DC Circuits PDF

# Current Resistor ## First page - **Current Resistor** - $I_1 - I_2$ - $I_1 + I_2$ - $-I_2$ - **Date:** - **No:** - $0.0855 - (-0.1418) = 6.22734A$ -  - **V** = $I_2 (I_1 - I_2)$ - = $12 (2.72 - 2.06)$ - **V** = $7.92V$ - **V** = $I_2 - I_3 = 2....

# Current Resistor ## First page - **Current Resistor** - $I_1 - I_2$ - $I_1 + I_2$ - $-I_2$ - **Date:** - **No:** - $0.0855 - (-0.1418) = 6.22734A$ -  - **V** = $I_2 (I_1 - I_2)$ - = $12 (2.72 - 2.06)$ - **V** = $7.92V$ - **V** = $I_2 - I_3 = 2.06 - 4.11 = -2.05A$ - **For Loop 1** - $-12 + 2I_1 + 12I_1 - 12I_2 + I_1I_3 = 0$ - $15I_1 - 12I_2 - I_3 = 12$ (Equation 1) - **For Loop 2** - $10 + 3I_2 - 3I_3 + 12I_2 - 12I_1 +2I_2 = 0$ - $-12I_1 + 7I_2 - 3I_3 = -10$ (Equation 2) - **For Loop 3** - $-24 + I_1I_3 - I_1 + 3I_2 - 3I_2 + 4I_3 = 0$ - $-I_1 - 3I_2 + 8I_3 = 24$ (Equation 3) - **LONG METHOD** - **Use** Equation 1 & 2 - $(15I_1 -12I_2 - I_3 = 12)3$ - $(-12I_1 + 17I_2 - 3I_3 = -10)$ - $45I_1 - 36I_2 - 3I_3 = 36$ - $-12I_1 + 17I_2 - 3I_3 = -10$ - $57I_1 - 53I_2 - 40 = 0 (4)$ - **NEXT PAGE** - **VECO** ## Second Page - **KCL** - $I_1 + I_3 = I_2$ - $2 + 5 = 12$ - $I_2 = 7A$ - **PAGE continuation** - $57I_1 - 53I_2 = 46$ - $-99I_1 + 127I_2 = 8$ - $I_1 = 53I_2 + 46 = 53(2.06) + 46 = 155.2$ - $I_1 = 155.18 / 57 = 2.72A$ - $15I_1 - 12I_2 - I_3 = 12$ - $I_3 = 4.08A$ ## Third page - **Series Parallel w/ KVL** -  - $R_{56} = 8+ 2 = 10\Omega$ - $R4||R56 = 8\Omega$ - $R3456 = 3+ 5 = 8\Omega$ - $R2||83456 = 4\Omega$ - $R_T = R_1 + R_{3456}$ - $I_T = 6V/6\Omega = 4A$ - $I_T = I_{23456} = 4(4)$ - $V_{23456} = 16V = V_2 = V_{3456}$ -  - **Loop 1** - $-24 + 2I_1 + 8I_1 - 8I_2 = 0$ - $10I_1 - 8I_2 = 24$ (Equation 1) - **Loop 2** - $8I_2 - 8I_1 + 3I_2 + 10I_2 - 10I_3 =0$ - $-8I_1 + 21I_2 - 10I_3 = 0$ (Equation 2) -**Loop 3** - $8I_3 + 10I_3 - 10I_2 + 2I_3 = 0$ - $-10I_2 + 20I_3 = 0$ -**NODAL ANALYSIS** -  - **node a** - $V_1/R_1 + V_1-V_2/R_2 = I$ (Equation 1) - **node b** - $V_a - V_1/R_3 + V_a/R_4$ ## Fourth Page - $I =2A, R_1 = 2\Omega, R_2 = 4\Omega, R_3 = 5\Omega, R_4 = 5\Omega$ - **Find** $V_1$ & $V_2$ - $V_1 - V_2 + V_2 -0 = 2A$ - $1/(2+4)V_1 + 1/5 V_2 = 2$ - $V_1 = 6.16V$ - $V_1 - V_1 + V_a + V_a =0$ - $(-1/5) V_1 + (-1/5)V_2 + V_a = 0$ - $V_2 = 1.3V$ -  - **Find** $V_1$ & $V_a$ - **Node a** - $V_1/R_1 + V_1-V_2/R_2 = (1/10 + 1/20) V_1 - 1/20 V_a = 2A$ - **Node b** - $V_a-V_1/R_3 + V_a/R_4 = 4/7V$ ## Fifth Page -  - **Node a** - $V_1/R_1 + V_1-V_3/R_2 = (1/10 + 1/20) V_1 - 1/20 V_2 = 2$ - **Node b** - $V_2-V_1/R_3 + V_2/R_4 = (1/5 + 1/20) V_2 - 1/20 V_1 = -6$ - $V_1 + 2V_2 - 20 = 20$ (Equation 1) - $4V_2 - 4V_1 + V_2 = -120$ - $-4V_1 + 5V_2 = -120$ (Equation 2) - **SUPERNODE ANALYSIS** -  - **Find** $V_1,V_2, V_3$ - **Node a** - $V_1 - V_3 / R_1 + V_1 - V_2 / R_2 = (1/10 + 1/20) V_1 - (1/20)V_2 - (1/10) V_3 = 2$ - **supernode** - $V_3 - V_1 / R_1 + V_3 - V_2 / R_3 + V_3 - V_1 / R_4 = (1/10 + 1/10 + 1/20)V_3 - (1/10)V_1 - (1/20)V_2 = 20$ - **Constraint Equation** - $V_a + 10 - V_3 = 0$ - $V_a - V_2 - 10 = 0$ - $5V_1 - 8V_3 + 2V_1 - 2V_2 = 60$ - $7V_1 - 2V_2 - 5V_3 = 60$ (Equation 1) - $4V_2 - 4V_1 + 5V_2 + 2V_3 + 10V_3 - 10V_1 = 0$ - $-14V_1 + 9V_2 + 12V_3 = 0$ (Equation 2) - $V_2 - V_3 = -10$ (Equation 3) ## Sixth Page -  - **Find** $V_1$ & $V_2$ - **Node a** - $V_1 - 10 / R_1 + V_1-V_2/R_2 = (1/10 + 1/5) V_1 - (1/5)V_2 = 2$ - $V_1 - 10 + (2/5)V_1 + 5V_1 - 5V_2 = 20$ - $8V_1 - 5V_2 = 30$ (Equation 1) - **Node b** - $V_2-V_1/R_1 + V_2-5(V_1-V_2)/R_2 = (1/2) V_2 - (1/2) V_1 + V_2 - 5 (V_1-V_2)/2 = (7/2)V_2 - (11/2)V_1 = -8 $ - $2V_1 + V_2 - 5(V_1 - V_2) = -8 $ - $-7V_1 + 8V_2 = -8$ (Equation 2) - $V_1 = 6.97V$ - $V_2 = 5.03V$ ## Seventh page -  - $V_2 = R_2 (V / R_1 + R_2)$ - $V_2 = 3(3/3+3)$ - $V_2 = 3V$ - $I_2 = 4(0.5/4+12)$ - $I_2 = 0.125A$ - $I_2 = 12 (0.125/12+13)$ - $I_2 = 0.0625A$ - $V_2 = 0.375V$ - $I = 60(17/60+40)$ - $I = 6.21A$ ## Eighth page -  - **Find** $V_D$ at $80\Omega$ Resistor - $V_{DP} - V_{DN} = 80(I_2)$ - $V_{DP} = 80(120/180+80)$ - $V_{DP} = 92A$ - **OR** - $R_T= 60 + 120 + 80 = 260$ - $I_T = 120 / 260 = 0.46A$ - $R_1 = R_3 = 20(5/20+5) = 4$ - $R_{12} = R_{23} = 4 (6.66/4+6.66) = 4 + 6.66$ - $I_T = 2.49(5/2.49 + 5)$ - **Find** $V_T$ - $V_T = I_T R = 5(20) = 100V$ - $V_T = I_T R = 100/7$ - $V = 2.857$ - **I** = $20 (5/3+5) = -2.857A$ ## Ninth Page - **SUPERPOSITION** -  - Eliminate the other source - **Voltage source:** short circuit - **Current source:** open circuit -  - **By Nodal** - **Node Vi** - $(V_1 - 20)/5 + V_1/10 + V_1/20 = 0$ - $6V_1 - 120 +3V_1 + V_1 = 0$ - $10V_1 = 120$ - $V_1 = 12V$ - **Find current by eliminating voltage** -  - $5(10)/10 + 10/20 = 40/20$ - $5+10/30 = 40/30$ - (1/3) + (1/3) = (4/3) - **PARALLEL SERIES** -  - **Node Vi** - $V_1/R_1 + V_1-V_2/R_2 = (1/10 + 1/20)V_1 - 1/20 V_2 = 0$ - $4V_1 - V_2 = 0$ ## Tenth page -  - $V_1/R_1 + V_2-V_1/R_2 = (1/20)V_1 + (1/5)V_2 - (1/20)V_1 = 20$ - $-2V_1 + V_2 + 2V_2 = 80$ - $-2V_1 + 3V_2 = 80$ - $I = 80 / 20 = 1.6A$ - **Wye - Delta Transformation** -  - **Wye to Delta** - $R_a = R_1R_2 + R_2R_3 + R_3R_1 / R_1$ - $R_b = R_1R_2 + R_2R_3 +R_3R_1 / R_2$ - $R_c = R_1R_2 + R_2R_3 + R_3R_1 / R_2$ - **Example 1** - $R_a = 20(40) + 40(10) + 10(20) / 20 = 70 \Omega$ - $R_b = 35\Omega$ - $R_c = 140\Omega$ ## Eleventh page -  - **Delta to Wye** - $R_1 = R_aRbRc / R_b + Rc + Ra$ - $R_2 = R_aRc / R_c$ - $R_3 = R_aRb / Rb$ - $R_1 = 70(140) / 70 + 140 + 35 = 20\Omega$ - $R_2 = 70(140) / 140 = 40\Omega$ - $R_3 = 70(35) / 35 = 10\Omega$ -  - $R_a = 10(50) + 50(30) + 30(10)/30 = 230\Omega$ - $R_b = 50/30 =46\Omega$ - $R_c = 76.7\Omega$ -  - **Find** $R_T$ - $R_T = 39.89 + 76.7 = 39.24\Omega$ ## Twelfth page -  - **Find** $R_T$ - $R_{12} = 18(6) / 18+6+12 = 3.2\Omega$ - $R_{23} = 12(18) / 12 + 18 + 6 = 2.9\Omega$ - $R_3 = 18(6) / 18 = 6\Omega$ - $R_T = 6+ 8.4 = 14.4 \Omega$ ## Thirteenth page -  - **Voltages** - $V_1= 3.64V$ - $V_2= 1.44V$ - $V_3 = 1.332V$ - $V_4 = 0.108V$ - $V_5 = 0.108V$ - **Current** - $I_1 = 0.36A$ - $I_2 = 6.29A$ - $I_3 = 0.07A$ - $I_4 = 0.07A$ - $I_5= 0.02A$ - $I_{345} = V_{345}/R_{345} = 1.44 / 20 = 0.07A$ - $I_{345} = I_{34} = I_{45}$ - $V_3 = I_3R_3 = 0.07(18) = 1.332V$ - $I_2 = V_2 / R_2 = 1.44 / 5 = 0.29A$ - $V_{345} = V_3 + V_{45}$ - $V_{45} = V_{345} - V_3 = 1.44 - 1.332 = 0.108V$ - $V_{45} = V_4 = V_5$ - $I_4 = V_4 / R_4 = 0.108 / 6 = 0.07 A$ - $I_5 = V_5 / R_5 = 0.108 / 6 = 0.02A$ - $R_T = R_{2345} + R_1 = 4 + 10 = 14\Omega$ - $I_{2345} = I_T$ - $V_1 = I_1R_1 = 0.36(10) = 3.64V$ - $V_{2345} = I_{2345}R_{2345} = 0.36(4) = 1.44V$ - $V_{2345} = V_2 = V_{345}$