4/27 Friday
Last HW: do not need to turn it. Solution will be posted on the web.
I have all the old homework if you need to collect them.
Final exam:
7-9pm, Monday, 4/30 at Lambert Fieldhouse F101 Calculator OK TA will have regular office hour on Monday (9am-4:30pm) I will have office hour 10am-12pm on Monday
Grading: Total 550 points
HW: 50 points (worst 4 grades dropped) Test 1, 2, 3: each for 100 points Final: 200 points Historical data: average grade of this course is 2.0-2.3
Voltage, Current, Power, Energy
Current i(t) is the rate of charge flowing through
where q(t) is the amount of charge having flown through from time t0 to t
Instantaneous power absorbed by an element is (passive sign convention)
where W(t) is total amount of energy absorbed from time t0 to t
Circuit Elements
Resistance (in )
(ohms law)
Inductance (in H)
Capacitance (in F)
Memoryless
With Memory doesnt jump
With Memory doesnt jump Store energy
Absorb/dissipate but doesnt store energy series parallel
Store energy
series parallel
series parallel
Circuit Elements
Independent voltage source
Independent current source + Dependent voltage source Dependent current source V: fixed I: arbitrary
I: fixed V: arbitrary
V: controlled by another signal I: arbitrary I: controlled by another signal V: arbitrary
Op-amp is a VCVS with a very high gain A
+ -
Basic Circuit Laws
KCL: the total current flowing into (out of) a node is zero
Generalization: the total current flowing into (out of) a Gaussian surface is zero.
KVL: the total voltage drop around a loop is zero
These are the basic circuit laws that apply to any circuit
Voltage division formula
Current division formula
Nodal and Loop (Mesh) Analysis
Both are analysis tools for an arbitrary circuit Together with the Ohms law, one can determine all the currents and voltages in a circuit with independent (dependent) sources and resistors. Sometimes it is easier to apply one of them than applying the other
Nodal Analysis
Variables
Circuit law used Simplified cases Complicated cases
Loop (Mesh) Analysis
Loop currents
Node voltages
KCL
Grounded voltage source Floating voltage source (super node)
KVL
Current source belonging to a single loop
Current source belonging to more than one loop
Analyzing Op-Amp Circuits
Assume vC(0)=0
Always use the ideal op-amp property: V+=V- and i+=i-=0 First find the voltage at one terminal, say, V+ Use V+=V- to find the voltage at the other terminal, say, VApply KCL to + terminal, using the fact that i+=0
Techniques for Simplifying Linear Circuit Analysis
Linearity Principle
In a linear circuit, the response/output is a linear function of all the independent sources
Superposition principle
To get the response, one can set all except one independent source to zero, compute the result, and then add them up Setting an independent voltage source to zero -> short circuit Setting an independent current source to zero -> open circuit
Source transformation
Thevenin and Norton Equivalent Circuits
Linear two-terminal network
Thevenin equivalent circuit
Linear two-terminal network
Norton equivalent circuit
To Compute Equivalent Circuits
Find Voc, the open circuit voltage. Voc=Vs Find Isc, the short circuit current. Isc=Vs/Rth Then set all the independent sources to zero, and the equivalent resistance is Rth A more general way, especially when there are dependent sources (including op-amps) within the two-terminal network, is to compute its v-i relation directly. Apply an arbitrary current I Compute the voltage drop as a function of I (pay attention to the polarity) Compare with V=Vs+IRth
Maximum Power Transfer Theorem
Theorem: for the simple two-terminal network show above in the box connected to a variable load RL, the maximum instantaneous power is transferred to the load when RL=Rth and the maximum instantaneous power is given by
First Order Circuits
Linear network
(resistors, independent and dependent sources)
Linear network
(resistors, independent and dependent sources)
Time constant =L/Rth
Time constant =RthC
First Order Circuits
The currents and voltages in a 1st order circuit are of the form
Elapsed time Time constant Initial value at time t0 Initial value is often given by the condition the switch has been closed (open) for a long time before it is opened (closed) at time t0.
Final value of x
One needs to use the condition that vC and iL doesnt jump to get the initial value after the switching.
For final value, replace the capacitor by open circuit, inductor by close circuit, and compute the final value.
Second Order Circuits
The voltage/current follows an inhomogeneous differential equation: with Its homogeneous version and given
has general solution
(case 1) (case 2) (case 3)
depending on the roots of the characteristic equation Inhomogeneous equation has one particular solution General solution to the inhomogeneous equation
Second Order Circuit
The key is to write the differential equation and/or characteristic equation for the voltage/current of interest
with
and
given
For series/parallel RLC circuit, you can use the formula sheet directly. Sometimes by setting independent sources to zero, you can convert a 2nd-order circuit to series/parallel RLC circuit. Initial value is often given by the condition the switch has been closed (open) for a long time before it is opened (closed) at time t0. One needs to use the condition that vC and iL doesnt jump to get the initial value after the switching. For final value, replace the capacitor by open circuit, inductor by close circuit, and compute the final value.
Sinusoidal Steady State Analysis
When all the excitations of a circuit are sinusoidal with the same frequency , all the steady state responses are sinusoidal with frequency
Phasor is a convenient tool for the SSS analysis of linear circuits consisting of R, L, and C
Phasor of x(t)=cos( t+):
Phasor Ohms Law:
impedance
admittance
Phasor Analysis
Basic circuit laws for phasors: KCL and KVL Ohms law in terms of impedance Nodal and mesh analysis Thevenin and Norton equivalent circuits Frequency response: low/high pass filter band pass/reject filter
Effective Value of Periodic Signals
The effective value (or root-mean square) of a periodic signal f(t) is:
For sinusoidal voltages/currents, the effective value is the peak value divided by the square root of 2 V
Veff
Average power consumed by the element is:
Vm I m cos( v i ) Pav Veff I eff cos(v i ) 2
Effective Phasors
Average power consumed by the element is:
where S is the complex power defined by
Complex power
P pf
v i
P (
1 2 ) 1 pf
Conservation of (complex) power - No conservation law for apparent power
Power factor: (leading or lagging)
Maximum power transfer:
VS2 PL RL 2 2 ( RS RL ) ( X s X L )
The maximal average power that ZL=RL+jXL can absorb is achieved when and And the maximal average power is
(DC version becomes a special case)
