# Electric circuits ?

Any help with this circuit?

I want to prove that the total power dissipated equals the total power developed. Relevance
• Notice that Vo = 3io*80 = 240io. 4Vo = 960io

Let the current leaving the 100V source = i. KVL cw around the left loop 100-10i-20(i-io)-960io = 0

-940io = 30i - 100

io = (-3i +10)/94

KVL around center loop:

+960io+20i-20io-40io+120-240io = 0

660io = -20i -120

io = -(i+6)/33

Solve for i by equating  (-3i +10)/94 = -(i+6)/33

i(-3/94+1/33) = -10/94-6/33

i = (-10/94-6/33)/(-3/94+1/33) = 178.8A out of 100V source

io = (-3*178.8+10)/94 = -5.6A or +5.6A into 120V source

Power delivered by sources:

100*178.8+184.4*5376-120*5.6 - 2*5.6* (-1344) = 102.4MW

Power consumed by resistance

178.8²*10+184.4²*20+5.6²*40+16.8²*80 = 102.4MW checks.

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• The current through the 80 ohm resistor is 3(Io)

Vo = 80(3Io) =(240)Io

Let I1 be the current through the 100 volt source and let it flow upwards

and I2 be the current in the dependent 4Vo source also flowing upward

Io = I1 + I2  therefore I1 = Io - I2

100 volts - 10I1 = 4Vo - 20I2

100 volts = 4Vo +10I1 - 20I2

100 volts = 4 Vo +10(Io -I2) - 20I2

100 volts = 4(240Io) + 10Io - 30I2

100 volts = 970Io - 30I2

10 volts = 97Io -3I2     call this equation 1

100 volts -10I1 -40Io +120 volts = Vo = 240(io)

220 volts = 280(Io) +10(Io -I2)

220 volts = 290(Io) - 10I2

22 volts = 29Io - I2  call this equation 2

Using both equations 1 and 2 solve for Io and I2

Io = -5.6 amps

I2 = -184.4 amps

I1 = -5.6 -(-184.4) = 178.8 amps

Vo = 3(Io)80= 240(-5.6)= -1,344 volts

The 80 ohm resistor dissipates 22,579.2 watts  (= 3(-5.6)( -1,344))

The 40 ohm resistor dissipates 1,254.4 watts  ( =( Io^2)(40))

The 10 ohm resistor dissipates 319,694 watts   (= (I1^2)(10))

And the 20 ohm resistor dissipates 680,067 watts  ( =(I2^2)(20))

The 100 volt source  provides  17,880 watts  ( = 100 volts x 178.8 amps)

The 120 volt source dissipates 672 watts since the current Io is flowing into the source. (-5.6 x 120)

The 2 Io dependent current source  provides 15,053 watts  (= 2(-5.6 amps) x - 1,344 volts

The 4 Vo dependent source provides 991.334 watts

( = 4Vo x I2) which is  4(-1,344 volts)(-184.4 amps)

double check the math

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