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5.2 Steady state electric circuit analysis
analysis have been carried out by fixing the number of turns in primary and secondary side. Other prefixed parameters in this study are coil shape, dimensions of the coil and circuit topology. The variable parameters are air gap distance, operating frequency and load. Skin effect is not considered because both coils are wound using multi-strand copper wire. Electric equivalent circuit models of different compensation topologies are developed to explain the mechanism of power transfer. The performance of four compensation topologies are validated and its results are analyzed. The analysis compares the efficiency of four basic topologies of CPT system with variation in frequency, distance and load.
The rest of the chapter is organized as follows: Section 5.2 describes the steady state electric circuit analysis. Section 5.3 presents the description of experimental set-up. Finally, experimental results are presented in Section 5.4 and its conclusions are given in Section 5.5.
5.2 Steady state electric circuit analysis
series-parallel as SP topology, parallel-series as PS topology and parallel-parallel as PP topology. In Figure 5.3,I1,IL1,IC1 denotes supply current, inductor current and capacitor current of the primary side andIL2,IC2 and IL represents inductor current, capacitor current and load current of secondary side respectively. Similarly, L1,C1 and V1 denotes inductor, capacitor and supply voltage in the pri- mary side and L2, C2 and RL denotes inductor, capacitor and load resistance in the secondary side respectively. The main criteria to increase the power transfer capability in all compensation is that
RL
C1
V1
C2
I1 IC1
IL IL1
M L2
IC2
L1
IL2
(a)
RL C1
V1
IC1
M
C2
L1 L2
IL
IL1 IL2 IC2
I1
(b)
RL
V1
IL
M IC1
C1
C2
IC2
IL1 IL2
L1 L2
I1
(c)
RL
V1
IL
IL1 IL2
M IC1
C1 C2
IC2
L1 L2
I1
(d)
Figure 5.3: Equivalent circuit model of contactless coils (a) SS topology (b) SP topology (c) PS topology (d) PP topology.
the primary side of the system should operate at secondary side resonance frequency. When operating at secondary resonance frequency, the self inductance of the secondary winding is fully compensated by the primary compensation capacitance. Therefore, the impedance of the secondary as seen by the primary is purely resistive in nature. Thus, these capacitors essentially store and supply reactive power to and from the secondary and primary windings, reducing the amount of reactive power drawn from the supply.
5.2.2 Mutual inductance coupling model
The coupling between the primary and secondary coil and its operation can be analyzed using several modeling methods. Among various modeling methodologies, transformer model and mutual inductance coupling model are the most commonly used methods. The transformer model uses the concept of transformed voltage and reflected current to describe the coupling effects. The transformed voltage and reflected currents are simply defined by the turns ratio. Here, the coupling and leakage TH-1345_TPERJOY
inductance must be separated by circuit analysis. This coupling model is well suited for closely coupled CPT systems because the leakage inductance is usually negligible. In contrast, mutual inductance coupling model uses the concept of induced and reflected voltages to describe the coupling effect between the primary and secondary networks. Both the induced and reflected voltages are expressed in terms of the mutual inductance. This model does not require the coupling and leakage inductance to be separated for circuit analysis. Hence this model is well-suited for loosely coupled CPT systems, where the leakage inductance is too large to be ignored. In this chapter, mutual inductance model is used to analyze the coupling between the primary and secondary coils of CPT systems. The effect of the secondary must be considered together with the primary winding in the analysis of the primary network. Therefore, the effect of secondary can be represented by the equivalent reflected impedance.
Compensation elements are required to compensate leakage inductance of the coil. The compensation elements can be connected in series or parallel on primary and secondary sides of the coil. For simplification purpose, neglecting coil resistances, the analysis of four compensation topologies using mutual inductance coupling models are presented in the following section.
5.2.3 Series-series (SS) compensation topology
Assuming all currents are sinusoidal, the steady state equations for primary and secondary series compensation can be written on phasor form as given by (5.1) and (5.2)
V1 =I1 1
jωC1
+jωL1I1−jωM I2 (5.1)
jωM I1 =jωL2I2+I2 1
jωC2
+I2RL (5.2)
where mutual inductance of the coil (M) is given by (5.3). From (5.1)-(5.2) it is observed, both the induced voltage and reflected voltage are represented in terms of mutual inductance (M) between the coils to describe the coupling effect between the primary and secondary network. The mutual inductance can be related with the magnetic coupling (k) as given by (5.3).
k= M
√L1L2 (5.3)
The magnetic coupling of contactless coil is usually poor. In (5.3), the coupling between the primary and secondary of the coil depends on the leakage inductance (L1, L2) and magnetizing inductance (M). The leakage inductance in contactless coil is much larger than the magnetizing inductance. On
5.2 Steady state electric circuit analysis
simplifying (5.2), currentI2 in series secondary coil can be written as (5.4).
I2= jωM I1 hjωL2+
1 jωC2
+RLi (5.4)
On substituting secondary side currentI2, given in (5.4) into (5.1), we get:
V1 =I1
1 jωC1
+jωL1I1+
ω2M2 jωL2+
1 jωC2
+RL
I1 (5.5)
It can be observed from (5.5), from the point of view of the first coil, the secondary coil is seen as a transformed impedance or reflected impedance (Zr) of the secondary network as seen from the primary side. Hence, for series compensated secondary:
Zr SS= ω2M2 h
jωL2+
1 jωC2
+RL
i = ω2M2
Z2S (5.6)
where, ZsS is the secondary side impedance. The total impedance (Zt) for series and parallel com- pensated system seen by the power supply is obtained from (5.5) is given by (5.7).
Zt SS = 1
jωC1
+jωL1+ ω2M2 jωL2+
1 jωC2
+RL
(5.7)
Now the simplified version of contactless system is shown in Figure 5.4(a) and Figure 5.4(b) Therefore,
V1
Zr
C1 I1
L1
(a)
I1
V1 Zt
(b)
Figure 5.4: Simplified equivalent circuit (a) with reflected impedance (b) with total impedance.
the current observed from the source is given by (5.8).
I1 = V1 Zt SS
(5.8)
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In order to obtain high power transfer capability, the operating frequency of the system should be equal to the secondary resonant frequency (ωo), which is given by (5.9)
ωo= 1
√L2C2 (5.9)
On substituting (5.9) into (5.6), we get the reflected impedance as given by (5.10).
Zr SS = ωo2M2 RL
= ω2oM2
ZsS (5.10)
It can be observed from (5.10), if the secondary is series compensated Zr SS is purely resistive. Thus to obtain high efficiency, the primary side of the system should operate at resonance frequency of the secondary side of the system. Therefore for SS compensation, C1 and C2 can be given by (5.11).
C1= 1
ωo2L1 ; C2 = 1
ωo2L2 (5.11)
Then, the total impedance of the system is given by (5.12).
Zt SS = ωo2M2 RL
(5.12) On substituting (5.11) into (5.2), the relation between I2 and I1 can be obtained, where I2 leads I1 at some angle.
I2
I1 = jωoM
RL (5.13)
I2 =G1I1; G1 = jωoM
RL (5.14)
Using (5.1) - (5.14), the input power (Pi), output power (Po) and total efficiency of the system are calculated.
Pi=I12Zt SS (5.15)
Po =I22RL=G21I12RL (5.16) Using these equations, efficiency for SS compensation is given by (5.17).
ηSS = G21RL
ZtSS (5.17)
5.2 Steady state electric circuit analysis
5.2.4 Series-parallel (SP) compensation topology
The steady state equations for primary series and secondary parallel compensation can be written on phasor form as given by (5.18) and (5.19)
V1 =I1
1 jωC1
+jωL1I1−jωM I2 (5.18)
jωM I1 =jωL2I2+I2
RL
1 jωC2
RL+
1 jωC2
(5.19)
Similar to the procedure explained in Section 5.2.3, simplifying (5.19), currentI2 in parallel secondary coil can be written as (5.20).
I2= jωM I1
"
jωL2+ RL
1
jωC2
RL+
1 jωC2
!# (5.20)
By substituting secondary side current I2, given in (5.20) into (5.18), we get V1 =I1
1 jωC1
+jωL1I1+
ω2M2 jωL2+
RL
RLjωC2+1
I1 (5.21)
It can be observed from (5.21), from the point of view of the first coil, the secondary side parallel compensated coil is seen as a transformed impedance or reflected impedance (Zr SP) of the secondary network as seen from the primary side. Therefore,Zr SP and Zt SP is given by (5.22) and (5.23).
Zr SP = ω2M2 hjωL2+ R RL
LjωC2+1
i = ω2M2
Z2P (5.22)
Zt SP = 1 jωC1
+jωL1+ ω2M2 jωL2+
RL
jωC2RL+1
(5.23)
The current observed from the source is given by (5.24) I1 SP = V1 Zt SP
(5.24) In order to obtain high power transfer capability, the operating frequency of the power supply should be equal to the secondary resonant frequency (ωo) and the secondary side compensation capacitor C2 is given by (5.25)
C2 = 1
ωo2L2 (5.25)
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When operating the CPT system at this frequency, the self inductance of the secondary winding is fully compensated by the primary compensation capacitor and therefore the impedance of the secondary as seen by the primary is purely resistive in nature. Therefore, on substituting (5.25) into (5.22) and (5.23), the secondary side impedance (Z2P) and the reflected impedance (Zr SP) is given by (5.26) and (5.27).
Z2P = ωo2L22 RL−jωoL2
(5.26) Zr SP = M2RL
L22 − jωoM2
L2 (5.27)
It can be observed from (5.27), if the secondary is parallel compensated it has a reactive component.
This introduces a phase shift in the system. This phase shift (−ML 2
2 ) should be compensated in the primary side of the system. For this reason, a large primary capacitance is usually used in parallel compensated CPT system. Therefore, in order to operate the SP compensated CPT system under resonance frequency and to obtain high efficiency, the compensated capacitors, C1 and C2 are given by (5.28).
C1 = 1 ωo2
L1−ML22
; C2= 1
ωo2L2 (5.28)
The total impedance (Zt SP) at resonance is given by (5.29) Zt SP = M2RL
L22 (5.29)
On substituting (5.28) into (5.19), the relation between I2 and I1 can be obtained, whereI2 leads I1 at some angle.
I2
I1 = jωM ωo2L22
RL−jωoL2
(5.30)
I2=G2I1= jωoM(RL−jωoL2)
ωo2L22 (5.31)
Using (5.18) - (5.31), the input power (Pi), output power (Po) and total efficiency of the system are calculated.
Pi =I12Zt SP (5.32)
Po =I22RL=G22I12RL (5.33) Using these equations, efficiency for SP compensation is given by (5.34).
ηSP = G22ωo2L22
Zt SP (RL−jωoL2) (5.34)
5.2 Steady state electric circuit analysis
5.2.5 Parallel-series (PS) compensation topology
The steady state equations for primary parallel and secondary series compensation can be written on phasor form as given by (5.35) and (5.36)
V1=IC1
1 jωC1
=IL1jωL1−jωM I2 (5.35)
jωM IL1 =jωL2I2+I2 1
jωC2
+I2RL (5.36)
The secondary side currentI2 in PS compensation can be obtained from (5.37).
I2 = jωM IL1
jωL2+
1 jωC2
+RL
(5.37)
By substituting the secondary side currentI2, given in (5.37) into (5.35), we get V1 =jωL1IL1+ ω2M2
jωL2+
1 jωC2
+RL
IL1 (5.38)
As explained above, from the point of view of the first coil, the secondary coil is seen as a transformed impedance or reflected impedance (Zr P S) of the secondary network as seen from the primary side is given by (5.39).
Zr P S= ω2M2 h
jωL2+jωC1 2 +RL
i = ω2M2
Z2s (5.39)
On substituting (5.37) in (5.35), the current through primary side inductor (IL1) and capacitor (IC1) is obtained.
IL1= V1 jωL1+ωZ2M2
2s
(5.40) IC1 = V1
1 jωC1
(5.41) From Figure 5.5, the total impedance (Zt P S) of the PS compensation is given by (5.42). The total
V1
C1 I1
Zr PS IC1 IL1
L1
Figure 5.5: Parallel compensated primary.
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current observed from the primary side is given by (5.43).
Zt P S= 1
jωC1+jωL1+Z1
r P S
(5.42)
I1P S= V1
Zt P S (5.43)
To obtain high efficiency in PS compensated CPT system, the system should operate under resonance condition. The resonance capacitance for parallel and series compensated primary is same as the basic resonance condition as given by (5.44).
C1= 1
ωo2L1 (5.44)
On substituting (5.44) into (5.42), the total impedance at resonance condition is given by Zt P S= Z2SL12
M2 −jωoL1 (5.45)
It has been seen from (5.45) has some constraints on (Z2S) for the system to be in resonance. While for the system to be in parallel compensated, Z2S has some resistive and reactive part, say
Z2S =RL+jX (5.46)
The criteria for resonance is, the inductance due to parallel primary side should be compensated in the secondary side coil such that the primary side impedance is resistive in nature. This condition is achieved when the reactance is in the form given below:
X=ωo
M2
L1 (5.47)
On substituting (5.46) in (5.42) and (Zt P S) can be obtained as given by (5.48) Zt P S = RLL12
M2 (5.48)
Thus, when the primary side is parallel compensated a phase shift is introduced to the secondary side that brings the secondary side out of resonance. Therefore, the compensation capacitance in PS is defined as given below:
C1 = 1
ωo2L1 ; C2 = 1 ωo2
L2−ML12 (5.49)
5.2 Steady state electric circuit analysis
Using these condition (5.49), the secondary side impedance (Z2S) given in (5.39) can be given by (5.50).
Z2S=jωo
M2
L1 +RL (5.50)
The secondary current (I2) can be given in (5.36) can be modified by substitutingIL1andZ2S given in (5.50) and (5.40). Therefore, I2 at resonance can be written as given by (5.51). The primary current (I1) can be written as given by (5.52).
I2 = M L1RL
V1 (5.51)
I1 = M2
RLL12V1 (5.52)
Using (5.37) and (5.52), the input power (Pi), output power (Po) and total efficiency of the system are calculated.
5.2.6 Parallel-parallel (PP) compensation topology
The steady state equations for primary and secondary parallel compensation can be written on phasor form as given by (5.53) and (5.54)
V1=IC1
1 jωC1
=IL1jωL1−jωM I2 (5.53)
jωM IL1=jωL2I2+I2
1
1
1 jωC2
+R1
L
(5.54)
The secondary side currentI2 in PP compensation can be obtained from (5.55).
I2 = jωM
jωL2+R RL
LjωC2+1
IL1 = jωM
Z2p IL1 (5.55)
By substituting the secondary side currentI2 (given in (5.55)) into (5.53) we get:
V1 =jωL1IL1+ ω2M2 jωL2+R RL
LjωC2+1
!
IL1 (5.56)
As explained above, from the point of view of the first coil, the secondary coil is seen as a transformed impedance or reflected impedance (Zr P P) of the secondary network as seen from the primary side and is given by (5.57).
Zr P P = ω2M2 jωL2+jωCR2RL
L+1
= ω2M2
Z2p (5.57)
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From (5.56), the current through primary side inductor (IL1) and capacitor (IC1) is obtained. The total impedance (Zt P P) of PP compensation is given by (5.60).
IL1 = V1 jωL1+ωZ2M2
2p
(5.58)
IC1 = V1
1 jωC1
(5.59)
Zt P P = 1
1 jωL1+ω2M2
Z2P
+jωC1 (5.60)
The total current observed from the primary side is given by (5.61).
I1= V1
Zt P P (5.61)
In case of parallel compensated primary and secondary, the system has reactive components, which introduces phase shifts on both sides of the system (as explained above). Hence for parallel compen- sated system, the compensation capacitors are defined as given by (5.62). This is the reason a large primary and secondary capacitor is usually added on both sides of CPT systems.
C1 = 1 ωo2
L1− ML22 ; C2 = 1 ωo2
L2−ML12 (5.62)
On substituting (5.61), the secondary side impedance (Z2S), reflected impedance (Zr P P) and total impedance (Zt P P) can be derived as given by (5.63)-(5.65).
Z2p =
ωo2L2
L2−ML12
+jωoML2
1RL RL−jωo
L2−ML12 (5.63)
Zr P P =
ω2M2
RL−jωo
L2− ML12
ωo2L2
L2−ML12
+jωoM2 L1RL
(5.64)
Zt P P = ωo2L12−ωo2L1ML2
2 −jωoL1Zr P P +jωoM2 L2Zr P P
jωoM2
L2 +Zr P P (5.65)
The primary current and load current is given by (5.66) and (5.67).
I1 = V1
Zt P P = jωoM2
L2 +Zr
ωo2L12−ωo2L1ML2
2 −jωoL1Zr+jωoML2
2Zr V1 (5.66)
IL= VL
RL
= ZpI2 RL
= I2
(RLjωC2+ 1) (5.67)