Question

The instantaneous voltage across a circuit element is V(t)=500sin(wt + 30^o)and the instantaneous current entering the positive terminal of the circuit element is i(t)=100cos(wt +10^o)A..For both the current and voltage, determine (a) maximum value, (b) the rms value, and (c) the phasor expression, using as the reference value.

Short answer

(a) The maximum value for the voltage is 400 V and for the current is 100 A .

(b) The rms value for the voltage is 282.84 V and for the current is 70.71 A.

(c)The phasor expression for the voltage is 282.84$<$-60^oV and for the current is 70.71$<$10^oA .

Step-by-step solution

Write the given data of the question

The instantaneous value for the voltage is given as V(t)=400 sin( wt + 30^O)V.

The instantaneous value for the current is given as I(t)=100 cos(wt +10^O)A. .

Write the concept for polar form and rectangular form;

If x (t) =Acoswt is taken as the reference for the phasor diagram.

The polar form or phasor expression of y (t)=Acos (wt-0) is,

y (t)= A/√2 ∠ =θ ........(1)

Here,-0 means that Y (t) lags by 0 angle, is the maximum or peak value of and is the rms value of y (t) and A/√2 is the rms value of the (t).

_$\sqrt{}$

Calculate maximum value of the voltage and current.

(a)

The instantaneous voltage expression is,

v(t) =400 sin (wt + 30^o) V........(2)

Now as $\mathrm{\theta }$=cos (90-$\mathrm{\theta }$)

Use this concept in the above equation (2).

v(t) =400 cos (wt + 30^o-90^o)

=400 cos (wt -60^o)

Using the concept of the equation (1), the maximum value for the voltage is 400 V.

The instantaneous current expression;

i (t) =100 cos (wt+10^o)A …… (3)

Since the current expression is already in cosine form.

Using the concept of the equation (1), the maximum value for the current is 100 A .

Therefore, the maximum value for the voltage is 400V and for the current is 100A .

Calculate the rms value for the voltage and current;

(b)

From the equation (2), the expression for the voltage isv(t) =400 sin (wt + 30^o) V .

Now using the concept of equation (1)

The rms value for the voltage is,

v_{rms =400/√2}

_{=282.84 V}

From the equation (3), the expression for the current is,

i (t) =100 cos (wt+10^o)A

The rms value of the current is,

i_{rms = 100/√2}

_{=70.71 A}

Therefore, the rms value for the voltage is 282.84 Vand for the current is 70.71A.

Calculate the phasor expression for the voltage and current.

(c)

From the equation (2), the expression for the voltage is v(t) =400 sin (wt + 30^o)V

Now using the concept of equation (1), the phasor expression for the voltage is,

v_{rms =400/√2} ∠ -60^o

=282.84 V∠ -60^o

From the equation (3), the expression for the current is,

i (t) =100 cos (wt+10^o)A

Now using the concept of equation (1), the phasor expression for the current is,

i_{= 100/√2} ∠ 10^o

=70.71 A

Therefore the phasor expression for the voltage is and for the current is 282.84 ∠ -60^o V .and for the current is 70.71 ∠ 10^oA