EBN 111
𝑞 𝑣
Current: 𝑖 = or 𝑖=
𝑡 𝑅
𝑤 If current enters through
Voltage: 𝑉 = or 𝑉 = 𝑖𝑅
𝑞 -ve then p=-vi
𝑤 𝑣2
Power: 𝑝 = or 𝑝 = 𝑣𝑖 or 𝑝 = 𝑖2𝑅 or 𝑝=
𝑡 𝑅
𝑡
Charge: 𝑄 ≜ ∫𝑡 𝑖 𝑑𝑡
0
𝑡 𝑡
Energy: 𝑤 = ∫𝑡 𝑝 𝑑𝑡 = ∫𝑡 𝑣𝑖 𝑑𝑡
0 0
𝑣𝑖
Cost: 𝑐𝑜𝑠𝑡 = × (𝑡𝑖𝑚𝑒 𝑖𝑛 ℎ𝑟𝑠) × 𝑐𝑜𝑠𝑡 𝑖𝑛 𝑟𝑎𝑛𝑑𝑠 𝑝𝑒𝑟 ℎ𝑟
1000
ℓ 𝑣
Resistance: 𝑅 = 𝜌 or 𝑅=
𝐴 𝑖
1 𝑖
Conductance: 𝐺 = or 𝐺=
𝑅 𝑣
Nodes, Branches and Loops: 𝑏 = 𝑙 + 𝑛 − 1
KCL: 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑠 𝑖𝑛 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑠 𝑜𝑢𝑡
KVL: 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑑𝑟𝑜𝑝𝑠 = 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑟𝑖𝑠𝑒𝑠
Resistors in series: 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + ⋯ + 𝑅𝑁
𝑅1 𝑅2
𝑣1 = 𝑣 𝑣2 = 𝑣
𝑅1 +𝑅2 𝑅1 +𝑅2
𝑅1 𝑅2
Resistors in parallel: 𝑅𝑒𝑞 = only for 2 resistors in //
𝑅1 +𝑅2
1 1 1 1
= + + ⋯+
𝑅𝑒𝑞 𝑅1 𝑅2 𝑅𝑁
𝑅2 𝑖 𝑅1 𝑖
𝑖1 = 𝑖2 =
𝑅1 +𝑅2 𝑅1 +𝑅2
𝑅𝑏 𝑅𝑐 𝑅𝑎 𝑅𝑐 𝑅𝑎 𝑅𝑏
Delta to Wye: 𝑅1 = 𝑅2 = 𝑅3 =
𝑅𝑎 +𝑅𝑏 +𝑅𝑐 𝑅𝑎 +𝑅𝑏 +𝑅𝑐 𝑅𝑎 +𝑅𝑏 +𝑅𝑐
𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1 𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1
Wye to Delta: 𝑅𝑎 = 𝑅𝑏 =
𝑅1 𝑅2
𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1
𝑅𝑐 =
𝑅3
, Nodal Analysis: Use KCL to find voltage of nodes.
Steps:
1. Select a node as a reference node.
2. Assign voltages 𝑣1, 𝑣2 , … 𝑖1 𝑣1 𝑖2
3. Point currents going out of the nodes.
4. Current sources flowing toward the node will be 𝑖3
negative in the equation and current sources flowing
away from the node are positive.
5. If there is a voltage source in between the reference and a node then
the nodes voltage is equivalent to the source.
𝑣
6. Create an equation at each node using ohms law by saying 𝑖 = , 𝑣 =
𝑅
𝑛𝑜𝑑𝑒 𝑦𝑜𝑢 𝑎𝑟𝑒 𝑎𝑡 − 𝑡ℎ𝑒 𝑛𝑜𝑑𝑒 𝑦𝑜𝑢 𝑎𝑟𝑒 𝑔𝑜𝑖𝑛𝑔 𝑡𝑜
7. If there is a supernode where a voltage source is in between two nodes
then switch off the source and use KCL.
8. Use simultaneous equations to solve for node voltages.
Mesh Analysis: Use KVL to find currents in loops.
1. Current always flows in the clockwise direction inside the loop.
2. If a current source flows in the opposite direction
to 𝑖1 then it is negative and vise versa. The current
source is the current in the loop. 𝑖1
3. When resistors are inbetween two loops then in the
equation you will say
𝑅(𝑙𝑜𝑜𝑝 𝐼 ′ 𝑚 𝑖𝑛 − 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑙𝑜𝑜𝑝 = 𝑅(𝑖1 − 𝑖2 )
4. When writing the equations use the formula 𝑣 = 𝑖𝑅
5. If there is a supermesh where there is a current source being shared
between two meshes then you switch off the current source and assume
meshes 1 and 2 is one single mesh(BUT still use the different mesh
currents).
6. Create a simultaneous and solve.
𝑞 𝑣
Current: 𝑖 = or 𝑖=
𝑡 𝑅
𝑤 If current enters through
Voltage: 𝑉 = or 𝑉 = 𝑖𝑅
𝑞 -ve then p=-vi
𝑤 𝑣2
Power: 𝑝 = or 𝑝 = 𝑣𝑖 or 𝑝 = 𝑖2𝑅 or 𝑝=
𝑡 𝑅
𝑡
Charge: 𝑄 ≜ ∫𝑡 𝑖 𝑑𝑡
0
𝑡 𝑡
Energy: 𝑤 = ∫𝑡 𝑝 𝑑𝑡 = ∫𝑡 𝑣𝑖 𝑑𝑡
0 0
𝑣𝑖
Cost: 𝑐𝑜𝑠𝑡 = × (𝑡𝑖𝑚𝑒 𝑖𝑛 ℎ𝑟𝑠) × 𝑐𝑜𝑠𝑡 𝑖𝑛 𝑟𝑎𝑛𝑑𝑠 𝑝𝑒𝑟 ℎ𝑟
1000
ℓ 𝑣
Resistance: 𝑅 = 𝜌 or 𝑅=
𝐴 𝑖
1 𝑖
Conductance: 𝐺 = or 𝐺=
𝑅 𝑣
Nodes, Branches and Loops: 𝑏 = 𝑙 + 𝑛 − 1
KCL: 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑠 𝑖𝑛 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑠 𝑜𝑢𝑡
KVL: 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑑𝑟𝑜𝑝𝑠 = 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑟𝑖𝑠𝑒𝑠
Resistors in series: 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + ⋯ + 𝑅𝑁
𝑅1 𝑅2
𝑣1 = 𝑣 𝑣2 = 𝑣
𝑅1 +𝑅2 𝑅1 +𝑅2
𝑅1 𝑅2
Resistors in parallel: 𝑅𝑒𝑞 = only for 2 resistors in //
𝑅1 +𝑅2
1 1 1 1
= + + ⋯+
𝑅𝑒𝑞 𝑅1 𝑅2 𝑅𝑁
𝑅2 𝑖 𝑅1 𝑖
𝑖1 = 𝑖2 =
𝑅1 +𝑅2 𝑅1 +𝑅2
𝑅𝑏 𝑅𝑐 𝑅𝑎 𝑅𝑐 𝑅𝑎 𝑅𝑏
Delta to Wye: 𝑅1 = 𝑅2 = 𝑅3 =
𝑅𝑎 +𝑅𝑏 +𝑅𝑐 𝑅𝑎 +𝑅𝑏 +𝑅𝑐 𝑅𝑎 +𝑅𝑏 +𝑅𝑐
𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1 𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1
Wye to Delta: 𝑅𝑎 = 𝑅𝑏 =
𝑅1 𝑅2
𝑅1 𝑅2 +𝑅2 𝑅3 +𝑅3 𝑅1
𝑅𝑐 =
𝑅3
, Nodal Analysis: Use KCL to find voltage of nodes.
Steps:
1. Select a node as a reference node.
2. Assign voltages 𝑣1, 𝑣2 , … 𝑖1 𝑣1 𝑖2
3. Point currents going out of the nodes.
4. Current sources flowing toward the node will be 𝑖3
negative in the equation and current sources flowing
away from the node are positive.
5. If there is a voltage source in between the reference and a node then
the nodes voltage is equivalent to the source.
𝑣
6. Create an equation at each node using ohms law by saying 𝑖 = , 𝑣 =
𝑅
𝑛𝑜𝑑𝑒 𝑦𝑜𝑢 𝑎𝑟𝑒 𝑎𝑡 − 𝑡ℎ𝑒 𝑛𝑜𝑑𝑒 𝑦𝑜𝑢 𝑎𝑟𝑒 𝑔𝑜𝑖𝑛𝑔 𝑡𝑜
7. If there is a supernode where a voltage source is in between two nodes
then switch off the source and use KCL.
8. Use simultaneous equations to solve for node voltages.
Mesh Analysis: Use KVL to find currents in loops.
1. Current always flows in the clockwise direction inside the loop.
2. If a current source flows in the opposite direction
to 𝑖1 then it is negative and vise versa. The current
source is the current in the loop. 𝑖1
3. When resistors are inbetween two loops then in the
equation you will say
𝑅(𝑙𝑜𝑜𝑝 𝐼 ′ 𝑚 𝑖𝑛 − 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑙𝑜𝑜𝑝 = 𝑅(𝑖1 − 𝑖2 )
4. When writing the equations use the formula 𝑣 = 𝑖𝑅
5. If there is a supermesh where there is a current source being shared
between two meshes then you switch off the current source and assume
meshes 1 and 2 is one single mesh(BUT still use the different mesh
currents).
6. Create a simultaneous and solve.
