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Delta(Δ)-Wye(Y) Transformations:Δ)-Wye(Y) Transformations:)-Wye(Δ)-Wye(Y) Transformations:Y) Transformations:
Chapter 1 Charge = Coulombs (integral of current, area under the current curve) Current = Coulombs/Second (derivative of Charge, slope of charge, change in charge over time, dq/dt) Voltage = Joules/Coulomb Power: P=IV, P=(V)^2/R, P=R(I)^ Amp-hours = Current * time Watt-hours (Δ)-Wye(Y) Transformations:Energy) = Power*time Watt-hours * $$$ = cost due to electricity Chapter 2 Resistance = ρ(l/A) p = resistivity constant, l=length, A = cross sectional area) p = resistivity constant, l=length, A) p = resistivity constant, l=length, A = cross sectional area = cross sectional area Ohm's Law : V=IR, I=V/R, R=V/I Short Circuit = 0 ohms (Resistance), any current can flow through a short Open Circuit = ∞ ohms (Resistance), no current can flow through an open circuit Branches, nodes, loops: B = L + N - 1 Series : Two Elements share a single node. Same Current for all resistors in series, different Voltages (Voltage Division). Parallel: Two Elements connected to same two nodes. Same Voltage for all resistors in parallel, different Currents (Current Division). Linearity : V/I = R creates a perfectly linear relationship for certain circuits. Kirchhoff's Current Law (Δ)-Wye(Y) Transformations:KCL): Current in=Current out Kirchhoff's Voltage Law (Δ)-Wye(Y) Transformations:KVL): The sum of Voltages around a loop = 0 Voltage Division (Δ)-Wye(Y) Transformations:Series Resistors): Two Resistors are considered in series if they have the same current pass through them. If we have a Vs and two resistors in series, here is the equations:
V1 = i(R1)
V2 = i(R2)
-V + V1 + V2 = 0
-V + i(R1) + i(R2) = 0
The current through all resistors in series is the same, so using Ohm’s law:
V1 = [(R1)/(R1 + R2)]*V
V2 = [(R2)/(R1 + R2)]*V
V = V1 + V2 = i(R1 + R2)
Current Division (Δ)-Wye(Y) Transformations:Parallel Resistors):
Chapter 3 Nodal Analysis : Focuses on current flowing into and out of each node using KCL. Because V=I/R, we are actually going to find the node voltages in the end (v1, v2, v3, etc.). Steps for Nodal Analysis :
- Identify nodes in the circuit (use coloring method if necessary)
- A) p = resistivity constant, l=length, A = cross sectional areapply KCL at each node (except for the ground node)
- Solve the KCL equations using a matrix to find the unknown node voltages. Tips: (^) IF there is a voltage source between two nonreference nodes (v1 and v2), that becomes a supernode. Treat the supernode as 1 node, and write a constraint equation using KCL (current coming in = current flowing out, A) p = resistivity constant, l=length, A = cross sectional areaND voltage amount = v1-v2 , using the + on the Vs as the positive node voltage and - on the Vs as the negative node voltage). (^) If there is a voltage source (Vs) between a nonreference node (V1) and a reference node (Ground), do [(Vs-V1)/resistance] (^) If there is a Voltage Source (Vs) right next to a nonreference node (v1) and it looks like it is the same voltage, it probably is! Do Nodal Analysis If the circuit contains: Many elements in parallel Current Sources Supernodes Circuits with fewer Nodes than meshes If the Node Voltage is what is being solved for Non-Planar Circuits Mesh Analysis: Uses KVL to find unknown "Mesh Currents" (only applies in planar circuits). Steps for Mesh Analysis:
- A) p = resistivity constant, l=length, A = cross sectional areassign mesh currents to meshes
- A) p = resistivity constant, l=length, A = cross sectional areapply KVL to each of the meshes (loops)
- Solve the resulting equations to get the mesh currents Tips: (^) The mesh currents use KVL calculated CLOCKWISE around the mesh/loop. (^) If the mesh current goes WITH the Voltage sorce (from --> (- +) -->), then the voltage source should be negative in the KVL equation. (^) If the mesh current goes AGAINST the Voltage source (from --> (+ -) -->), then the voltage source should be positive in the KVL equation. (^) IF there is a resistor bordering two meshes, make the 1st mesh current you are working on positive and the bordering mesh current negative. (^) If there is a current source isolated to a mesh, that mesh current = the current source. (^) If there is a current source bordering two meshes, that is classified as a SUPERMESH. For the constraint equation, use the mesh currents bordering the current source and their direction to determine which is positive and which is negative. If a mesh current is going the same direction as the current source, then it is positive in the constraint equation. If a mesh current is going against the current source, then it is negative in the constraint equation. (^) A) p = resistivity constant, l=length, A = cross sectional areafter the constraint equation is written for the SUPERMESH, you then REMOVE (create an open circuit) the wire containing the current source and resistor it is attached to. Then, apply KVL on the remaining Mesh, using I1 and I2 accordingly (whatever resistors applied to I1 before the supermesh branch was removed, still applies to I1, and vice versa for I2). Do Mesh Analysis if the circuit contains: Many elements in series Voltage sources Supermeshes A) p = resistivity constant, l=length, A = cross sectional area circuit with fewer meshes than nodes If a branch/mesh current is what is being solved for Chapter 4 Linearity: A) p = resistivity constant, l=length, A = cross sectional areas Voltage goes up, Current goes up proportionally. The response of a circuit to a sum of sources will be the sum of the individual responses from each source separately. Linearity: (^) V=iR (^) k(iR) = k(V) (^) V= (i1 + i2)R = (i1)R + (i2)R = V1 + V Superposition : If there are two or more independent sources there are Three ways to solve for the circuit parameters: Nodal A) p = resistivity constant, l=length, A = cross sectional areanalysis, Mesh A) p = resistivity constant, l=length, A = cross sectional areanalysis, Superposition (^) Superposition Principle states that the voltage/Current through an element in a linear circuit is the total sum of the voltages/currents through that element due to each independent source acting alone. Steps for Superposition:
- Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using techniques in Chapter 2/
- Repeat step 1 for each of the other independent sources
- Find the total contribution (voltage or current) by adding all the contributions (voltages or currents) due to each independent source.
Chapter 5 Concepts (^) 5 Terminals found on all Op-amps: a. Inverting input b. Non-inverting input c. Output d. Positive and negative power supplies (^) Output voltage of an op-amp can be found using the following equation: Vo = A) p = resistivity constant, l=length, A = cross sectional areaVd = A) p = resistivity constant, l=length, A = cross sectional area(V+-V-) a. A) p = resistivity constant, l=length, A = cross sectional area is the Open Loop Gain, which is different from the closed loop gain. o (^) Feedback: The output of the op-amp is fed back to the inverting terminal, giving the op-amp "Negative Feedback". o Voltage Saturation: The output voltage of the op-amp cannot exceed the input voltages. Therefore, when an output voltage should exceed the possible voltage range, the output remains at either the minimum or maximum supply voltage. o Ideal Op-Amps: a. Infinite Open Loop Gain (A) p = resistivity constant, l=length, A = cross sectional area) b. Infinite Input Resistance (Ri) c. Zero output Resistance (Ro) d. Zero input current to the inverting/noninverting terminals (Io) e. Output current is NOT zero Input voltages on the inverting/noninverting terminals are equal Inverting Op-amp (ideal) (^) Ideal Op-A) p = resistivity constant, l=length, A = cross sectional areamp Rules apply (^) Vo = (-Rf/R1)Vi Equivalent circuit: Non-inverting Op-Amp (Δ)-Wye(Y) Transformations:Ideal) (^) Ideal Op-A) p = resistivity constant, l=length, A = cross sectional areamp Rules apply (^) Vo = Vi * [1 + (Rf/R1)] Voltage Follower (Δ)-Wye(Y) Transformations:Ideal, non-inverting op-amp) Vi = Vo Summing Op-Amp Uses the inverting amplifier and several inputs (each with their own resistor), the summing amplifier can be used to create a simple digital to analog converter (DA) p = resistivity constant, l=length, A = cross sectional areaC). I1 = [(V1-Va)/R1] I2 = [(V2-Va)/R2] I3 = [(V3-Va)/R3] Ia = I1 + I2 + I Ia = [(Va-Vo)/Rf] Vo = - [(Rf/R1)V1 + (Rf/R2)V2 + (Rf/R3)V3] Difference Amplifier -Vo is proportional to the difference between the two inputs. -Va = Vb due to negative feedback. Common Mode Rejection: A) p = resistivity constant, l=length, A = cross sectional area difference amplifier rejects any signal that is common to the two inputs, following this equation: Instrumentation Amplifier -Places 2 non-inverting amplifiers before the difference amplifier to increase the impedance of the difference amplifier
If this condition is true for a difference op-amp, the output is then: Chapter 6 Concepts Capacitors o Capacitor Formulas: (^) Capacitance Formula: C=(Δ)-Wye(Y) Transformations:EA)/d* (^) E = permittivity of dialectric (^) A) p = resistivity constant, l=length, A = cross sectional area = cross sectional area of plates (^) D = distance between plates (^) Charge stored in a capacitor is equal to the capacitance * Voltage: q = CV* (^) Take the derivative of this equation to find current in a capacitor: I = C(Δ)-Wye(Y) Transformations:dv/dt)* (^) This is also the "ICE" part of the "ELI THE ICE MA) p = resistivity constant, l=length, A = cross sectional areaN" acronym (^) To find voltage in a capacitor due to current: o (^) Capacitor Properties: (^) When the voltage is NOT changing (constant), the current through the capacitor is zero (open circuit at DC conditions). (^) Voltage on the capacitor plates CA) p = resistivity constant, l=length, A = cross sectional areaN NOT change instantaneously. (^) If the voltage on the capacitor does not equal the applied voltage, charge will flow until the cap reaches the applied voltage. (^) Parallel Capacitors: A) p = resistivity constant, l=length, A = cross sectional areact like series resistors, or like conductance values. C1 + C2 + C3 = Ceq (^) Series Capacitors: A) p = resistivity constant, l=length, A = cross sectional areact like Parallel Resistors, or like conductance values. [(1/C1) + (1/C2) + (1/C3)]^-1 = Ceq Inductors o (^) Inductor formulas: (^) For a solenoid, the inductance formula is: L=(N^2uA) p = resistivity constant, l=length, A = cross sectional area)/l (^) U = permeability of the core material (^) N = number of turns of the wire (^) A) p = resistivity constant, l=length, A = cross sectional area = cross sectional area (^) L = length (^) If current is passed through an inductor, the voltage across it is directly proportional to the rate of change of current flowing through it: V = L(Δ)-Wye(Y) Transformations:di/dt)* (^) This is the "ELI" part of the "ELI THE ICE MA) p = resistivity constant, l=length, A = cross sectional areaN" acronym (^) The current stored in an inductor due to voltage: o (^) Inductor Properties: (^) If the current across an inductor is NOT changing (constant), the voltage across an inductor is zero (short circuit at DC conditions). (^) The current through an inductor CA) p = resistivity constant, l=length, A = cross sectional areaN NOT change instantaneously. (^) Series Inductors: Just like series resistors. L1 + L2 + L3 = Leq (^) Parallel Inductors: Just like parallel resistors. [(1/L1) + (1/L2) + (1/L3)]^-1 = Leq Combining Op-amps, Capacitors, and inductors: Integrator Differentiator
(^) The key to working with this type of situation is: (^) Start with the initial voltage across the capacitor and the time constant RC. (^) With these two items, the voltage as a function of time can be known. (^) From the voltage, the current can be known by using the resistance and Ohm’s law. (^) The resistance of the circuit is often the Thevenin equivalent resistance. Singularity Functions: Unit Step Function The switching time can be shifted to t=to by: Unit Impulse Function The derivative of the unit step function Unit Ramp Function Integration of the Unit Step Function (^) Current cannot change instantaneously (^) We are looking for the current through the inductor (^) Therefore, we must determine its value as a function of time (^) Initial current passing through the inductor at t=0: (^) I(0) = 0 (^) A) p = resistivity constant, l=length, A = cross sectional areas the inductor begins to release energy into the system, the natural response starts to occur: (^) Time constant: o (^) Step Response of RL Circuits (^) Steady-state current through an inductor: (^) Current cannot change instantaneously through an inductor: (^) i(0+) = I(0-) = Io (^) Complete response of current through an inductor: Chapter 8 Concepts Second-order Circuits (RLC circuits) o (^) Start by getting initial conditions I(Δ)-Wye(Y) Transformations:0) and dI(Δ)-Wye(Y) Transformations:0)/dt (for parallel RLC circuits, V(Δ)-Wye(Y) Transformations:0) and dV(Δ)-Wye(Y) Transformations:0)/dt ): (^) Capacitor: Open Circuit at long-term conditions. Voltage cannot change abruptly (^) V(0-) = V(0+) = Vo (^) Inductor: Short Circuit at long-term conditions. Current cannot change abruptly. (^) i(0-) = I(0+) = Io o (^) Figure out which damping case the circuit will require, solve characteristic eqn. for roots, apply them in correct formula, solve for I or V: (^) Series RLC Circuits: Parallel RLC Circuits: (^) A) p = resistivity constant, l=length, A = cross sectional areall the same equations as series RLC, except: (^) Change I(t) --> V(t) (^) Vs = V(∞) (^) Change: Damping Responses:
(the roots are real and negative) Damped (roots are real and equal) (roots are complex) o (^) Damping factor, Undampened Natural Frequency, damped natural Frequency (shown below in order) o (^) Solutions to characteristic equation: o (^) A) p = resistivity constant, l=length, A = cross sectional arealso can be found using quadratic formula: o Underdamped case roots: Overdamped Critically Damped Underdamp ed Chapter 9 (^) Sinusoidal Voltage: V(t) = Vm sin (ωt t ± ɸ) o (^) The function repeats every period, or every T seconds o T = 2π/ωt o (^) Frequency (Hertz): f = 1/T o (^) A) p = resistivity constant, l=length, A = cross sectional areangular frequency: ωt = 2π f (^) Complex numbers: o (^) Rectangular form: z = x + jy o Polar form: Z = r<ɸ o (^) Exponential form: Z = rejɸ (^) Rectangular to polar: (^) r = sqrt(x^2 +y^2 ) (^) ɸ = tan-1(y/x) (^) Polar to rectangular: Chapter 10 (^) Steps to analyze A) p = resistivity constant, l=length, A = cross sectional areaC Circuits: o (^) Transform the circuit to the phasor or frequency domain o (^) Solve the problem using circuit techniques o (^) Transform back to time domain. (^) Perform the following as in DC: o (^) Nodal A) p = resistivity constant, l=length, A = cross sectional areanalysis o (^) Mesh A) p = resistivity constant, l=length, A = cross sectional areanalysis o (^) Superposition o (^) Source Transformation o (^) Thevenin and Norton Equivalents o (^) Op A) p = resistivity constant, l=length, A = cross sectional areamp A) p = resistivity constant, l=length, A = cross sectional areanalysis
o (^) Resistive: When ɸv - ɸi = 0 ° , the voltage and current are in phase and the circuit is purely resistive (^) P = (1/2)VmIm = (1/2)Im^2 R = (1/2)|I|^2 R o (^) Reactive: When ɸv - ɸi = ± 90 ° , the circuit absorbs no power and is purely reactive (^) P = (1/2)VmImCos(± 90 ° ) = 0 (^) Maximum A) p = resistivity constant, l=length, A = cross sectional areaverage Power o (^) Pmax = (|VTh|^2 )/(8RTh) (^) RMS o (^) for a sinusoidal waveform, the RMS value is related to the amplitude as follows: o (^) RMS Power can be determines from either RMS current or voltage: (^) P = (IRMS)^2 R = (VRMS)^2 /R (^) Complex Power o (^) S = (1/2)VI* (^) I* = complex conjugate of current o (^) S = P + jQ
P = Real power --> S cos ( θv − θi )
jQ = Reactive power --> S sin ( θv − θi )
o (^) S = VRMSIRMS* o (^) S = |VRMS||IRMS|<(ɸv - ɸi) (^) Apparent power o (^) The product of RMS voltage and current will be called apparent power.
|S| = |VRMS||IRMS| = √ P^2 + Q^2
(^) Power Factor o (^) P/S = cos(ɸv - ɸi) --> cos-1(P/S) = Power Factor Ratio (between 0 and 1) (^) P = Real Power (^) S = A) p = resistivity constant, l=length, A = cross sectional areapparent power (^) Real Power (P) = A) p = resistivity constant, l=length, A = cross sectional areapparent Power (S) * Power Factor (PF) (^) Power Factor = cos(ɸv - ɸi) (^) A) p = resistivity constant, l=length, A = cross sectional areadding a capacitor o (^) To mitigate the inductive aspect of the load, a capacitor is added in parallel with the load. o (^) With the same supplied voltage, the current draw is less by adding the capacitor.
o C =
Qc
ωVV rms
2 =^
P ( tan θ 1 −tan θ 2 )
ωVV rms
2
