1. Phasor Relationships for Circuit Elements (7.4)
Dr. Holbert
September 4, 2001
ECE201 Lect-5

2. Phasor Relationships for Circuit Elements
Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.
A complex exponential is the mathematical tool needed to obtain this relationship.
ECE201 Lect-5

3. I-V Relationship for a Resistor+
i(t)
v(t) R –
Suppose that i(t) is a sinusoid:
i(t) = IM ej(wt+q)
Find v(t).
ECE201 Lect-5

4. Computing the Voltage
ECE201 Lect-5

5. Class Example
Learning Extension E7.5
ECE201 Lect-5

6. I-V Relationship for a Capacitor
v(t) + –
i(t)
Suppose that v(t) is a sinusoid:
v(t) = VM ej(wt+q)
Find i(t).
ECE201 Lect-5

7. Computing the Current
ECE201 Lect-5

8. Phasor Relationship Represent v(t) and i(t) as phasors: V = VM  q
I = jwC V
The derivative in the relationship between v(t) and i(t) becomes a multiplication by jw in the relationship between V and I.
ECE201 Lect-5

9. Example v(t) = 120V cos(377t + 30) C = 2mF What is V? What is I?
What is i(t)?
ECE201 Lect-5

10. Class Example
Learning Extension E7.7
ECE201 Lect-5

11. I-V Relationship for an Inductor+
i(t)
v(t) L –
V = jwL I
ECE201 Lect-5

12. Example i(t) = 1mA cos(2p 9.15•107t + 30) L = 1mH What is I?
What is V?
What is v(t)?
ECE201 Lect-5

13. Class Example
Learning Extension E7.6
ECE201 Lect-5

14. Circuit Element Phasor Relations (ELI and ICE man)
ECE201 Lect-5

15. Phasor Diagrams
A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).
A phasor diagram helps to visualize the relationships between currents and voltages.
ECE201 Lect-5

16. An Example
2mA  40 + + VC
1mF – V
w = 377 +
1kW VR – –
ECE201 Lect-5

17. An Example (cont.) I = 2mA  40 VR = 2V  40 VC = 5.31V  -50
ECE201 Lect-5

18. Phasor Diagram
Imaginary Axis
Real Axis V VC VR
ECE201 Lect-5

19. MATLAB Exercise
Let’s use MATLAB to plot an ac current and voltage, and then to graphically determine the lead-lag relationship
Start MATLAB on your computer
We begin by creating a time vector
>> t = 0 : : 0.025;
Next, we create the voltage and current
>> vt = 170 * cos(377*t+10*pi/180);
>> it = 100 * cos(377*t-65*pi/180);
ECE201 Lect-5

20. MATLAB Exercise Now we will graph v(t) and i(t)
>> plot(t,vt,'b',t, it,'r--');
>> xlabel('Time (sec)');
>> ylabel('Voltage (Volts) or Current (Amps)');
>> title('Household AC Voltage-Current');
>> legend('v(t)=170cos(377t+10)', 'i(t)=100cos(377t-65)');
ECE201 Lect-5

21. MATLAB Exercise From the graphs created:
Determine whether the current leads the voltage, or vice versa
Determine the amount of lead by the current or voltage
Compare the voltage-current lead-lag relationship obtained by graphical means above to an analytic solution which you should be able to compute
ECE201 Lect-5