Circuit analysis with a dependent current source

Question

I have been working through Boylestad and I am having trouble with this folling question.

^{simulate this circuit – Schematic created using CircuitLab}

h = 50.

this one i'm finding difficult because the current source depends on another part of this network. I tried applying Kirchhoff's law around the both loops to get loop currents \$I_1\$ and \$I_2\$. $$hI = I_1 - I_2$$

So \$I = I_1\$

Then I got $$hI_1 = I_1 - I_2$$

$$0 = I_1(1-h) - I_2$$

And I used kirchhoff's around the outer loop to get the second equations. $$20 = I_1(2000) + I_2(2000)$$

So I used these two equations using determinants but I didn't get the result in the book which was \$19.62\angle 53^\circ\$

Am I on the right path here or is there a better way to try and get the correct answer?

Now correctly completed with the help of @TimWescott and @relayman357 with the correct equations: $$I_1(1+h)-I_2=0$$ $$I_1(2000) + I_2(2000) = 20\angle 53^\circ$$

Then using determinant to determine \$I_2\$ to then find \$VR2\$.

Answer 1

Your nodal equation is wrong if i'm understanding the directions you assumed for your loop currents. If my figure below is correct, then your KCL should give,

$$i_1 + hi_1 = i_2$$ $$(h+1)i_1 - i_2 = 0$$

You can also try writing a single KVL equation around the entire perimeter - and recognize that the current going up through R2 is \$(h+1)I\$.

Answer 2

The current through R2 is \$(1+h)I\$, so the loop gives \$v(t) = I R_1 + (1+h) I R_2 \$ and so \$ I = {v(t) \over R_1 + (1+h) R_2 } \$, and \$v_{R_2}(t) = {(1+h)R_2\ v(t) \over R_1 + (1+h) R_2 } \$.

Substituting \$h=50, R_1=R_2 = 2k \$ gives \$v_{R_2} \approx 19.6 \angle 53^\circ\$RMS.