A C CIRCUIT SUMMARY PDF

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PREPARED BY LASISI S.A (PhD in-view) TOPIC: ALTERNATING CURRENT CIRCUIT (A.C. CIRCUIT) DURATION : 4 DAYS WEEK 1 1ST AND 2ND DAY 1 HOUR PER DAY

INTRODUCTION Alternating Current Circuits are circuits through which an Alternating Current flows. Alternating current is the current that flows in one direction during one part is a cycle and in opposite direction during the rest of the cycle. i.e. it varies sinusoidally or periodically. A sinusoidal current or alternating current is described by a function such as I =I 0 sin(wt )

, -----------------1

Where; I is t h e instantaneous current at time “ t ” . I 0 is t h e maximum∨amplitude∨ peak current . ω=2 πf =angular frequency .

f =frequency . Also, the voltage accompanied is given by V =V 0 sin (wt ) -------------- 2 If the current and the voltage are of the same frequency but out of phase, equations 1 and 2 can be written as; I =I 0 sin(wt )

-------3

V =V 0 sin (wt +Φ) ----------4 Where

Φ is t h e p h ase angle.

1. ROOT MEAN SQUARE VALUE Voltages and currents for AC circuits are generally expressed as r.m.s. values. For a sine wave, the relationship between the peak and the r.m.s. average is: r.m.s. value = 0.707 peak value-----------4 I r .m . s=

I0

√2

,

V r . m .s =

V0

√2

,

This is the value of a direct current that would produce the same heating effect as the same heating effect as the alternating current.

2. Resistance in an AC circuit (Resistive Circuit) The relationship V = IR applies for resistors in an AC circuit, so I = V/R = (V o/R) sin(ωt) = Io sin(ωt) In a circuit which only involves resistors, the current and voltage are in phase with each other, which means that the peak voltage is reached at the same instant as peak current.

3. Capacitance in an AC circuit (Capacitive circuit) A capacitor is a device for storing charging. It turns out that there is a 90° phase difference between the current and voltage. The current leads the voltage by 90° in a purely capacitive circuit.

A capacitor in an AC circuit exhibits a kind of resistance called capacitive reactance, measured in ohms. This depends on the frequency of the AC voltage, and is given X c=1/ωC =1/2 πfC -----------------------5 We can use this like a resistance (because, really, it is a resistance) in an equation of the form V = IR to get the voltage across the capacitor: V =IX c ,----------------------------------------------------6  Note that V and I are generally the r.m.s. values of the voltage and current.

4. Inductance in an AC circuit (Inductive Circuit) An inductor is simply a coil of wire (often wrapped around a piece of ferromagnet). This time, however, the current lags the voltage by 90°, so it reaches its peak 1/4 cycle after the voltage peaks. As with the capacitor, the effective resistance of the inductor is known as the inductive reactance. This is given by X L=ωL =2 πfL ----------------------------7

where L is the inductance of the coil (this depends on the geometry of the coil and whether it’s got a ferromagnetic core). The unit of inductance is the henry. The voltage across the inductor is given by: V = IXL-----------------------------------8

5. Energy in AC Circuit One of the main differences between resistors, capacitors, and inductors in AC circuits is in what happens with the electrical energy.  With resistors, power is simply dissipated as heat.  In a capacitor, no energy is lost because the capacitor alternately stores charge and then gives it back again. In this case, energy is stored in the electric field between the capacitor plates. The amount of energy stored in a capacitor is given by energy in a capacitor: Energy = ½ CV2-----------------------9  Energy in an inductor: Energy = ½ LI2---------------------------------10 6. Power in an A.C. Circuit The average power in an a.c. circuit is given by P=IVcosɸ ------------------------------------11 Where; I, V, are the effective (r.m.s.) values of the current and voltage respectively ɸ is the angle of lag or lead between them. Cosɸ is known as the power factor of the device.

NOTE: A power of zero means that the device is a pure reactance, inductance or capacitance. Thus no power is dissipated in an inductance or capacitance. However if I is the r.m.s. value of the current in a circuit containing a resistance, R, then the power absorbed in the resistance is given by P = I2R-----------------------------12 For an a.c. circuit, the instantaneous power is given by P =IV (instantaneous values.)-------------------------------13 Cos ɸ = R/Z = Resistance/Impedance------------------------------14 7. REACTANCE This is the opposition offered to the passage of an alternating current by either the capacitor (capacitive reactance) or the inductor (inductive reactance) or both. it is represented by the symbol X and measured in ohm It is expressed by the equation X =X L – X C∨ X= X C – X L ----------------------15

8. IMPEDANCE In practice AC circuits containing reactance also contain resistance, the two combine to oppose the flow of current. This combined opposition by the resistance and the reactance is called the IMPEDANCE, and is represented by the symbol Z and measured in ohms  LR CIRCUIT: A circuit containing only resistor and inductor in series Z =√ R + X L 2

tan Φ=

2

------------------------16a

XL (Phase angle)----------------16b R

 RC CIRCUIT : A circuit containing only resistor and capacitor in series 2 2 Z =√ R + X C ----------------------17a

tan Φ=

XC R

(Phase angle)--------------------------17b

 RLC CIRCUIT : A circuit containing resistor, inductor and capacitor in series X L− X C ¿ ¿ -----------------18a ¿2 2 R +¿ Z =√ ¿ tan Φ=

X L− X C R

(Phase angle)-------------------------------18b

9. RESONANCE IN RLC SERIES CIRCUIT Resonance is said to occur in A.C. series circuit when the maximum current is obtained from such a circuit. The frequency at which this resonance occurs is called the Resonance Frequency (f0). This is the frequency at which X L=X C ------------------------------------19 Hence solving the above equation we obtain that f0 is given by f 0=

1 2 π √ LC

----------------------20

since ω = 2πf, we can write the condition of resonance as: ω 0=

1 --------------------------21 √ LC

Hence, at resonance, X L=X C

,

R=Z , Current is maximum Impedance is minimum

10. APPLICATION OF RESONANCE The resonant circuit finds applications in electronics. It is used to tune radios and TVs. Its great advantage is that it responds strongly to one particular frequency. The other frequencies are very little effect.

WEEK 2 3RD AND 4TH DAY 1 HOUR PER DAY WORKED EXAMPLES 1. An a.c. voltage of amplitude 2.0 volts is connected to an RLC series circuit. If the resistance in the circuit is 5 ohms, and the inductance and capacitance are 3 mH and 0.05μF respectively, calculate i. the resonance frequency f0 ii. the maximum a.c. current at resonance 2. A generator produces current at a frequency of 60 Hz with peak voltage and current

amplitudes of 100V and 10A, respectively. What is the average power produced if they are in phase? 3. In the circuit below, calculate i. reactance of the capacitor ii. imoedance of the circuit iii. current through the circuit iv voltage across the capacitor v. average power used in the circuit (RC circuit) 4. A source of emf 240v and frequency 50HZ is connected to a series arrangement of a resistor, an inductor and a capacitor. When the current in the capacitor is 10A, the potential difference across the resistor is 140V and that across the inductor is 50V. calculate the i. potential difference across the capacitor ii. capacitance of the capacitor iii. inductance of the inductor (RLC circuit)

5.

A series RLC circuit comprises of a 100ohm resistor, a 3H inductor and a 4µF capacitor. 160 HZ . The A.C source of the circuit has an emf of 100V and a frequency of π i. Draw the circuit diagram of the arrangement . calculate; i. Capacitive reactance ii. inductive reactance iii. impedance of the circuit iv. Current in the circuit v. Average power dissipated in the circuit vi. Power factor 6.

Calculate the inductance L of the coil in the circuit shown below (L circuit)

7. An alternating source of 10V and frequency 100Hz is connected in series with a 5Ω resistor and 20mH. Calculate the i. reactance of the inductor ii. impedance of the circuit iii. potential difference across the inductor 8. Calculate the instantaneous value of such a current, if in a circuit it has r.m.s. value of 15a when its phase angle is 30º 9. A resistor of resistance 50Ω, a capacitor of capacitance 0.1µF and an inductor of inductance 0.1H are connected in series to a 1.50V rms alternating voltage supply. i. Draw the circuit diagram of the illustration ii. calculate the root mean square current iii calculate the resonant frequency...