Trigonometric Methods, using instantaneous value of current

In summary, the homework statement is asking for the equations for amplitude, period, frequency, and initial phase angle. The first equation is for amplitude, and it is given by: i=15 sin(100π.t+0.6). The second equation is for period, and it is given by: T=2π/ω. The third equation is for frequency, and it is given by: ω=2.π.f. The fourth equation is for initial phase angle, and it is given by: deg⁡〖=rad .180/π〗. Finally, the last equation is for i, and it is given by: i=15sin⁡(100πt+0
  • #1
rikiki
32
0

Homework Statement



The instantaneous value of current, i amp, at t seconds is given by:

i = 15 sin(100π.t + 0.6)

Find the value of;
a) amplitude
b) period
c) frequency
d) initial phase angel
e) value of i when t = 2.5s
f) time when current first reaches maximum value

Homework Equations



i=A*sin(ωτ+∅)

i = A.sin(2pi.f.t + ∅)

The Attempt at a Solution



I understand from looking around these are the correct formula to use, is someone please able to explain how to arrive at the formula from question statement i = 15 sin(100π.t + 0.6)? having some difficulties understanding the math involved. Thanks.
 
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  • #2
compare the formulas one has a 15 and the general formula has an A for amplitude so now you have the first answer

next do a comparison with the omega *t + theta term inside the sin function and match with the actual formula that should answer you other parts.

for initial phase angle consider the case when t=0

This problem is pretty straightforward and is just trying to get you to extract the facts from the formula so review the A * sin (omega *t + theta) formula to understand each parameter.

also you need to show some work before we can help.
 
  • #3
Thanks for your prompt reply, that makes the application of the formula very clear. I'm still struggling to find any literature on where the actual 'general' formula comes from. I don't know if your able to help at all with this? my notes on this seem pretty sparce and can't seem to find anything much on the web. Is there anywhere you can think of that will explain this in a bit more detail. Thanks for your help.
 
  • #5
Perfect. Thanks very much. I'll get going on the answers. Thanks for your help, much appreciated.
 
  • #6
answers to date, i'd be grateful if anybody is able to check they're ok.


2 a) Current=maximum amplitude ×sin (Angular frequency ×time+phase angle)
I_((t) )=Im ×sin⁡(ωt+ θ)
I_((t) )=15 ×〖sin 〗⁡(100πt+ 0.6)
∴ Amplitude = 15
Angular frequency=100π rad/s
Phase angle=0.6 radians

2 b) Period= 2π/(angular frequency)
T= 2π/ω
T= 2π/100π
T= 6.2831853071796/314.1592653589793
P= 0.02s


2 c) Angular frequency=2 × π ×f
ω=2.π.f
100π=2.π.f
314.1592653589793= 6.2831853071796 ×f
f= 314.1592653589793/6.2831853071796
f= 49.9999999999999
f=50Hz



2 d) Phase angle=0.6 radians or 34º
deg⁡〖=rad .180/π〗
deg⁡〖=0.6 × 180/π〗
deg⁡〖=0.6 × 57.2957795130823 〗
deg⁡〖= 34.3774677078494 〗
deg⁡〖=34º〗

2 e) i=15sin⁡(100πt+0.6)
i=15sin⁡(100π2.5+0.6)
i=15 sin⁡(785.9981633974483)
i= 13.7029862899072
i=13.7amps

2 f) I_((t) )=Im ×sin⁡(ωt+ θ)
I_((t) )=15 ×sin⁡(100π.t+ 0.6)
I_((t) )=15 ×sin⁡(314.1592653589793 × t + 0.6)


I'm having a few difficulties trying to find an equation to calculate t. Is there a way of using the phase relationship between voltage and current or is there a more mathematical way of calculating this? I've read about creating a derivative of the function? my maths is pretty poor, so if anyone could offer some here that would be much appreciated. Thanks.

I'm not sure if I've had a mare and need to recalculate some of my answers with my calculator in the radians setting?
 
  • #7
rikiki said:
answers to date, i'd be grateful if anybody is able to check they're ok.

2 e) i=15sin⁡(100πt+0.6)
i=15sin⁡(100π2.5+0.6)
i=15 sin⁡(785.9981633974483)
i= 13.7029862899072
i=13.7amps

2 f) I_((t) )=Im ×sin⁡(ωt+ θ)
I_((t) )=15 ×sin⁡(100π.t+ 0.6)
I_((t) )=15 ×sin⁡(314.1592653589793 × t + 0.6)I'm having a few difficulties trying to find an equation to calculate t. Is there a way of using the phase relationship between voltage and current or is there a more mathematical way of calculating this? I've read about creating a derivative of the function? my maths is pretty poor, so if anyone could offer some here that would be much appreciated. Thanks.

I'm not sure if I've had a mare and need to recalculate some of my answers with my calculator in the radians setting?
Hi,

I think from 2e onwards you need your calculator in the radians setting. This would give you an answer for when t=2.5 of 8.47amps.
Could someone confirm my suspicions?
 

1. What are Trigonometric Methods used for in relation to the instantaneous value of current?

Trigonometric Methods are used to analyze and calculate the instantaneous value of current in alternating current (AC) circuits. These methods use trigonometric functions, such as sine and cosine, to determine the magnitude and phase of the instantaneous current at any given point in time.

2. How do Trigonometric Methods differ from other methods of analyzing current?

Trigonometric Methods are specifically designed for analyzing alternating current, whereas other methods, such as Ohm's Law, are more suitable for direct current (DC) circuits. Trigonometric Methods take into account the changing direction and amplitude of the current, while Ohm's Law assumes a constant current flow.

3. What is the significance of the instantaneous value of current in AC circuits?

The instantaneous value of current is important in AC circuits because it helps us understand the behavior of the current over time. By calculating the instantaneous current, we can determine the maximum and minimum values, as well as the phase shift, which are crucial for designing and analyzing AC circuits.

4. Can Trigonometric Methods be used for all types of circuit analysis?

No, Trigonometric Methods are best suited for analyzing AC circuits. They can also be used for certain types of transient analysis, but they are not as effective for DC circuits. Additionally, Trigonometric Methods are not suitable for analyzing circuits with non-linear components.

5. Are there any limitations to using Trigonometric Methods for instantaneous current analysis?

One limitation of Trigonometric Methods is that they assume a purely sinusoidal current waveform. In reality, many AC circuits have non-sinusoidal waveforms, which can affect the accuracy of the calculations. Additionally, Trigonometric Methods may not be as effective for circuits with complex impedance or multiple frequency components.

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