RMS Value of Time Varying Waveforms MCQ Quiz - Objective Question with Answer for RMS Value of Time Varying Waveforms - Download Free PDF

Explanation:

RMS Value of Rectangular Wave:

The root mean square (RMS) value of a waveform is a measure of the effective value or the equivalent DC value of the waveform. It is particularly useful in electrical engineering as it helps in determining the amount of power delivered by a periodic waveform. For a rectangular wave with alternating positive and negative values, the RMS value can be derived as follows:

Given:

• A rectangular wave with period T.
• Amplitude of the wave is +V for a duration T₁ and -V for the remaining duration T₂ = T - T₁.

Derivation of RMS Value:

The RMS value of a periodic waveform is defined as:

RMS Value = √(1/T ∫₀^T [f(t)]² dt)

For the given rectangular wave:

• During the interval 0 ≤ t ≤ T₁, the value of the waveform is +V.
• During the interval T₁ < t ≤ T, the value of the waveform is -V.

Thus, the RMS value can be calculated as:

RMS Value = √(1/T [∫₀^T₁ (+V)² dt + ∫_T1^T (-V)² dt])

Since (+V)² = (-V)² = V², the expression simplifies to:

RMS Value = √(1/T [∫₀^T₁ V² dt + ∫_T1^T V² dt])

Breaking it into two parts:

RMS Value = √(1/T [V²∫₀^T₁ dt + V²∫_T1^T dt])

The integrals represent the time intervals T₁ and T₂ (where T₂ = T - T₁), so:

RMS Value = √(1/T [V²T₁ + V²T₂])

Substituting T₂ = T - T₁:

RMS Value = √(1/T [V²T₁ + V²(T - T₁)])

Simplifying further:

RMS Value = √(1/T [V²T])

RMS Value = √(V²)

RMS Value = V

Hence, the RMS value of the rectangular wave is V.

Correct Option:

Option 1: V

This option correctly represents the RMS value of the given rectangular wave.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: T₁ - T₂ / T

This option incorrectly assumes that the RMS value depends on the difference between T₁ and T₂. However, the RMS value is derived from the square of the waveform values over the entire period and is independent of the relative durations of T₁ and T₂, as long as the waveform alternates symmetrically.

Option 3: V / √2

This option is incorrect because V / √2 is the RMS value for a sinusoidal waveform, not a rectangular wave. The rectangular wave has a constant amplitude +V and -V, leading to an RMS value equal to the amplitude V.

Option 4: T₁ / T₂

This option is unrelated to the calculation of RMS value. The ratio of T₁ to T₂ does not influence the RMS value of the waveform.

Conclusion:

The RMS value of a rectangular wave with amplitude ±V is equal to the amplitude V, as derived. Understanding the RMS calculation for different waveforms is essential for analyzing their power delivery capabilities in electrical and electronic systems.

Calculation:

The peak value to r.m.s value means the current becomes 1/√2 times the peak value.

So from the equation of current:

i = i₀ sin(100πt)

To change from peak value to r.m.s value, we set:

1/√2 × i₀ = i₀ sin(100πt)

Which simplifies to:

sin(π/4) = sin(100πt)

Thus, solving for t:

t = 1/400 sec = 2.5 × 10^-3 sec

The correct answer is option 4: 2.5 × 10^-2 sec.

Concept

The RMS value of an alternating current is the steady (D.C) current that produces the same amount of heat as the A.C. 

The RMS value of any signal is given by:

\(RMS=\sqrt{{1\over T}\int_{-\infty}^{\infty}x^2(t)\space dt}\)

where, T = Time period of the signal

The relationship between RMS and peak value for an AC supply is given by:

\(V_{RMS}={V_m\over \sqrt{2}}\)

where, Vm = Peak value (maximum value)

Calculation

Given, Vm = 100 V

\(V_{RMS}={100\over \sqrt{2}} \space V\)

Peak Value of AC:

The peak value of an AC (alternating current) is actually higher than the RMS (root mean square) value. The 220 V AC mentioned is the RMS value, but the peak value can be calculated as √2 times the RMS value, which is approximately 311 V. This higher peak value makes the AC more dangerous in comparison to DC (direct current) with the same RMS value.

Concept 

The RMS value of an AC waveform is:

\(RMS={A\over \sqrt{2}}\)

Here, A represents the peak value of the AC waveform

Calculation

The given waveform represents a sine wave whose maximum value is 40 i.e. 40 sinωt.

In this waveform DC value of 60 is added.

\(f(t)=60+40sin\omega t\)

RMS = \(\sqrt{60^2+({40\over \sqrt{2}})^2}\)

RMS = 66.33 V

Note: The closest value to 66.33V is 70V but the answer provided was 57V which was wrong. So we have changed the answer key according to the correct solution.

Concept:

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:

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

  T = Time period

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.
• \(Form\;factor = \frac{{rms\;value}}{{average\;value}}\)

Calculation:

For a rectangular wave, the RMS value is equal to the average value. RMS value is also equal to peak value.

For given waveform RMS value is 100 V.

IMPORTANT EVALUATIONS:

Concept:

• The root mean square current /voltage (rms) of an AC circuit is the effective current/voltage of that circuit.
• The maximum value of the potential in an AC circuit is called the peak value of voltage.
• The average value of the potential and current of an AC circuit for a cycle is called average potential and average current.

\(Average\;current\;over\;half\;cycle\;\left( {{I_{avg}}} \right) = \frac{{2\;I}}{\pi }\)

Average current over complete cycle = 0

\(rms\;current\;\left( {{I_{rms}}} \right) = \frac{I}{{\sqrt 2 }}\)

Where I is maximum/peak current in the circuit.

Calculation:

Given that: i = 50 sin 100 t A

∴ The peak value of current, I = 50 A

Average current over half cycle = Average value of sinusoidal current:

 \(\Rightarrow \frac{{2\;I}}{\pi } = \frac{{2\times 50 }}{\pi }\;A\) 

∴ Average current over half cycle =  \( \frac{{100}}{\pi } ~A\)

Average current over complete cycle = 0 A

Mistake Points 

In the given question AC (Alternating Current) wave is mentioned, which consists of peak values 'I' in the positive half cycle and '- I' in the negative half cycle.

∴  Average current over the complete cycle (positive + negative half cycle) = 0 ampere

That's why the average value of AC wave is calculated over half cycle, which is equivalent to the full-wave rectifier's average value = \( \frac{{2\;I}}{\pi } \)

The average value for a half and full-wave rectifier are:

\({I_{avg}} = \frac{{{I}}}{\pi }\) for half-wave rectifier

\({I_{avg}} = \frac{{2{I}}}{\pi }\) for full-wave rectifier

Note that, half wave rectifier consists of the one-half cycle and inactive for other half cycle. So that its average value is half of the full-wave rectifier's average value.

Where the average value of AC wave is calculated over half cycle is equal to full-wave rectifier's average value.

Concept:

A function f(t) is said to be a periodic function, if f(t ± T) = f(t)

Where T is a time period

The average value of f(t) is given by,

\({f_{avg}} = \frac{1}{T}\mathop \smallint \nolimits_0^T f\left( t \right)dt\)

The RMS value of f(t) is given by,

\({f_{rms}} = \sqrt {\frac{1}{T}\mathop \smallint \nolimits_0^T {{\left( {f\left( t \right)} \right)}^2}} dt\)

Application:

Given that,

x_R and x_A are, respectively, the RMS and average values of x(t) = x(t – T)

y_R and y_A are, respectively, the RMS and average values of y(t) = kx(t)

k, T are independent of t.

The average value of x(t) is,

\({x_A} = \frac{1}{T}\mathop \smallint \nolimits_0^T x\left( t \right)dt\)

The average value of y(t) is,

\({y_A} = {\left( {kx} \right)_A} = \frac{1}{T}\mathop \smallint \nolimits_0^T kx\left( t \right)dt = k{x_A}\)

The RMS value of x(t) is,

\({x_R} = \sqrt {\frac{1}{T}\mathop \smallint \nolimits_0^T {x^2}\left( t \right)dt}\)

The RMS value of y(t) is,

\({y_R} = \sqrt {\frac{1}{T}\mathop \smallint \nolimits_0^T {{\left( {kx\left( t \right)} \right)}^2}dt}\)

\(= k\sqrt {\frac{1}{T}\mathop \smallint \nolimits_0^T {{\left( {x\left( t \right)} \right)}^2}dt} = k{x_R}\)

Power dissipated in the resistor \(P = I_{rms}^2R\)

The given waveform is a triangular waveform.

\({I_{rms}} = \frac{{{I_m}}}{{\sqrt 3 }} = \frac{9}{{\sqrt 3 }} = 3\sqrt 3 \;A\)

Concept: 

RMS value of a sine wave

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

Calculation:

V_rms = 100 V

V_m = 100√2

Vm = 140 V

2Vm = 280 V (Peak to Peak)

Additional Information

Form Factor:

The form factor of an alternating current waveform is the ratio of the RMS value to the average value.

\(Form~Factor=\frac{RMS~Value}{Average~Value}\)

Peak factor

• The peak factor is the peak amplitude of the waveform divided by the RMS value of the waveform.

\(Peak~Factor=\frac{Peak~Amplitude}{RMS~Value}\)

• For square waveform form factor and peak, the factor is the same and the value is one.

Concept:

For a periodic waveform given by x(t) with the fundamental period as ‘T,’ the RMS value is defined as:

\(RMS = \sqrt{{\frac{1}{T}\mathop \smallint \limits_0^T x{{\left( t \right)}^2}dt}}\)

Calculation:

Sequence is repeating at time period of T = 4 sec.

∴ x(t) = ( 50 / 4) t = 12.5 t

Where x(t) is the function of the curve between \(0\leq t \leq T\)

RMS value of waveform, 

 \(X_{RMS} =\sqrt{{\frac{1}{4}\int_{0}^{4} (12.5t)^2 dt }}\)

\(\large{=\sqrt{{(\frac{12.5^2}{4})\frac{t^3}{3}\Biggr|_{0}^{4}}}}\)

\(\large{=\frac{12.5\times 4}{√3}}\)

\(\large{= \frac{50}{√3}}\)

Thus, rms current \(I=\frac{50}{\sqrt 3}\)A

Additional Information

The term RMS ONLY refers to time-varying sinusoidal voltages, currents, or complex waveforms where the magnitude of the waveform changes over time.

RMS value is defined as the “amount of AC power that produces the same heating effect as an equivalent DC power".

Concept:

For a sinusoidal alternating waveform

1. Peak to peak value of voltage (Vp-p) = 2 (Peak value)  ---(1)

2. The peak value of voltage (Vp) = √2 Vrms  ---(2)

3. RMS value of voltage \(\left( {{V_{rms}}} \right) = \dfrac{{{V_p}}}{{\surd 2}}\)

4. The average value of voltage \(\left( {{V_{avg}}} \right) = \dfrac{{2{V_p}}}{\pi }\)

Calculation:

Given:

Vrms = 2 V

From equation (2);

Vp = √2 × 2 V

From equation (1);

Vp-p = 2 × √2 × 2

∴ Vp-p = 5.656 V

Considered a sinusoidal Alternating wave of voltage and current

From the waveform:

\( v = {V_m}\sin \left( {\omega t} \right) \)

And,

\(i = {I_m}\sin \left( {\omega t + ϕ } \right)\)

Where Vm and Im are the maximum value of instantaneous voltage and current respectively.

v, i is the instantaneous value of voltage and current at any instant t.

ω is the angular frequency in radian/second.

And, ω = 2πf  

f is frequency in Hz.

Root-Mean-Square (RMS) Value:

• The RMS value of an alternating current is given by that steady (DC) current which when flowing through a given circuit for a given time produces the same heat as produced by the alternating current when flowing through the same circuit for the same time.
• It is also known as the effective or virtual value of the alternating current, the former term being used more extensively.
• RMS values of an alternating voltage or current represent the real magnitude.
   

RMS value of the alternating quantity is given as,

\(V_{RMS}=\frac{V_m}{\sqrt2}\)

\(I_{RMS}=\frac{I_m}{\sqrt2}\)

Average Value:

The average value of an alternating current is expressed by that steady current which transfers across any circuit the same charge as is transferred by that alternating current during the same time.

The average value of the alternating quantity is given as,

\(V_{avg}=\frac{2V_m}{\pi}\)

\(I_{avg}=\frac{2I_m}{\pi}\)

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 = \(\frac{{Maximum\;Value}}{{1.41}}=0.707× Maximum \ value\)

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

Application:

Given, RMS Value = 100 A

From the above concept,

Maximum value = 1.41 × RMS Value

or, Maximum value = 1.41 × 100 = 141 A

Concept

RMS (Root mean square) value:

• RMS value is based on the heating effect of waveforms.
• 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.

The average or mean value of alternating current is the 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