Passive DC electrical circuits review:

 

Review of Electrical Circuits

Electrical Circuits Lab

HW

Quiz

 

 

Electrical Circuits Review:  This review will begin with an overview of basic electrical elements:

 

Resistors: 

A resistor is a dissipative element that converts electrical energy to heat and follows Ohm’s law as:

V=IR

with R=rL/A, r the resistivity of the material, L and A the length and cross-sectional area of the material.  Typical resistors are used in logic circuits and can handle up to ¼ Watt of power (P=I^2/R). 

 

Equivalent resistance for resistors in series:

R=R1+R2

Resistors in parallel:

R=R1R2/(R1+R2)

 

Capacitors:

A capacitor is a passive element that stores energy in the form of an electric field as a result of a separation of electrical charge.  DC current does not flow through a capacitor.  Instead, charges are transferred from one side of the capacitor to the other through the conducting surrounding circuit, thus causing a displacement current.  This is governed by the equation:

I(t)=C*dV/dt

With C the capacitance in farads.

 

Equivalent capacitance for capacitors in series:

Ceq=C1C2/(C1+C2)

Capacitors in parallel:

Ceq=C1+C2

 

Capacitors come in several types, such as ceramic, electrolytic, etc.

 

 

 

 

 

 

 

 

                           Ceramic                              Electrolytic capacitors

 

Note: Electrolytic capacitors are polarized and have a positive and negative end, and must be placed in the circuit in the proper direction.

 

 

Note: Capacitance values are either printed directly with a label, ex. 70 pF or in a 3-digit code as: X1X2X3 with the capacitance given as: X1X2 10X3 pF. 

Note: Use caution when working with capacitors, they can retain their charge long after power is removed from a circuit (ex, TV).

 

Inductors:

An inductor is an element that stores energy in the form of a magnetic field, and can be created in a simple form as a coil of wire.  The voltage to current relationship is given as,

V(t) = L*dI/dt

with L the inductance measured in Henry’s.  The current is given as,

.

Note from this equation that the current cannot change instantaneously, but increases or decreases over time as a function of the voltage.  This is an important consideration to keep in mind in any circuit that contains an inductive load.  A good example is an electric motor which has a large inductance, and therefore cannot be turned on or off in a very short period of time.  This is also true of relays and solenoids.  This is true even in the simplest of circuits and must be considered in high frequency circuits.  One phenomenon common in switching circuits is called inductive kick and can cause arcing in the switch.  This can be corrected with a diode as shown in the circuit below.

Text Box: Circuit experiencing inductive kick Circuit corrected with a diode
 

 


Equivalent inductance for inductors in series:

Leq=L1+L2

Inductors in parallel:

Leq =L1L2/(L1+L2)

 

Evaluating Circuits:

Circuits are evaluated using Kirchoff’s laws.  Kirkhoff’s voltage law (KVL) states that the sum of the voltages around a closed loop is zero.

 

To apply KVL to a circuit, first assume current directions through every element and assign the appropriate voltage polarity across each element assuming the voltage drops across an element in the direction of the current.  Then, apply KVL in any direction and complete the loop. 

 

Example, Circuit Analysis:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Kirchoff’s current law (KCL) can also be used to evaluate circuits, and states that the sum of the currents flowing into a closed surface or node is zero.

Voltage Divider:

A voltage divider can be used to divide a source voltage to a lower, desired rate.  Consider for example the circuit below powered by a 9 V battery with a 5 V output desired to drive an IC. 

 

 

 

 

 

 

 

 

 

 

Voltage divider

 

The voltage at each location is given as:

VR1=R1Vs/(R1+R2)

VR2=R2Vs/(R1+R2)

 

Voltage dividers are appropriate only in circuits that use a small amount of current, for example when a reference voltage is needed.  For applications requiring more current, a voltage regulator should be used.

 

Alternating Current Circuit Analysis:

A sinusoidal ac voltage V(t) is:

V(t) = V*sin(wt+f)

with V the amplitude, w the frequency and f the phase angle.  The frequency and period of the waveform are related as,

f=1/T=w/2p

 

Steady-state analysis of AC circuits proceeds using phasor analysis, in which the voltage and current through each element in the circuit is described as a complex number.  For example:

Vei(wt+f) = V[cos(wt+f)+sin(wt+f)] = Vx + Vyi

In a steady state condition, the voltage across each element in the circuit will oscillate at the driving frequency, w and will have a constant voltage and phase shift from the input.  Thus, the magnitude and phase shift are variables to be determined.  The impedance of resistors, capacitors and inductors are given as:

ZR = R

Zx = -i/(wC)

ZL = iwL

Note that the inductor will act as a short circuit in a dc circuit and as an open circuit in an ac circuit with a high frequency.  Conversely, the capacitor will act as an open circuit in dc circuit and as a closed circuit in an ac circuit. 

 

Example: AC circuit analysis

 

 

 

 

 

 

 

 

 

 

 

Impedance matching:

When connecting devices or circuits, care must be taken to match the impedance of each device.  For example, consider a function generator driving a higher impedance system as in the figure below.  The 50 W resistor is added in parallel with the high impedance circuit to match impedances.  Without this resistor, frequency elements from the function generator will be reflected backward to the driver.  A good mechanical analogy is a wave transferring from one medium to another and being reflected at sharp boundaries. 

 

Text Box:

 


Designing a circuit to avoid noise interference:

Noise interference in circuits is one of the more common difficulties in designing a robust, working system.  Remember that we are often designing circuits to amplify and observe small voltage signals, or pass digital information with assumed clean edges.  Noise can be generated in a system due to many factors including motors, switches and electromagnetic interference perhaps from AC line voltage. 

 

Here are some suggestions to alleviate noise in a circuit:

  1. Design a single-point ground system, with a common ground bus.  
  2. Isolate and/or separate signal circuits and high-power circuits.  In some cases optoisolators are used. 
  3. Eliminate inductive coupling that can result from a ground loop. 
  4. Shield signal leads with grounded metal covers or shielded cable and or twisted pair cable.
  5. Use short leads to reduce inductive coupling between leads.
  6. Use bypass capacitors over motor leads or other high voltage devices and ground.  This will create an short circuit path for high frequency noise that may exist on power supply lines.