio of the resonant frequency to the band width.
i.e.^ Using the above equations we get,
Q = (^) √ so , √
