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Laws
Laws
In series circuits:
total resistance (all resistances together): RT = R1 = R2 = R3...
RT is always a larger value than any resistance value
total voltage (all voltages together): VT = VR1 = VR2 = VR3...
supply voltage/source voltage = voltage from a power source to a circuit/device
current is the same in all resistors: I = IT = VT÷RT
E = ET = IT x RT (Ohm’s law from battery point of view)
Kirchoff’s voltage law for series circuit
VT (total voltage) = supply voltage
If resistance increases, voltages is the same, current drops (electricity struggles more to flow).
If resistance drops (decreases), voltages is the same, current increases (electricity flows easier).
If voltage drops, resistance is the same, current drops.
If voltage increases, resistance is the same, current increases.
(irl) Voltage drop is proportional to a load’s RT percentage and tempreature, like many real loads (motors, wires, thermistors).
E.g. Copper resistance increases by 20% (0.4%/°C x 50°C = 20%) new resistance = 12Ω and new voltage = 1A × 12Ω = 12V (20% higher).
Note: "Resistance stays constant" rules is for schematics, voltage drop proportional to RT percentage is irl.
Kirchhoff's voltage law (KVL)/Kirchhoff's 2nd law/The loop rule says VT around a closed loop in a circuit equals zero, reflecting energy conservation.
KVL formula: V1 − V2 − V3 = 0
E.g. Consider a simple DC circuit with a battery and 2 resistors in series:
Battery (V₁): 12 V
Resistor 1 (R₁): 4 Ω (drops V₂)
Resistor 2 (R₂): 2 Ω (drops V₃)
VT drops around a closed loop = 0):
Assume current goes clockwise.
Voltage drop R₁: V₂ = I × R₁ = I × 4 Ω
Voltage drop R₂: V₃ = I × R₂ = I × 2 Ω
KVL equation: 12 V − 4I −2I = 0
Current:
6I = 12
2 A = I
Voltages:
V₂ = 4Ω x 2A = 8 V
V₃ = 2Ω x 2A = 4 V
KVL verification: 12V - 8V - 4V = 0
Notations
Notations
VT = total voltage = input voltage
RT = total resistance
IT = total current
gnd = ground
Example 1 (2 resistors)
Example 1 (2 resistors)
Label battery polarities.
If there's a current path from the battery's positive to its negative terminal.
Recall: Current is the same all along the circuit: IT = IR1 = IR2
Label resistors' polarities. The battery current is limited by both resistors.
Calculate total resistance: RT = R1+R2 = 100+300 = 400 Ω
Calculate total current: IT = IR1 = IR2 = V/RT = 12/400 = 0.03 A = 30 mA
Calculate total power: PT = 0.03x12 = 0.36 W = 360 mW or
PR1 + PR2 = 360 mW
Voltage per load:
VR1 = 0.03x100 = 3 V
VR2 = 0.03x300 = 9 V
Power per load:
PR1 = 0.03x3 = 0.09 W = 90 mW
PR2 = 0.03x9 = 0.27 W = 270 mW
Example 2 (3 resistors)
Example 2 (3 resistors)
Find RT, IT, PT, and, voltage and power per resistor.
Find the path and label the polarities.
RT = R1+R2+R3 = 2 kΩ+4.7 kΩ + 680 Ω = 7.38 kΩ
IT = IR1 = IR2 = IR3 = 2.71 mA
VR1 = IR1xR1 = IT x R1 = 2.71 mA x 2 kΩ = 5.42 V
VR2 = IR2xR2 = IT x R2 = 2.71 mA x 4.7 kΩ = 12.74 V
VR3 = IR3xR3 = IT x R3 = 2.71 mA x 680 Ω = 1.84 V
PR1 = IR1xVR1 = IT x VR1 = 2.71 mA x 5.42 V = 14.69 mW
PR2 = IR2xVR2 = IT x VR2 = 2.71 mA x 12.74 V = 34.53 mW
PR3 = IR3xVR3 = IT x VR3 = 2.71 mA x 1.84 V = 4.99 mW
PT = ITxE = 2.71 mA x 20 V = 54.2 mW
Verify the values
Verify the values
E = IT x RT = 2.71 mA x 7.38 kΩ = 20 V
E = VR1 + VR2 + VR3 = 5.42 V + 12.74 V + 1.84 V = 20 V
PT = PR1 + PR2 + PR3 = 14.69 mW + 34.53 mW + 4.99 mW = 54.21 mW
Example 3 (equal value resistors)
Example 3 (equal value resistors)
RT = R1 + R2 + R3 = 2 kΩ + 2 kΩ + 2 kΩ = 6 kΩ
IT = E/RT = 24/6k = 0.04 A = 4 mA
VR1 = VR2 = VR3 = IT x R1 = 4 mA x 2 kΩ = 8 V
Note: As current is the same in a series circuit, so is each resistor's voltages.
Voltage divider rule
Voltage divider rule
Voltage divider/division rule (VDR) is a form of KVL divides the VT of a series of voltage drops across resistors.
VDR formula:
RT = 2 kΩ + 1 kΩ + 4 kΩ = 7 kΩ
IT = E/RT = 28/7k = 4 mA
VR1 = IT x R1 = 4 mA x 2 kΩ = 8 V
VR2 = IT x R2 = 4 mA x 1 kΩ = 4 V
VR2 = IT x R3 = 4 mA x 4 kΩ = 16 V
See R1 has twice R2's resistance and its voltage is also twice R2.
See R3 four times R2's resistance and its voltage is also four time R2.
A series circuit's supply voltage is divided between resistors in direct proportion to resistors.
VRn = (Rn/RT)E
Example 1
Example 1
Notes: Voltage drops are proportional to a load’s percentage of total resistance.
All circuit's points have the same currents.
Find each circuits' voltages, resistance, current, and their scripts.
VT = 64 + 40 = 104 V
RT= 10 + 47 + 50 + 57 = 164 Ω
P = V²/R = 104²/10 = TBA
I = V/R
IT = 104/164 = 0.63414634146 A
IT = 0.63414634146 x 1000 = 634.15 mA
voltage per resistor:
V10 = 634.15 x 10 = 6.34 V
V47 = 634.15 x 47 = 29.8 V
V50 = 634.15 x 50 = 31.71 V
V57 = 634.15 x 57 = 36.15 V
same circuit from earlier for less scrolling
Single script voltage analysis
Single script voltage analysis
[1] Voltage analysis/Voltage node is a circuit analysis method to find a node's voltage in a circuit.
In electric circuits, a "script" is a code set describing a circuit's interactions, often for advanced circuit simulation software to model complex circuit functions, including how signals propagate and change over time, acting as a "program" for the circuit simulation to follow.
A circuit with more than 1 battery has a different script.
Single scripts are wrt gnd. Point f is at gnd reference.
wrt: with respect to
It's required to add polarity to a circuit, which is easy.
--no matter the direction, it's the same value and polarity.
So is the negative of the battery.
This circuit has 2 ways to reach each resistors.
Va = from ground reference to 'a' = 64 V OR 36.15 (V10) – 40 + 31.71 (V50) + 29.8 (V47) + 6.34 (V10) = 64 V
Vb = 36.15 (V57) – 40 + 31.71 (V50) + 29.8 (V47) = 57.66 V OR 64 – 6.34 = 57.66 V
Vc = 36.15 – 40 + 31.71 = 27.86 V OR 64 – 6.34 – 29.8 = 27.86 V
Vd = 36.15 – 40 = -3.85 V OR 64 – 6.34 – 29.8 – 31.71 = -3.85 V
Ve = from ground reference to 'e' = 36.15 V OR 64 – 6.34 – 29.8 – 31.71 + 40 = 36.15 V
Vf = stays at ground reference = 0 V
Double scripts
same circuit from earlier for less scrollingDouble scripts
Many ways to calculate a script:
E.g. For double script 'Vce',
1. We can use points' voltages (here is 'c' and 'e'):
Vc – Ve = 27.86 (Vc point) – 36.15 (Ve point) = -8.29 V
Note: With points, we can directly substract/add the points without having use points we passed by, such as poind 'd' isn't used even thought it's on the path.
2. We can use the same path as 1#, but with resistors and batteries.
31.71 (V50) - 40 (40V battery) = -8.29 V
3. We can use batteries and resistors in the other way, with different signs:
-40 (40V battery) + 31.71 (V50) = -8.29 V
4. Or opposite way of the circuit:
-36.15 (V57) + 64 (64V battery) – 6.34 (V10) – 29.8 (V47) = -8.29 V
The rest of the circuit's double scripts:
Vbd = Vb – Vd = 57.66 – (-3.85) = 61.51 V
Vcd = Vc – Vd = 27.86 – (-3.85) = 31.71 V
Vae = Va – Ve = 64 – 36.15 = 27.85 V OR -40 + 31.71 + 29.8 + 6.34 OR -36.15 + 64 = 27.85
Vda = Vd – Va = -3.85 – 64 = -67.85 V OR -6.34 – 29.8 – 31.71 OR -64 + 36.15 - 40 = -67.85
Example 3
Example 3
RT = 20 + 30 + 42 + 36 = 128 Ω
IT = 70.31 mA
double scripts voltages:
Vbf = Vb - Vf =
Vae = Va - Ve =
Vdg = Vd - Vg =
Vda = Vd - Va =
Vfa = Vf - Va =
voltage per resistor:
VR1 = 1.41 V
VR2 = 2.53 V
VR3 = 2.95 V
VR4 = 2.11 V
single scripts:
Va = 8 V
Vb = 6.59 V
Vc = 3.59 V
Vd = 641 mV
Ve = 4.64 V
Vf = 2.11 V
Vg = 0 V
Example 4
Example 4
RT = 220 + 570 + 140 = 930 Ω
IT = 24/930 = 0.0258064516 A
IT = 0.0258064516 x 1000 = 25.81 mA
Confirm the circuit's voltage is 24...
V220 = 0.0258 x 220 = 5.68 W
V570 = 0.0258 x 570 = 14.71 W
V140 = 0.0258 x 140 = 3.61 W
or use the voltage divider rule: (Rx/RT) * E
VR1 (220/930) * 24 = 5.68 V
VR2 (570/930) * 24 = 14.71 V
VR3 (140/930) * 24 = 3.61 V
VT = 5.676 + 14.706 + 3.612 = 23.994 W = 24 V--confirmed
P220 = 0.0258² x 220 = 0.146 W
P570 = 0.0258² x 570 = 0.379 W
P140 = 0.0258² x 140 = 0.093 W
PT = 0.146 + 0.379 + 0.093 = 0.618 W or 25.81 x 24 = 619.44 W
power dissipated per load = circuit's total power
Single script
Single script
Label polarities:
Vd = 0
Va = 24 V
Vb = 24 - 5.68 (R1) = 18.32 or the other way around--3.61 + 14.71 = 18.32 V
Vc = 140 V
Double script
Double script
Vda = Vd + Va = 0 + 24 = 24 V
Vab = Va + Vb = 24 - 18.32 = 5.68 V
Vac = Va + Vc = 24 - 5.68 = -766
References
References
[Q2]
[Q3]
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