Lecture 3

Fundamentals of Electrical Circuits-2

(AC circuits, resistors in AC circuits, peak and RMS voltage
and current, inductors in AC circuits, capacitors in AC Circuits)

ElecEng2MM3 – Lecture 3 1
 Resistors in AC Circuit
v(t) Vp
Let, v = V p cos(2πft )
⎛ v ⎞ Vp
i = ⎜ ⎟ = cos(2πft ) = I p cos(2πft ) t
⎝ R⎠ R !
P(t) = vi = V p cos(2πft ) ⋅ I p cos(2πft )
i = I p cos(2"ft )
Vp Vp
= V p cos(2πft ) ⋅ cos(2πft ) Ip = !
R R
2
Vp v = V p cos(!2"ft )
= cos2 (2πft )
R P(t)
Pmin = 0
2 !
Vp
Pmax = Pav
R
1 T
Pav = ∫ 0 P(t)dt t
T
ElecEng2MM3 – Lecture 3 2
 AC Circuit Power
1
∫
T
Pav = P(t)dt P(t)
T 0

2
1 Vp Pav
∫0 R (2πft ) dt
T
= cos 2
!
T
t
1+ cos2θ
Since, cos θ =
2
!
2 +ve
2 !
Vp 1 1+ cos(4 πft)
∫
T
Pav = dt !ve
R T 0 2
T
2
Vp 1 dt T
∫ ∫
T T
= {I1 + I2 } Here, I1 = = And, I2 = cos(4 πft)dt = 0
R T 0 2 2 0

2 2 The second integral vanishes since the

Vp 1 T Vp integral of a cosine function over a full
= =
R T 2 2R 2 period is zero.
Vp
Pav =
ElecEng2MM3 – Lecture 3 2R 3
 Peak vs. Root Mean Square (RMS)
Pav can be written in terms of the root mean square (rms) voltage
First, consider the mean square voltage over a period T
2
1 V
( )
T
〈v (t )〉 = ∫0 p π =
2 2 2 p
V cos 2 ft dt
T 2
Vp is the peak value for the amplitude of voltage
The rms voltage is defined as the square-root of the mean square voltage
1
⎡1 ⎤
1
Vp
V cos ( 2π ft ) dt ⎥ =
T 2
= 〈v (t )〉 = ⎢ ∫
2 2 2
Vrms 2
p
⎣T 0
⎦ 2
Or Vp = Vrms ⋅ 2
ElecEng2MM3 – Lecture 3 4
 Peak vs. Root Mean Square (RMS)
2
Vp
Since Pav = , Here, Vp = Vrms ⋅ 2
2R 2
Vrms
Then, the average power can be written as Pav =
R
So, RMS value is defined as the voltage or current of an “equivalent” DC
voltage or current source that generates the same power.
2
Vrms 1 T á v 2 (t )ñ
Prms = = Pac = ò0 dt
R T R 1
Ip
Similarly, the RMS current can be written as I rms = 〈i ( t )〉 =
2 2
2
I p = I rms ⋅ 2
2
Vp I R
Or Pav = I
2
Since I p = , Pav = p
= I 2rms R rms R
R 2
ElecEng2MM3 – Lecture 3 5
 Inductors in AC Circuits

Consider vin = Vp cos(2π ft)

t
di 1
vin = vind =L or i = ∫ vin ( t ) dt
dt L0

t
1
i = ∫ VP ⋅ cos ( 2π ft ) dt
L0

Vp
= sin ( 2π ft )
2π fL

ElecEng2MM3 – Lecture 3 6
 Inductors in AC Circuits
⎛ π⎞
Using the trigonometric relation, cos ⎜ θ − ⎟ = sin θ
⎝ 2⎠
Vp Vp ⎛ π⎞
So, i = ⋅sin ( 2π ft ) = ⋅ cos ⎜ 2π ft − ⎟
2π fL 2π fL ⎝ 2⎠
or Vp is peak voltage and Ip is peak current.
⎛ π⎞ Ip
i = I p ⋅ cos ⎜ 2π ft − ⎟ Vp
⎝ 2⎠
Amplitude:
Vp
Ip =
2π fL
V
This equation for current is similar to the Ohm’s Law, I =
R
Vp Vp
Ip =
2π fL X L
= ∴ X L = 2 π fL XL is the reactance of the inductor, having similar
role as resistance.
ElecEng2MM3 – Lecture 3 7
 Inductors in AC Circuits
Phase:
Let θ ( t ) = 2π ft, v = Vp ⋅ cos (θ ( t ))
⎛ π⎞
i = I p ⋅ cos ⎜ θ ( t ) − ⎟
⎝ 2⎠
For simplicity, let us assume that X L = 1 Ω and I P = Vp .
π
We see that the current is a replica of the voltage except that it is delayed by .
2
π
In an inductor, the current lag the voltage by .
2
π
To draw the current curve, simply delay the voltage curve by radians.
2
π π
Since , θ ( t ) = 2π ft, radians correspond to a delay of ⋅ sec
2 2 ⋅ 2π ⋅ f
ElecEng2MM3 – Lecture 3 8
 Summary: Inductors in AC Circuits

vL = Vp ⋅ cos(2π ft)

VP æ p ö VP æ pö
iL = × cosç2pft - ÷ = × cosç2pft - ÷
2pfL è 2 ø XL è 2ø

In summary, inductor changes the amplitude of the current by 1/XL and it changes the phase
π
by radians with respect to voltage.
2

In contrast, a resistor does not change the phase. It means that the phase of the current remain
the same as that of the voltage in a resistor.
ElecEng2MM3 – Lecture 3 9
 Capacitors in AC Circuits (Homework)

Consider vin = Vp cos(2π ft)

Cdv(t)
\i(t) = = -2pfCVp sin(2pft)
dt
Using the trigonometric relation,
p
-sin(q ) = cos(q + )
2 i v
p
Then, i(t) = -2pfCVp sin(2pft) = 2pfCVp cos(2pft + )
2 "!/2 0 !/2 ! 3!/2 t
Or, i = I p cos(2pft +
p
) Ip is the peak current.
2
ElecEng2MM3 – Lecture 3 10
 Capacitors in AC Circuits
Amplitude:

I p = 2pfCVp

This equation for current is similar to the Ohm’s Law, Vp

Ip i v
Vp Vp 1
Ip = = , XC =
1 XC 2pfC "!/2 0 !/2 ! 3!/2 t
2pfC
V XC is the reactance of the capacitor, having same role as resistance.
I=
R

ElecEng2MM3 – Lecture 3 11
 Capacitors in AC Circuits
Phase:
Let θ ( t ) = 2π ft, v = Vp ⋅ cos (θ ( t ))
i v
æ pö
i(t) = I p cosçq (t) + ÷ "!/2 t
è 2ø 0 !/2 ! 3!/2

For simplicity, let us assume that XC = 1 W and I P = Vp .

π
We see that the current is a replica of the voltage except it is leading by .
2
π
In a capacitor, the current leads the voltage by .
2
π π
Since , θ ( t ) = 2π ft, radians correspond to a lead of ⋅ sec
2 2 ⋅ 2π ⋅ f
ElecEng2MM3 – Lecture 3 12
 Capacitors in AC Circuits
vC = Vp cos(2pft)
Vp æ p ö Vp æ pö
iC = cosç2pft + ÷ = cosç 2pft + ÷
1 è 2 ø XC è 2ø
2pfC
In summary, capacitor changes the amplitude of the current by 1/XC and it
π
changes the phase by radians with respect to voltage.
2
In contrast, a resistor does not change the phase. It means that the phase of the
current remain the same as that of the voltage in a resistor.

ElecEng2MM3 – Lecture 3 13