Chapter 7: Q. 106 (page 487)

In an alternating current (ac) circuit, the
instantaneous power p at time t is given by
$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t)$
Show that this is equivalent to
$P (t) = v_{m} i_{m} \sin (ω t) \sin (ω t - ϕ)$

Short Answer

Expert verified

By function $p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t)$ , the function $P (t) = v_{m} i_{m} \sin (ω t) \sin (ω t - ϕ)$ can be obtained by using difference formula for sine function as

$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t) p (t) = v_{m} i_{m} \sin (ω t) (\cos ϕ \sin (ω t) \cos (ω t) - \sin ϕ \cos (ω t)) p (t) = v_{m} i_{m} \sin (ω t) (\sin (ω t - ϕ))$

Step by step solution

Step 1. Given data

The given function is

$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t)$

The function that needs to be proven is

$P (t) = v_{m} i_{m} \sin (ω t) \sin (ω t - ϕ)$

Step 2. proof

Use difference formula for the sine function

$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t) p (t) = v_{m} i_{m} \cos ϕ \sin (ω t) \sin (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t) p (t) = v_{m} i_{m} \sin (ω t) (\cos ϕ \sin (ω t) \cos (ω t) - \sin ϕ \cos (ω t)) p (t) = v_{m} i_{m} \sin (ω t) (\sin (ω t - ϕ))$

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In an alternating current (ac) circuit, the
instantaneous power p at time t is given by
$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t)$
Show that this is equivalent to
$P (t) = v_{m} i_{m} \sin (ω t) \sin (ω t - ϕ)$

Short Answer

Step by step solution

Step 1. Given data

Step 2. proof

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In an alternating current (ac) circuit, theinstantaneous power p at time t is given byp(t)=vmimcosϕsin2(ωt)-vmimsinϕsin(ωt)cos(ωt)Show that this is equivalent toP(t)=vmimsin(ωt)sin(ωt-ϕ)

Short Answer

Step by step solution

Step 1. Given data

Step 2. proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Study anywhere. Anytime. Across all devices.

In an alternating current (ac) circuit, the
instantaneous power p at time t is given by
$p (t) = v_{m} i_{m} \cos ϕ \sin^{2} (ω t) - v_{m} i_{m} \sin ϕ \sin (ω t) \cos (ω t)$
Show that this is equivalent to
$P (t) = v_{m} i_{m} \sin (ω t) \sin (ω t - ϕ)$