The voltage across the terminals of a circuit is v(t)= 30+20 cos(60 π t+45^∘) +1

step 1 Hello everyone, let us do the following question. Here in this question we have given the voltag

The voltage across the terminals of a circuit is v(t)= ...

The voltage across the terminals of a circuit is \[\begin{aligned} v(t)=& 30+20 \cos \left(60 \pi t+45^{\circ}\right) \\ &+10 \cos \left(60 \pi t-45^{\circ}\right) \mathrm{V} \end{aligned}\] If the current entering the terminal at higher potential is \[\begin{aligned} i(t)=& 6+4 \cos \left(60 \pi t+10^{\circ}\right) \\ &-2 \cos \left(120 \pi t-60^{\circ}\right) \mathrm{A} \end{aligned}\] find: (a) the rms value of the voltage, (b) the rms value of the current, (c) the average power absorbed by the circuit.

The voltage across the terminals of a circuit is
\[\begin{aligned}
v(t)=& 30+20 \cos \left(60 \pi t+45^{\circ}\right) \\
&+10 \cos \left(60 \pi t-45^{\circ}\right) \mathrm{V}
\end{aligned}\]
If the current entering the terminal at higher potential is
\[\begin{aligned}
i(t)=& 6+4 \cos \left(60 \pi t+10^{\circ}\right) \\
&-2 \cos \left(120 \pi t-60^{\circ}\right) \mathrm{A}
\end{aligned}\]
find:
(a) the rms value of the voltage,
(b) the rms value of the current,
(c) the average power absorbed by the circuit.

Key Concepts

Root Mean Square (RMS) Value

The RMS value is a statistical measure of the magnitude of a varying quantity and is especially useful in analyzing AC circuits. It represents the equivalent DC value that delivers the same power to a load. For signals that include both DC and sinusoidal components, the overall RMS value is computed by summing the squares of the individual components (taking into account any orthogonality between different AC terms) and then taking the square root of the mean value.

Signal Decomposition

In circuit analysis, a complex waveform is often decomposed into its DC (constant offset) and AC (time-varying) components. This separation allows for separate analysis of the continuous component and the oscillatory components using appropriate techniques such as RMS calculation and phasor analysis for sinusoids.

Orthogonality of Sinusoidal Functions

Sinusoidal functions of different frequencies are orthogonal over their common period, meaning their product integrates to zero over that period. This principle simplifies the analysis of signals by allowing the calculation of power or RMS values of composite signals by treating each frequency component independently, without cross-term interference.

Average Power in AC Circuits

The average power absorbed by a circuit is determined by calculating the time average of the product of the instantaneous voltage and current. When the signals contain both DC and AC components, the average power consists of contributions from the DC product, as well as contributions from the AC components that are in phase. Cross terms involving AC components of different frequencies or phases typically average out to zero due to the orthogonality of sinusoidal functions.

Transcript

Please give Ace some feedback