Symbol for charges:
Q = constant charge
q = time varying charge

Electric Current: The motion of charges constitutes electric current.   i = dq/dt

Franklin's guess:

• Current is the movement of +ve charges.
• Electricity travels from +ve to -ve

• Free electrons are the carrier of current.
• In circuit theory, there is no difference between equal charges moving in opposite direction.
 ------------> +ve charge
----------------------------------------
           <------------ -ve charge

+ve charge = -ve charge 
 

Power: is the rate at which energy is expended.     p = dw/dt=vi

Passive Sign Convention: If the current reference direction arrow points to the positive end of the voltage reference direction, the current and voltage so defined are said to satisfy the passive sign convention.

Independent Voltage Source: is a 2 terminal element that maintains a specified voltage between its terminal regardless of the rest of the circuit. The voltage is completely independent of the current through it. If terminal voltage reference direction and the source function direction are opposite then insert a minus in the element law: v=v_s

Iindependent Current Source: is a two terminal element through which a specified current flows.

Element Law : is an equation involving the terminal voltage/current.

For Circuit Analysis: two things are necessary
1. The elements it contains.
2. How the elements are connected.

Node: is a point of connection of two or more circuit elements.

Kirchoff's Current Law (KCL):
I.   The algebraic sum of the currents entering a node is zero.
II.  The algebraic sum of the currents leaving any node is zero.
III. The sum of currents entering any node equal to the sum of currents leaving the node.

Generalized Form of KCL:
The algebraic sum of currents entering any closed surface is zero.

Kirchoff's Voltage Law (KVL):
I.   The algebraic sum of voltage drops around any closed path is zero.
II.  The algebraic sum of voltage rises along any closed path equal zero.
III.  The sum of voltage drops equals the sum of voltage rises along any closed path

Linear Resistor: obeys ohm's law and has an i-v graph like equation of line.
Non-linear Resistor: have different i-v graphs and laws. In non-linear resistors, the resistance vary with the current flowing through it.

Instantaneous Power for resistors:

                             p =  vi =Ri² = v² / R

Short Circuit: is a resistor of zero ohm resistance. A perfect conductor.
Open Circuit: is a resistor of zero siemens conductance. A perfect insulator.

Equal Sub-Circuits: Two 2-terminal subcircuits are equivalent if they have the same terminal law.

Elements in Series:
By KCL, elements connected in series must all carry the same current.

                ---------/\/\/\-------------/\/\/\---------------/\/\/\-------------
                            R1                 R2                     R3

                Req = R1 + R2 + R3
 

• Two elements are in parallel if they form a loop containing no other element.
• Elements in parallel have the same voltage across them.
• Putting resistors in paralell reduces the overall resistance below that of any of them individually.
• N equal resistors in parallel reduces the resistance by a factor of N

Current Sources in Series: Current sources of different ratings are not connected in series.

Voltage Sources in Series: Voltage: sources can be connected in series. Net voltage is obtained by summing the sources with same polarity and subtracting the total of the sources with the opposite pressure. The net polarity is the polarity of the larger sum.

Voltage Sources in Parallel:Voltage sources can be placed in parallel only if they have the same voltage rating. The reason is to increase the current and therefore the power.

Linear circuit: is any circuit containing nothing ut linear elements and independent sources.

Superposition: The overall response of a circuit containing several sources is the sum of responses to each individual source with other sources killed.

1. Designate sources, current or voltage, as a, b, c, etc.
2. Keep the source a, redraw the circuit without the other sources.
3. Find the V_a.
4. Keep the source b, redraw the circuit without the other sources.
5. Find V_b.
6. To find V, add V_a, V_b, V_c ... together.     V = V_a + V_b

• The use of superposition is limited to calculating currents and voltages in linear circuits. 
• Only independent sources generate components
• Even in linear circuits, power does not superpose, only voltage and current do.
 

1. Assign a distinct current in the clockwise direction to each independent, closed loop of the network. It is not absolutely necessary to choose the clockwise direction for each loop current. Any direction can be chosen for each loop current with no loss in accuracy as long as remaining steps are followed properly.
2. Indicate polarities within each loop for each resistor as determined by the assumed direction loop current for that loop.
3. Apply KVL around each closed loop in the clockwise direction.
4. If a resistor has two or more assumed currents through it, the total current through the resistor is the assumed current of the loop in which KVL is being applied, plus the assumed currents of the other loops passing through in the same direction, minus the assumed currents through in the opposite direction.
5. The polarities of a voltage source is unaffected by the direction of the assigned loop current. 
6. Solve the resulting simultaneous linear equations for the assumed currents.

Nodal Analysis:

1. Determine the umber of nodes within the network.
2. Pick a reference node and label each remaining node with a subscripted value of voltage: V1, V2, and so on.
3. Apply KCL at each node except the reference. Assume all unknown currents leave the node for each application of KCL. In other words, for each node, don't be influenced by the direction an unknown current for another node may have  had . Each node is to be treated as a separate entity, independent of the application of KCL to the other nodes.
4. Solve the resulting equations for the nodal voltages.

Linear Capacitor: is a device whose charge-valtage relationship is a straight line intersecting at the origin..                  q = CV
 

• Abrupt or instantaneous changes in the voltage across capacitor is not possible. 
• The voltage across a capacitor is always continuous even though the current may be discontinuous.
• A capacitor is like an open circuit to a DC voltage.
• Capacitor is a passive element.

Terminal Law of a Capacitor:

q = CV

i = dq/dt = Cdv/dt
 

Capacitors in Series:

                         1/C = 1/C1 + 1/C2...=1/Cn 

If just two capacitors: 
                                             Cs = C1C2 / C1+C2

Capacitors in Parallel:  The equivalent of parallel capacitors is a single capacitor whose capacitance is the sum of capacitance of the parallel capacitors.

                                             Cs = C1 + C2 +...+Cn

Inductor: is a 2 terminal device consists of a coiled conducting wire around a core.

Terminal law of an Inductor:       V = Ldi/dit
 

If either reference direction (but not both) is reversed , a negative sign must be introduced:

                                                  V = - Ldi/dit
 

• The terminal law of inductor shows that if current  i is constant the voltage V is zero.
• As current i increases, a voltage is developed across the terminals of inductor.
• An inductor acts like a short circuit to a DC current.
• The current through an inductor is always continuous.

Inductors in Series: The equivalent of series inductors is a single inductor whose inductance is the sum of individual inductor:

                                        Ls = L1 + L2 +... +Ln
Inductors in  Parallel: The equivalent of parallel inductors is a single inductor whose inverse inductance is the sum of the inverses of the parallel inductance's. 

              1/Lp = 1/L1 + 1/L2 +...+1/Ln

If just two inductors in parallel then :    Lp = L1L2 / L1 + L2
 

• Time constant: Time constant is the time required for the natural response to become zero. In another words,  time constant is the time required for a natural response to decay by a factor of 1/e is defined as time constant of the circuit.
• Natural response: is the response ( v, i) of the circuit that is governed by the circuit elements themselves and not by some independent source forcing a different behavior during t > 0. The natural response is synonymous with the response in the absence of independent sources. 
• In DC steady state, where all currents and voltages are constant, we may analyze the circuit by replacing inductors with short circuit and capacitors by open circuits.

when    v = constant                                   when  i = constant
then      i = Cdv/dt = 0                                then   v = L di/dt = 0

1. Identify the variable X₁ for which the solution is desired.
2. Write one DE in terms of the desired variable X₁ and a second variable X₂.
3. Obtain an additional equation for the second variable in terms of the desired variable X₁ as X₂ = f (X₁).
4. Substitute X₂ = f (X₁) in the equation of step2.
5. If an integral term is included in the equation resulting from step 4, take the derivative of the equation to obtain the second order differential equation.

 
v(t) = vsin(wt)

v = amplitude
w = radian frequency
 

Frequency:       f = 1 / T = W/2pi

Sum of two sinusoid of the same frequency: 

A cos(wt) + B sin(wt) = [A² + B² ]^1/2 cos(wt-theta) 
theta = arctan(B/A)

Rectangular form :        A = a + jb
Polar form:                   A = |A|  /__theta
|A| =                             [A² + B² ]^1/2
theta = arctan(B/A)

Rectangular form :    e^jt = cos(t) + jsin(t)
Polar form:                 e^jt = 1  /__theta

complex exponential form:  A = |A|  e^jt

Source                             Complex Source                                  Source Phasor

Acos(wt+Q)                       Ae^j(wt+Q)                                                A /_ Q
Acost(wt+Q-90)                Ae^j(wt+Q-90)                                            A /_ Q-90

Impedance's:

Resistor:     Zr = R
Inductor:    Z_L = jwL = wL /_ 90
Capacitor:    Z_c = 1/jwC = -j/wC = 1/WC /_-90

Z = R + jX

R = resistive component
X = reactive component

Y = 1/Z
Y = G + jB

G = Re(Y)                B = Im(Y)
 

Caution:

1. Phasors are used to find forced response only.
2. Phasor analysis may be used to find steady-state response with only stable circuits.

Impedance Z(s) of an RLC element is the ratio of V(s) / I(s) = Z(s) when all initial condition are zero.

Resistor:     Zr = R
Inductor:    Z_L = sL
Capacitor:    Z_c = 1/sC 
• s replaces t in the unknown current and voltages.
• Independent source functions replaced by their s-domain transform pair.


V_R(s) = RI_R(s)
V_L(s) = sLI_L(s) - Li_L(0)
V_C(S) = 1/sC I_C(s) + 1/s v_c(0)

H(s) = V_o(s) / V_i(s)           all Initial Conditions = 0

• both currents
• both voltages
• or one of each

To Determine H(s):

• set all initial conditions to zero
• Draw s-domain circuit

Uses of Transformer:
-Electrical isolation. Isolation is useful both for safety and for establishing separate reference voltage levels in primary and secondary circuits.

 
                       ---> i₁                <--- i₂
              _________            _________
                                  |           |
                               + |  *    *  |+
                                  )          (
                          V1   )          (    V2
                                  )          (
                                - |           | -
              _________|           |_________ 

1. Assign current and voltage reference direction that satisfy the passive sign convention.
2. Apply this rule: If both current reference arrows point into the dotted ends or both into the undotted ends of the inductors, use plus sign for both mutual inductance term, otherwise use the minus sign.
 
v₁ = L₁  di₁/dt +- M di₂/dt
v₂ = L₂ di₂/dt +- M di₁/dt