RMS Value of Time Varying Waveforms MCQ Quiz in मल्याळम - Objective Question with Answer for RMS Value of Time Varying Waveforms - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Crest Factor ‘or’ Peak Factor is defined as the ratio of the maximum value to the R.M.S value of an alternating quantity.

C.F. ‘or’ P.F. = \(\frac{{Maximum\;Value}}{{R.M.S\;Value}}\)

For a sinusoidal waveform, the value of the crest factor is 1.41.

∴ RMS Value = \(\frac{{Maximum\;Value}}{{1.41}}=0.707\times Maximum \ value\)

Hence, For a sinusoidal waveform, the RMS value of current will be 0.707 times the maximum value of current.

Form Factor:

The form factor is defined as the ratio of the RMS value to the average value of an alternating quantity.

F.F. (Form factor) = \(\frac{{R.M.S\;Value}}{{Average\;Value}}\)

For a sinusoidal waveform, the value of the form factor is 1.11.

:

RMS (Root mean square) value:

• RMS value is based on the heating effect of wave-forms.
• The value at which the heat dissipated in the AC circuit is the same as the heat dissipated in the DC circuit is called the RMS value provided, both the AC and DC circuits have equal value of resistance and are operated at the same time.

• RMS value 'or' the effective value of an alternating quantity is calculated as:

  \({V_{rms}} = \sqrt{\frac{1}{T}\mathop \smallint \limits_0^T {v^2}\left( t \right)dt} \)

  T = Time period

RMS value = V_m sin θ = V_m sin 45° 

= 0.707 V_m 

Note:

• Average or mean value of alternating current is that value of steady current which sends the same amount of charge through the circuit in a certain interval of time as is sent by alternating current through the same circuit in the same interval of time

• The ratio of the maximum value (peak value) to RMS value is known as the peak factor or crest factor.
• \(Peak\;factor = \frac{{maximum\;value}}{{rms\;value}}\)
• The ratio of RMS value to the average value is known as the form factor.
• \(Form\;factor = \frac{{rms\;value}}{{average\;value}}\)

Concept:

A general sinusoidal voltage is expressed as:

v(t) = Vm sin (ωt + ϕ)   ---(1)

Vm = Maximum amplitude of the wave

ϕ = Phase angle

ω = angular frequency, given by:

\(ω = 2πf = \frac{2π}{T}\)

T = Time period of the wave

The rms value of any general expression is calculated as:

\({V_{rms}} = \sqrt{\frac{1}{T}\mathop \smallint \limits_0^T {v^2}\left( t \right)dt} \;\;\)

For the general sinusoidal wave of equation (1), the RMS value is:

\(V_{rms}=\frac{V_m}{√2}\)

Calculation:

V(t) = 200 sin 377t V

Comparing this with the general expression of Equation (1), we get:

Vm = 200

ω = 377

2πf = 377

\(f=\frac{377}{2\pi }\)

= 60 Hz

The time period will be:

\(T=\frac{1}{f}=\frac{1}{60}\)

T = 0.0167 s

The RMS value will be:

\(V_{rms}=\frac{V_m}{√2}=\frac{200}{√2}\)

The correct answer is RMS voltage = 200/√2 V and frequency = 60 Hz

Concept:

For a sinusoidal alternating waveform

• Peak to peak value of voltage (V_p-p) = 2 (Peak value)
• The peak value of voltage (V_p) = √2 V_rms
• RMS value of voltage \(\left( {{V_{rms}}} \right) = \frac{{{V_p}}}{{\surd 2}}\)
• The average value of voltage \(\left( {{V_{avg}}} \right) = \frac{{2{V_p}}}{\pi }\)

Calculation:

V_rms = 500 V

V_p = √2 × 500 V

V_p-p = 2 × √2 × 500

∴ V_p-p = 1414 V

Concept:

To find the average and RMS (root mean square) values of the given AC current, we'll use the following formulas,

\(\begin{align*} I_{avg} &= \frac{1}{T} \int_0^T i(t) dt \\ \end{align*} \)

\(\begin{align*} I_{rms} &= \sqrt{\frac{1}{T} \int_0^T [i(t)]^2 dt} \end{align*} \)

The integral of  \(sin(wt)\) over one complete cycle is zero, resulting in \(I_{avg}=0\) for a DC signal, the average value is equal to the amplitude of the signal.

Calculation:
​​\(i = 10 + 10 sin 314t\)

\(\therefore\ I_{avg} =10+0=10A\)

For a sinusoidal signal, : \(RMS=\frac{A}{\sqrt2}=\frac{10}{\sqrt2}=7.07\)

\(\therefore\ I_{rms} =\sqrt{10^2+{7.07}^2}=12.24 ~A\).

Concept:

RMS value of current for different frequency functions can be determined as

\({I_{rms}} = \sqrt {I_{rms1}^2 + I_{rms2}^2 + I_{rms3}^2 + \; \ldots \ldots \ldots .} \)

Calculation:

Given-

i (t) = 3 + 4 sin (ωt) + 4 sin (2 ωt) A

\({I_{rms1}} = {3}{{}}~A\)

\({i_{rms2}} = \frac{4}{{\surd 2}} A\)

\({i_{rms2}} = \frac{4}{{\surd 2}}A\)

\({I_{rms}} = \sqrt {{{\left( {3} \right)}^2} + {{\left( {\frac{4}{{\sqrt 2 }}} \right)}^2}+{{\left( {\frac{4}{{\sqrt 2 }}} \right)}^2}} \)

\({I_{rms}} = \sqrt {{{25}}} = {5}~A\)

RMS (Root mean square) value:

• RMS value is based on the heating effect of wave-forms.
• The value at which the heat dissipated in AC circuit is the same as the heat dissipated in DC circuit is called RMS value provided, both the AC and DC circuits have equal value of resistance and are operated at the same time.

• RMS value 'or' the effective value of an alternating quantity is calculated as:

V = a0 + a1 sin (ω1t + θ1) + a2 sin (ω2t + θ2) + a3 sin (ω3t + θ3) +...........

Here a0 = DC value = average value of current

RMS value Vrms = \( \sqrt {a_0^2 + \frac{1}{2}\left( {a_1^2 + a_2^2 + a_3^2 + \ldots } \right)} \)

Calculation:

For the given 

u(t) = 3 + 4 cos (3t) 

urms = \( \sqrt {a_0^2 + \frac{1}{2}\left( {a_1^2 + a_2^2 + a_3^2 + \ldots } \right)} \)

Rms value of given voltage is,

\( = \sqrt {9 + {{\left( {\frac{4}{{\sqrt 2 }}} \right)}^2}} = \sqrt {9 + 8} = \sqrt {17} \;V\)

Different Forms of Alternating Voltage:

The standard form of an alternating voltage is given by:

v = V_m sin θ = V_m sin ωt  = V_m sin (2πft) = Vm sin \(\frac{2\pi}{T}\)t

Where,

v is the instantaneous value of voltage
V_m is the maximum value of voltage
(θ = ωt = 2πft = \(\frac{2\pi}{T}\)t) is Co-efficient of sine of time angle

From the above equation following point can be noted:

• The maximum value of alternating voltage is given by the co-efficient of sine of the time angle i.e. Maximum value of voltage = Co-efficient of sine of time angle
• The frequency f of alternating voltage is given by dividing the co-efficient of time in the angle by 2π i.e.,
  f = Co-efficient of time in the angle/2π

Average Value: The average value of any waveform is the ratio of the area under the curve to the time or length of the base (X-axis).

\(A_v=\frac{Area}{t}\)

Method 1: Calculating by using Conventional Method:

Here, total area under one cycle = Area 1 + Area 2

Total Time = 0.6 sec

Area 1 = (15 × 0.2) = 3 Vsec

Area 2 = (-3 × 0.4) = -1.2 Vsec

Total Area = (3 - 1.2) = 1.8 Vsec

Hence, Average value = \(\frac{Area}{t}=\frac{1.8}{0.6}=3\ V\)

Method 2: Calculating by using Method of integration:

Area = \(\int_0^{0.2}15dt+\int_{0.2}^{0.6}-3dt\) = 1.8 Vsed

Hence, Average value = ​\(\frac{Area}{t}=\frac{1.8}{0.6}=3\ V\)

Given 

DC current / Average current = 20 A.

Peak value of sinusoidal current = 20 A

Calcualation:

RMS value of resultant current =  \(\sqrt{I^{2}_{0}\;+I_{1rms}^{2}\;+I_{2rms}^{2}\;+I_{3rms}^{2}.....}\)

I_1rms = \(\frac{20}{\sqrt{2}}\) A.

\({{\rm{I}}_{{\rm{rms \;resultant}}}} = \sqrt {{{20}^2} + {{\left( {\frac{{20}}{{\sqrt 2 }}} \right)}^2}} = \sqrt {400 + 200} = 24.5{\rm{A}}\)